1. Introduction

In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms path integral, curve integral, and curvilinear integral are also used; contour integral as well, although that is typically reserved for line integral in the complex plane [1].

The function to be integrated may be a scalar field or a vector field. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length) or, for a vector field, the scalar product of the vector field with a differential vector in the curve. This weighting distinguishes the line integral from simpler integrals defined on intervals. Many simple formula in physics (for example, \(W = F \cdot s\)) have natural continuous analogs in terms of line integrals ( \(W = \int_{c}F~\cdot~ ds\)). The line integral finds the work done on an object moving through an electric or gravitational field, for example [1].

The main purposes of this study are to

• show the areas where line integral is applicable;
• point out the role of line integral in solving practical problems.

2. Preliminaries

2.1. Space curves

Since the concept of vector valued function is useful to define space curve, let us define vector valued function before we define space curve.

Definition 1.(Vector valued function [2]) A function \(r : t\longrightarrow\mathbf{V}\) is said to be a vector valued function if \(t \in ~\mathbb{R}\) and \(\mathbf{V}\) is \(n-\)dimensional vector.

Example 1. \(r(t) = 2ti-t^{2}j\) and \(r(t) = i-tj + (\sin t)k\) for any real number \(t\) are vector valued functions.

Definition 2.(Space curve [2]) A space curve (or simply curve) is the range of a continuous vector valued function on an interval of real numbers.

We will generally use \(C\) to denote a curve and \(r\) to denote a vector valued function whose range is a curve \(C.\) In that case we say that \(C\) is parameterized by \(r\) or that \(r\) is a parametrization of \(C.\)

There are so many types of curves. Some of them are listed as follows;

Definition 3.(Simple curve [3]) A curve \(C\) is said to be simple if it has no self-intersection except possibly at the end points of the interval of a parameter \(t.\)

Definition 4.(Closed curve [3]) A curve \(C\) is said to be closed if it has a parametrization \(r(t)\) on the closed interval \([a, b]\) such that \(r(a) = r(b).\) In other words, a curve \(C\) is said to be closed if its initial and terminal points coincides.

Example 2.

1. A curve \(C\) parameterized by \(r(t) = (\cos t )i + (\sin t )j\) for \(0\leq t\leq 2\pi\) is a closed curve since \(r(0) = r(2\pi) = i.\)
2. A curve \(C\) parameterized by \(r(t) = ti + t^{2}j-k\) for \(0\leq t\leq 1\) is not a closed curve since \(r(0) \neq r(1).\)

Definition 5.(Simple closed curve [3]) A curve \(C\) is said to be simple closed if it is both simple and closed curve.

Definition 6.(Smooth curve [3]) A vector valued function \(r(t)\) is said to be smooth on an interval \(I\) if \(r'(t)\) is continuous on \(I\) and \(r'(t) \neq 0\) for each interior point of \(I.\) A curve \(C\) is said to be smooth if its parametrization \(r(t)\) is smooth.

Example 3.

1. Consider a curve \(C\) parameterized by \begin{equation*} r(t) = t^{2}i + 2tj + t^{3}k. \end{equation*} The derivative of \(r(t)\) with respect to \(t\) is \begin{equation*} r'(t) = 2ti + 2j + 3t^{2}k. \end{equation*} Since \(r'(t)\) is continuous and \(r'(t) \neq 0\) for any \(t,\) then \(r(t)\) is smooth on \((-\infty,~\infty)\). Because of this, a curve \(C\) is smooth on \((-\infty,~\infty)\).
2. Consider a curve C\('\) parameterized by \begin{equation*}r(t)=(t-sint)i + (1-cost)j, \ \ \ -2\pi \leq t \leq 2\pi \end{equation*} The derivative of \(r(t)\) with respect to \(t\) is \begin{equation*}r'(t) = (1-\cos t)i + (\sin t)j.\end{equation*} Even though \(r'(t)\) is continuous on \([-2\pi,~2\pi],\) \(r(t)\) is not smooth on \([-2\pi,~2\pi]\) because \(r'(t) = 0\) at \(t = 0.\) Because of this, a curve \(C'\) is not smooth on \([-2\pi,~2\pi].\)

Definition 7.(Piecewise smooth curve [3]) A vector valued function \(r(t)\) is said to be piecewise smooth on an interval \(I\) if \(I\) is expressible as the union of finite number of subintervals such that \(r(t)\) is smooth on each of the subintervals and the one-sided derivatives of \(r(t)\) exist at each interior point of \(I.\) A curve \(C\) is said to be piecewise smooth if its parametrization \(r(t)\) is a piecewise smooth.

For 2 in Example 3, \(r(t)\) is piecewise smooth because \(r(t)\) is smooth on subintervals \([-2\pi, 0]\) and \([0, 2\pi]\), and one-sided derivatives of \(r(t)\) exist on \((-2\pi,~2\pi).\) Because of this, that curve is a piecewise smooth on \([-2\pi,~2\pi].\)

Remark 1. Any smooth curve can be piecewise smooth but the converse is not always true.

Definition 8.(Counter clockwise oriented curve [4]) A curve \(C\) is said to be counter clockwise oriented if it is always lying to the left of the path while moving along \(C.\)

Definition 9. (Clockwise oriented curve [4]) A curve \(C\) is said to be clockwise oriented if it is always lying to the right of the path while moving along \(C.\)

Definition 10. (Positively and Negatively oriented curves [4])

• A curve \(C\) is said to be positively oriented if the values of parameter \(t\) is increasing while moving along \(C.\)
• A curve \(C\) is said to be negatively oriented if the values of parameter \(t\) is decreasing while moving along \(C.\)

Remark 2. If the curve is simple closed, then it is

• positively orientation corresponds to counter clockwise orientation,
• negatively orientation corresponds to clockwise orientation.

It is possible to express different curves using a parameter. Now, let us see smooth parametrization of important curves which we will use frequently.

Line segment: Suppose \(L\) is a line segment from point \(A(x_{0},~y_{0},~z_{0})\) to another point \(B(x_{1},~y_{1},~z_{1})\). Then the parametric equation of this line segment is

\[ \begin{cases} x=x_{0} + (x_{1}- x_{0})t \\ y=y_{0}+(y_{1}-y_{0})t\\ z=z_{0}+(z_{1}-z_{0})t \end{cases}\] for \(0\leq t\leq 1.\) Therefore, the smooth parametrization of this line segment is \[[x_{0}+(x_{1}-x_{0})t]i +[y_{0}+(y_{1}-y_{0} )t]j+[z_{0}+(z_{1}-z_{0})t]k\;\;\;\;\;\;\; \text{for }\;\;\;\;\;\;0\leq t\leq 1.\]

Circle: Suppose \(C\) is a circle with equation \((x-h)^{2}+(y-k)^{2}=r^{2}\). Then the parametric equation of this circle is

\[ \begin{cases} x=h+r\cos t \\ y=k+r\sin t \end{cases}\] for \(0\leq t\leq 2\pi. \) Therefore, the smooth parametrization of this circle is \[[h+rcost]i+[k+r\sin t]j\;\;\;\;\;\;\; \text{for }\;\;\;\;\;\;0\leq t\leq 2\pi.\]

Ellipse: Suppose \(E\) is an ellipse with equation \(\frac{(x-h)^{2}}{a^{2}} +\frac{(y-k)^{2}}{b^{2}} =1\). Then the parametric equation of this ellipse is

\[\begin{cases} x=h+a\cos t \\ y=k+b\sin t \end{cases}\] for \(0\leq t\leq 2\pi. \) Therefore, the smooth parametrization of this ellipse is \[[h+acost]i+[k+b\sin t]j\;\;\;\;\;\;\; \text{for }\;\;\;\;\;\;0\leq t\leq 2\pi.\]

Graph of a function: Suppose \(C\) is a graph of a function \(y = g(x)\) for \(a\leq x \leq b\). Then the parametric equation of this graph is

\[\begin{cases} x=t\\ y=g(t) \end{cases} \] for \(a\leq t\leq b. \) Therefore, the smooth parametrization of this graph is \[ti+g(t)j \;\;\;\;\;\;\; \text{for }\;\;\;\;\;\;a\leq t\leq b.\]

Region formed by a plane \(z = m\) and a circle \((x-h)^{2}+(y-k)^{2}=r^{2}\): The smooth parametrization of this region is

\[[h+r\cos t]i+[ k+r\sin t]j+mk\;\;\;\;\;\;\; \text{for }\;\;\;\;\;\;0\leq t \leq 2\pi.\]

Sphere: Suppose \(S\) is a sphere with equation \((x-h)^{2} + (y-k)^{2} + (z-w)^{2} =\rho^{2}\). Then the parametric equation of this sphere is

\[ \begin{cases} x=h+\rho \cos\beta \sin\theta \\ y=k+\rho \sin\beta \sin\theta\\ z=w+\rho \cos\theta \end{cases} \] for \(\beta \in [0,~ 2\pi]~and~\theta \in [0,~ \pi] \). Therefore, the smooth parametrization of this sphere is \[[h+\rho \cos\beta \sin\theta]i+[k+\rho \sin\beta \sin\theta]j +[w+\rho \cos\theta] k\;\;\;\;\;\;\; \text{for }\;\;\;\;\;\;\beta \in [0,~ 2\pi]~and~\theta \in [0,~ \pi].\]

Definition 11. (Length of a curve [2]) Let \(C\) be a curve with a piecewise smooth parametrization \(r(t)\) defined on \([a, b],\) then the length \(L\) of \(C\) is \begin{equation*} L=\int_{a}^{b}|r'(t)|dt=\int_{a}^{b}|\frac{dr}{dt}|dt. \end{equation*}

Remark 3.If \(r(t) = x(t)i + y(t)j + z(t)k,\) then \begin{equation*} L=\int_{a}^{b}\sqrt{[x'(t)]^{2}+[y'(t)]^{2}+[z'(t)]^{2} }dt. \end{equation*}

Example 4. Consider a curve parameterized by \(r(t) = (\cos t)i + (\sin t)j + tk\) for \(0\leq t \leq 2\pi\). The derivative of \(r(t)\) with respect to \(t\) is \begin{equation*} r'(t)=(-\sin t)i+(\cos t )j+k. \end{equation*} The length of this curve on \([0, 2\pi]\) is \begin{equation*} L=\int_{0}^{2\pi}\sqrt{(-\sin t)^{2}+(\cos t)^{2}+1^{2}} dt =\int_{0}^{2\pi}\sqrt{2}dt=2\sqrt{2}\pi. \end{equation*}

2.2. The concepts of scalar and vector fields

Definition 12. (Scalar field [3]) A function \(f : P\rightarrow Q\) is said to be a scalar field if \(P\in\mathbb{R}^{n}\) and \(Q\in\mathbb{R}\).

Example 5. \(f(x, y, z) = xyz\) is a scalar field.

Definition 13. (Vector field [3]) A function\( F : P\rightarrow\mathbf{V}\) is said to be a vector field if \(P\in\mathbb{R}^{n}\) and \(\mathbf{V}\) is \(n-\)dimensional vector.

Example 6. \(F(x, y, z) = x^{2} i+y^{2} j+z^{2} k\) is a vector field.

Remark 4.A vector field \(F\) formed by the gradient of a scalar function \(f\) is said to be gradient vector field and denoted by \begin{equation*} F = \triangledown f=(\frac{\partial}{\partial x}i + \frac{\partial}{\partial y}j + \frac{\partial }{\partial z}k)f=f_{x}i+f_{y}j+f_{z}k. \end{equation*}

Definition 14. (Conservative vector field [3]) A vector field \(F\) is said to be conservative if there exists a differentiable function \(f\) such that \(F = \triangledown f.\) The function \(f\) is called potential function for \(F.\)

Example 7. A function \(F(x, y, z) = 3x^{2} i+3y^{2} j+3z^{2}k\) is conservative because there exists a potential function \(f(x, y, z) = x^{3} +y^{3}+z^{3}\) for \(F\) such that \(F = \triangledown f.\)

Before we see test of conservativeness, let us define the curl of a vector field \(F.\)

Definition 15. (Curl of a vector field [3]) Let \(F = Mi + Nj + Pk\) be a vector field such that all the partial derivatives of \(M, N\) and \(P\) exist, then the curl of \(F,\) denoted by curl \(F,\) is given by \[\text{curl}~F=\triangledown \times F = \begin{vmatrix} i & j & k \\ \frac{\partial}{\partial x} & \frac{\partial }{\partial y} & \frac{\partial }{\partial z} \\ M & N & P \end{vmatrix} .\]

Remark 5. If \(F = Mi +Nj,\) then \(\text{curl} F = (N_{x}-M_{y})k\).

Example 8. Consider a vector field \(F(x, y, z) = 3x^{2}i + xyj - y^{2}zk\). Then the curl of \(F\) is given by \[ \begin{vmatrix} i & j & k \\ \frac{\partial }{\partial x} & \frac{\partial }{\partial y} & \frac{\partial }{\partial z} \\ 3x^{2} & xy & -y^{2} z \end{vmatrix} = -2yzi + yk.\]

Now let us see test of conservativeness and procedures of finding potential function of conservative vector field.

Test of conservativeness [3]

The concept of curl of a vector field \(F\) is useful to test \(F\) is conservative or not i.e., \(F\) is conservative if curl \(F = 0.\) A vector field in Example 8 is not conservative since \(\text{curl} F\) is not always \(0.\)

Procedures of finding potential function of conservative vector field

We are able to find a potential function for a given conservative vector field \(F.\) The procedures of finding a potential function \(f\) for a given conservative vector field \(F(x, y, z) = Mi + Nj + Pk\) are listed as follows;

Step-1 ( Variable separation) In this step, we express the function \(f\) as a sum of three functions in the form of \[f(x,~y,~z)=k(x,~y,~z)+g(y,~z)+h(z).\] So, \(f_{x}=k_{x},~f_{y}=k_{y}+g_{y}\) and \(f_{z}=k_{z}+g_{z}+h_z.\) Since \(F = \triangledown f,\) then \(f_{x}=k_{x}=M,~f_{y}=k_{y}+g_{y}=N\) and \(f_{z}=k_{z}+g_{z}+h_{z}=P\).

Step-2 Integrate \(f_{x}=k_{x}=M\) with respect to \(x.\) That is \(k(x, y, z)\) = \(\int Mdx\).

Step-3 Integrate \(f_{y}=k_{y}+g_{y}=N\) with respect to \(y.\) That is \(g(y,z)=\int(N-k_{y})dy\).

Step-4 Integrate \(f_{z}=k_{z}+g_{z}+h_{z}=P\) with respect to \(z.\) That is \(h(z)=\int(P-k_{z}-g_{z})dz\).

Step-5 Finally, we write \(f(x, y, z) = k(x, y, z) + g(y, z) + h(z)\) using the above results.

Example 9. Consider a vector field \(F(x, y, z)= xy^{2}z^{2} i+x^{2} y z^{2} j+x^{2} y^{2} zk\). Then \( M=x y^{2} z^{2},~ N=x^{2} y z^{2}\) and \(P=x^{2} y^{2} z\).

Firstly, let us test \(F\) is conservative or not. \[ \text{curl} F = \begin{vmatrix} i & j & k \\ \frac{\partial }{\partial x} & \frac{\partial }{\partial y} & \frac{\partial }{\partial z} \\ xy^{2}z^{2} & x^{2} y z^{2} &x^{2} y^{2} z \end{vmatrix} = 0.\] Hence, this vector field is conservative.

• Step-1 Suppose \(f(x, y, z) = k(x, y, z) + g(y, z) + h(z).\)
• Step-2 \(k(x, y, z) = \int Mdx=\int x y{^2} z^{2} dx=\frac{1}{2} x^{2} y^{2} z^{2} + c_{1}.\)
• Step-3 \(g(y, z) = \int(N-k_{y} )dy = \int0dy = 0 + c_{2} = c_{2}.\)
• Step-4 \(h(z) = \int(P-k_{z}-g_{z} )dz=\int0dz=0 + c_{3} = c_{3}\).
• Step-5 The potential function for the given conservative vector field is \[f(x, y, z) = \frac{1}{2} x^{2} y^{2} z^{2}+c\] where \(c = c_{1} + c_{2} +c_{3}\).

3. Line integrals

Under this section, we will see detail description of line integral in Cartesian and Complex planes.

3.1. Line integral in Cartesian plane

There are two types of line integral in Cartesian plane. These are line integral of scalar and vector fields.

Definition 16.(Line integral of scalar field [3]) Let \(f\) be a continuous scalar field on a smooth curve \(C\) parameterized by \(r(t) = x(t)i + y(t)j + z(t)k\) on \([a, b],\) then the line integral of \(f\) on \(C\) is given by \begin{equation*} \int_{C} f(x,y,z)ds = \int_{a}^{b}f(x(t),~y(t),~z(t)) |r'(t)|dt. \end{equation*}

Example 10.

1. Consider \(\int_{C} (x+y+ z)ds\) where \(C\) is the line segment from \((0, 1, 1)\) to \((2, 2, 3).\) The smooth parametrization of the line segment from \((0, 1, 1)\) to \((2, 2, 3)\) is \begin{equation*} r(t) = 2ti + (1 + t)j + (1 + 2t)k; \ \ \ 0\leq t \leq 1\,, \end{equation*} implies that \(| r' (t)|=3\) and \(x(t) + y(t) + z(t) = 2 + 5t.\) Hence, \begin{equation*} \int_{C} (x+y+ z)ds = \int_{0}^{1}3(2+5t) dt= \frac{27}{2}. \end{equation*}
2. Consider \(\int_{C} (x+y)ds\) where \(C\) is the upper half of the circle \(x^{2}+y^{2}=25\) in counter clockwise orientation. The smooth parametrization of this half circle is \[r(t) = (5\cos t)i + (5\sin t)j;\;\;\;\;\;\;\;0 \leq t \leq \pi\,,\] implies \(| r' (t)|=5\) and \(x(t) + y(t) = 5\cos t + 5\sin t.\) Hence, \[\int_{C} (x+y)ds = \int_{0}^{\pi}5(5\cos t+5\sin t) dt=50.\]

Definition 17.(Line integral of vector field [3]) Let \(F = Mi + Nj + Pk\) be a continuous vector field defined on a smooth curve \(C\) parameterized by \(r(t) = x(t)i + y(t)j + z(t)k\) on \([a, b],\) then the line integral of \(F\) on \(C\) is given by \[ \int_{C} F\cdot dr=\int_{a}^{b} F(x(t),~y(t),~z(t)) \cdot r'(t)dt.\]

Example 11. Consider \(F(x, y) = 3yi + 3xj\) and \(C\) is the ellipse \(\frac{x^{2}}{9}+\frac{y^{2}}{4}=1\) in the \(xy-\)plane. The smooth parametrization of this ellipse is \[r(t) = (3\cos t)i + (2\sin t)j; \;\;\;\;\;\;0\leq t \leq 2\pi\,,\] implies \[F(x(t),y(t))\cdot r'(t)=[(6\sin t)i +(9\cos t)j]\cdot [(-3\sin t)i +(2\cos t)j]=18\cos 2t.\] Hence, the line integral of \(F\) on \(C\) is \( \int_{C} F(x,y)\cdot dr=\int_{o}^{2\pi}(18 \cos 2t) dt=0\).

Remark 6. If \(a\) and \(b\) are constants, and \(F\) and \(G\) are continuous vector field on a smooth curve \(C\) parameterized by \(r(t),\) then the following properties hold;

1. \(\int_{C} [aF\pm bG]\cdot dr=a\int_{C} F\cdot dr\pm b\int_{C} G\cdot dr\),
2. \(\int_{-C} F\cdot dr=-\int_{C} F\cdot dr\),
3. \(\int_{C} F\cdot dr=\int_{C_{1} } F\cdot dr+\int_{C_{2} } F\cdot dr +...+\int_{C_{n}} F\cdot dr ~ if ~C= \bigcup _{i=1}^{n} C_{i}\),
4. if \(F\) is conservative and \(C\) is from \(P\) to \(Q,\) then \(\int_{C} F\cdot dr=f(Q)-f(P)\) where \(f\) is a potential function for \(F.\) In other words, the line integral \(\int_{C} F\cdot dr\) is independent of the path if \(F\) is conservative,
5. if \(F = Mi + Nj + Pk,\) then \(\int_{C} [Mdx+Ndy+Pdz]=\int_{C} F\cdot dr\).

Example 12.

Consider \(F(x, y) = xi + yj \) and \( C\) is the triangle in a plane with vertices \(A(0, 0), B(1, 0)\) and \( C(0, 1)\) in counter clockwise orientation, see Figure 4.

From Figure 4, \(C = C_{1} ~\bigcup ~C_{2} ~\bigcup~ C_{3}\), where \(C_{1}:x=t\) and \(y = 0; \;\;\;\;0\leq t \leq 1\), \(C_{2}:x=1-t\) and \(y = t;\;\;\;\;0\leq t \leq 1\), and \(C_{3}:x=o\) and \(y = 1\;\;t;\;\;\;\;0\leq t\leq 1\).

On \(C_{1},\) \[\int_{C_{1} } F\cdot dr=\int_{0}^{1}t dt=\frac{1}{2}.\] On \(C_{2},\) \[\int_{C_{2}} F\cdot dr=\int_{0}^{1}(2t-1) dt=0.\] On \(C_{3},\) \[\int_{C_{3 }} F\cdot dr=\int_{0}^{1}(t-1)dt=-\frac{1}{2}.\] Hence, \[\int_{C} F\cdot dr=\int_{C_{1 }} F\cdot dr+\int_{C_{2 }} F\cdot dr+ \int_{C_{3 }} F\cdot dr=\frac{1}{2}+0-\frac{1}{2}=0.\] In other way, since \(F\) is conservative, \( f(x, y) = \frac{x^{2}}{2}+\frac{y^{2}}{2}\) is a potential function for \(F\) and \( C\) is closed path, then \(\int_{C} F\cdot dr=f(0,0)-f(0,0)=0\).

3.2. Line integral in complex plane

The line integral in complex plane is known as contour integral.

Definition 18.(Contour integral [5]) If a function of complex variable \(f(z)\) is continuous on a smooth curve \(C\) parameterized by \(z(t) = x(t) + y(t)i\) on an interval \([a, b],\) then the contour integral of \(f\) on \(C\) is given by \[\int_{C} f(z)dz=\int_{a}^{b}f(z(t)) z'(t)dt.\]

Remark 7. Contour integral grants all properties of line integral in Cartesian plane.

Example 13. Consider \(f(z) = z^{2}\) and \(C\) is given by Figure 5.

From Figure 5, \(C = C_{1} ~\bigcup ~C_{2} ~\bigcup~ C_{3}\), wehere \(C_{1} : z(t) = t; \;\;\;0\leq t\leq 1\), \(C_{2}: z(t) = 1+ti; \;\;\;0\leq t\leq 1\) and \(C_{3}: z(t) = (1-t) + (1-t)i; \;\;\;0\leq t\leq 1\).

On \(C_{1},\) \[ \int_{C_{1 }} f(z)dz=\int_{0}^{1}t^{2} dt=\frac{1}{3}.\] On \(C_{2},\) \[\int_{C_{2 }} f(z)dz=\int_{0}^{1}(1+ti)^{2 }i dt=-1+\frac{2}{3} i.\] On \(C_{3},\) \[\int_{C_{3 }} f(z)dz=\int_{0}^{1}[(1-t)+(1-t)i]^{2} (-1-i)dt=\frac{2}{3}-\frac{2}{3} i.\] Hence, \[\int_{C} f(z)dz=\int_{C_{1 }} f(z)dz+\int_{C_{2}} f(z)dz+ \int_{C_{3}} f(z)dz=0.\]

4. Applications of line integral

Line integral has physical and geometrical applications. Now, let us see selective applications of line integral in detail.

4.1. Geometrical applications of line integral

In mathematics, line integral is used, in particular, for computations of

• length of a curve,
• area of a region bounded by a closed curve
• volume of a solid formed by rotating a closed curve about the \(x-\) axis.

4.1.1. Length of a curve [6]

Let \(C\) be a piecewise smooth curve described by the position vector \(r(t);\alpha \leq t\leq \beta\), then the length of the curve is given by the line integral\begin{equation*} L=\int_{C} ds=\int_{\alpha}^{\beta}|\frac{dr}{dt}| dt = \int_{\alpha}^{\beta} \sqrt{(\frac{dx}{dt})^{2}+(\frac{dy}{dt})^{2}+(\frac{dz}{dt})^{2}} ~dt \end{equation*} where \(\frac{dr}{dt}\) is the derivative of \(r(t)\) with respect to \(t,\) and \( x(t), y(t), z(t)\) are the components of the position vector \(r(t).\)

If the curve \(C\) is two-dimensional, the latter formula can be written in the form of

\begin{equation*} L=\int_{C} ds=\int_{\alpha}^{\beta}|\frac{dr}{dt}| dt = \int_{\alpha}^{\beta} \sqrt{(\frac{dx}{dt})^{2}+(\frac{dy}{dt})^{2}} ~dt. \end{equation*} If the curve \(C\) is the graph of a continuous and differentiable function \(y = f(x)\) on \([a, b]\) in the \(xy-\)plane, then the length of the curve is given by \begin{equation*} L=\int_{a}^{b}\sqrt{1+(\frac{dy}{dx})^{2}} ~dx. \end{equation*} Finally, if the curve \(C\) is given by the equation \(r = r(\theta ),\) \(\alpha \leq \theta \leq \beta\) in polar coordinates, and the function \(r(\theta)\) is continuous and differentiable in the interval \([\alpha, \beta],\) then the length of the curve is defined by the formula \begin{equation*} L=\int_{\alpha}^{\beta}\sqrt{(\frac{dr}{d\theta})^{2}+r^{2}}~ d\theta. \end{equation*}

Example 14.

1. Consider a space curve \(C\) parameterized by \(r(t) = (3t, 3t^{2}, 2t^{3})\) where \(0\leq t\leq 1\). The derivative of \(r(t)\) with respect to \(t\) is \(r'(t) = (3, 6t, 6t^{2})\). The length of this curve is \begin{equation*} L= \int_{C} ds= \int_{0}^{1} \sqrt{(3)^{2}+(6t)^{2}+(6t^{2} ){^2} } dt=\int_{0}^{1}(3+6t^{2} ) dt=5. \end{equation*}
2. Consider a parabola \(y = x^{2}\) for \(0\leq x\leq 1\). The length of this curve is \begin{equation*} L=\int_{0}^{1} \sqrt{1+(2x)^2 } dx=\int_{0}^{1}\sqrt{1+4 x^{2}} dx ,\end{equation*} implies \begin{equation*} L=\frac{1}{2} \int_{0}^{arctan2} sec^{3} t dt=\frac{\sqrt{5}}{2}+\frac{1}{4} ln(\sqrt{5}+2)\approx 1.48. \end{equation*}
3. Consider a cardioid given in polar coordinates by the equation \(r = 5(1 + \cos\theta)\), see Figure 6. The derivative of \(r(\theta)\) with respect to \(\theta\) is \(r' (\theta) = -5 \sin\theta\). The length of this cardioid is \begin{align*}L&= \int_{0}^ {2\pi}\sqrt{(-5 \sin\theta )^{2}+[5(1+\cos\theta)]^{2} } d\theta\\ &= 5\sqrt{2}\int_{0}^ {2\pi}\sqrt{1+\cos\theta} d\theta\\ &= 10\int_{0}^ {2\pi} \left|\cos(\frac{\theta}{2})\right| d\theta \\ &= 10\left[\int_{0}^ {\pi}\cos\left(\frac{\theta}{2}\right) d\theta - \int_{\pi}^ {2\pi}\cos\left(\frac{\theta}{2}\right) d\theta\right] \;\;\;\text{since}\;\;\; \left|\cos\left(\frac{\theta}{2}\right)\right| = \begin{cases} \cos\left(\frac{\theta}{2}\right) ~ \text{if}~ 0\leq \theta \leq \pi\\ -\cos\left(\frac{\theta}{2}\right) ~ \text{if} ~\pi \leq \theta \leq 2\pi \end{cases}\\ &= 40.\end{align*}

4.1.2. Area of a region bounded by a closed Curve [6]

If \(C\) is a closed smooth piecewise curve in the \(xy-\)plane (Figure 7), then the area of the region \(R\) bounded by the curve is \(S=\oint_{C} xdy=-\oint_{C} ydx=\frac{1}{2} \oint_{C} [xdy-ydx].\)

Remark 8. It is supposed here that the contour \(C\) is traversed in the counterclockwise direction.

If the closed curve \(C\) is given in parametric form \(r(t) = (x(t), y(t))\) on \([\alpha,~ \beta],\) then the area of the corresponding region can be calculated by the formula \begin{equation*}S=\int_{\alpha}^{\beta}x(t) \frac{dy}{dt} dt=-\int_{\alpha}^{\beta}y(t) \frac{dx}{dt} dt=\frac{1}{2} \int_{\alpha}^{\beta}[x(t) \frac{dy}{dt}-y(t) \frac{dx}{dt}]dt \,. \end{equation*}

Example 15. Consider a region bounded by the ellipse \(x = a\cos t, y = b\sin t; \;\;\;0\leq t \leq 2\pi\), see Figure 8.

The area of this region is

\begin{equation*} S=\int_{0}^{2\pi}(a\cos t) (b\cos t)dt=ab\int_{0}^{2\pi}\cos^{2} tdt=\pi ab. \end{equation*}

We can also get the following result using the other two formulas;

• \(S=-\int_{0}^{2\pi}(b\sin t)(-a\sin t )dt=ab\int_{0}^{2\pi}\sin^{2} tdt=\pi ab\),
• \(S=\frac{1}{2} \int_{0}^{2\pi}[(a\cos t) (b\cos t)-(b\sin t)(-a\sin t )]dt=\frac{1}{2} ab\int_{0}^{2\pi}dt=\pi ab\).

4.1.3 Volume of a solid formed by rotating a closed curve about the \(x-\)axis [6]

Let \(R\) be a region in the half-plane \(y\geq0\) bounded by a closed smooth piecewise curve \(C\) traversed in the counterclockwise direction. Suppose that the solid \(\Omega\) is formed by rotating the region \(R\) about the \(x-\)axis (Figure 9), then the volume of the solid is given by \begin{equation*} V=-\pi \oint_{C} y^{2}dx=-2\pi \oint_{C} xydy=-\frac{\pi}{2} \oint_{C} [2xydy+y^{2}dx]. \end{equation*}

Example 16. Consider a solid formed by rotating the region \(R\) bounded by the curve \(y = 2-\sin x\) and the lines \(x = 0, x = 2\pi, y = 0\) about the \(x-\)axis (Figure 10).

The volume of the solid is

\begin{align*} V&=-\pi \oint_{C} y^{2} dx=-\pi [\int_{OA} y^{2} dx+\int_{AB} y^{2} dx+\int_{BD} y^{2} dx+\int_{DO} y^{2} dx]\\ &=-\pi(0+0-9\pi+0)=9\pi^{2} \end{align*} where, since \(y = 0\) on \(OA\), so \(\int_{OA} y^{2} dx=\int_{0}^{2\pi}o^{2} dx=0\). Since, the smooth parametrization of \(AB\) is \(2\pi i +2tj\), so \[\int_{AB} y^{2} dx=\int_{2\pi}^{2\pi}y^{2}dx=\int_{0}^{1}(2t)^{2}\times 0~ dt =0.\] Since \(y = 2- \sin x\) on \(BD\), so \[\int_{BD} y^{2} dx=\int_{2\pi}^{0}(2-sinx )^{2} dx=-9\pi.\] Since, the smooth parametrization of DO is \(0i + (2 \)-\( 2t)j\), so \[\int _{DO} y^{2} dx=\int_{0}^{0}y^{2} dx=\int_{0}^{1}(2-2t)^{2}\times 0 ~dt =0.\]

4.2. Physical applications of line integral

In physics, the line integral is used, in particular, for computations of

• mass of a wire,
• center of mass and moments of inertia of a wire,
• work done by a force on an object moving in a vector field,
• magnetic field around a conduct (Ampere\('\)s law),
• voltage generated in a loop (Faraday\('\)s law of magnetic induction).

4.2.1. Mass of a wire

Suppose that a piece of a wire is described by a curve \(C\) in three dimensions. The mass per unit length of the wire is a continuous function \(\rho(x, y, z).\) Then the total mass of the wire is expressed through the line integral of scalar function as [7] \begin{equation*} m=\int_{C} \rho(x,y,z)ds. \end{equation*} If \(C\) is a curve parameterized by the vector function \(r(t) = (x(t), y(t), z(t))\) on \([\alpha,~\beta],\) then the mass can be computed by the formula \begin{equation*} m =\int_{\alpha}^{\beta}\rho(x(t),y(t),z(t))\times\sqrt{(\frac{dx}{dt})^{2}+(\frac{dy}{dt})^{2}+(\frac{dz}{dt})^{2} )} dt\,. \end{equation*} If \(C\) is a curve in the \(xy-\)plane, then the mass of the wire is given by \[m=\int_{C} \rho(x,y)ds.\] Or in parametric form \begin{equation*} m =\int_{\alpha}^{\beta}\rho(x(t),y(t))\times\sqrt{(\frac{dx}{dt})^{2}+(\frac{dy}{dt})^{2} } dt\,. \end{equation*}

Example 17. Consider a wire lying along the arc of the circle \(x^{2}+y^{2}=1\) from \(A(1, 0)\) to \(B(0, 1)\) with the density \(\rho(x, y) = xy\) (Figure 11).

The parametric equation of this part of a circle is \[ \begin{cases} x=\cos t\\ y=\sin t \end{cases} \] for \(0\leq t \leq \frac{\pi}{2}\). Then the mass of this wire is \begin{equation*} m =\int_{0}^{\frac{\pi}{2}} \cos t ~\sin t\sqrt{(-\sin t )^{2}+(\cos t )^{2} } dt=\frac{1}{2} \int_{0}^{\frac{\pi}{2}}\sin 2t ~dt=\frac{1}{2}\,. \end{equation*}

4.2.2. Center of mass and moments of inertia of a wire [7]

Let a wire is described by a curve \(C\) with a continuous density function \(\rho(x, y ,z).\) The coordinates of the center of mass of the wire are defined as \begin{equation*} \overline{x}=\frac{M_{yz}}{m},~ \overline{y}=\frac{M_{xz}}{m} ~and~ \overline{z}=\frac{M_{xy}}{m}\,, \end{equation*} where \begin{equation*} M_{yz} = \int_{C} x\rho(x,y,z)ds,~M_{xz}=\int_{C} y\rho(x,y,z)ds ~and~ M_{xy}=\int_{C} z\rho(x,y,z)ds \end{equation*} are so-called first moments. The moments of inertia about the \(x-\)axis, \(y-\)axis and \(z-\)axis are given by the formulas [7]

• \(I_{x}=\int_{C} (y^{2}+z^{2})\rho(x,y,z)ds\),
• \(I_{y}=\int_{C} (x^{2}+z^{2})\rho(x,y,z)ds\),
• \(I_{z}=\int_{C} (x^{2}+y^{2})\rho(x,y,z)ds.\)

Example 18. Consider a circle \(x^{2} + y^{2} = a^{2}\) with the density \(\rho\) = 1. The parametric equation of this circle is\[ \begin{cases} x=a\cos t \\ y=a\sin t \end{cases} \] for \(0 \leq t \leq 2\pi\). Then the moment of inertia \(I_{x}\) about the \(x-\)axis is \begin{equation*} I_{x}=\int_{0}^{2\pi}(a\sin t)^{2}\times 1\times \sqrt{(-a\sin t)^{2}+(a\cos t)^{2}} dt=a^{3} \int_{0}^{2\pi}\sin^{2}t dt = \pi a^{3}\,. \end{equation*}

4.2.3. Work done by a force on an object moving in a velocity field [7]

Work done by a force \(F\) on an object moving along a curve \(C\) is given by the line integral \[W=\int_{C} F~.~dr\] where \(F\) is the vector force field acting on the object, dr is the unit tangent vector (Figure 12). The notation \(F . dr\) means dot product of \(F\) and \(dr.\)

Remark 9. The force field \(F\) is not necessarily the cause of moving the object. It might be some other force acting to overcome the force field that is actually moving the object. In this case the work of the force \(F\) could result in a negative value.

If a vector field is defined in the coordinate form \(F = (P(x, y, z), Q(x, y, z), R(x, y, z)),\) then the work done by the force is calculated by the formula

\begin{equation*} W=\int_{C} F ~. ~dr=\int_{C} [Pdx+Qdy+Rdz]. \end{equation*} If the object is moved along a curve \(C\) in the \(xy-\) plane, then the following formula is valid \begin{equation*} W=\int_{C} F~.~dr=\int_{C} [Pdx+Qdy] \end{equation*} where \(F = (P(x, y), Q(x, y)).\)

If a path \(C\) is specified by a parameter \(t\) (\(t\) often means time), the formula for calculating work becomes

\begin{equation*} W = \int_{\alpha}^{\beta}[P(x(t),y(t),z(t)) \frac{dx}{dt} +Q(x(t),y(t),z(t)) \frac{dy}{dt}+R(x(t),y(t),z(t)) \frac{dz}{dt} ]dt\,, \end{equation*} where \(t\) goes from \(\alpha\) to \(\beta\). If a vector force field \(F\) is conservative, then the work on an object moving from \(A\) to \(B\) can be found by the formula \begin{equation*} W=u(B)-u(A)\,, \end{equation*} where \(u(x, y, z)\) is a scalar potential function of a vector force field \(F.\)

Example 19. Consider a force field \(F(x, y) = (xy, x + y)\) on an object moving from the origin \(O(0, 0)\) to the point \(A(1, 1)\) along the path \(C.\)

1. Assume \(C\) is the line segment \(y = x.\) The parametric equation of this line segment is \[ \begin{cases} x=t \\ y=t \end{cases} \] for \(0 \leq t \leq 1\). The work along the line segment \(y = x\) is \begin{equation*} W_{1}=\int_{C} [xydx+(x+y)dy]=\int_{0}^{1}[t^{2} +2t]dt=\frac{4}{3}. \end{equation*}
2. Assume \(C\) is the curve \(y = \sqrt {x}\). The parametric equation of this line segment is \[ \begin{cases} x=t \\ y=\sqrt{t} \end{cases} \] for \(0 \leq t \leq 1\). The work along the line segment \(y = \sqrt{x}\) is \begin{equation*} W_{2}=\int_{C} [xydx+(x+y)dy]=\int_{0}^{1}[t^{\frac{3}{2}} +\frac{1}{2}(\sqrt{t}+1)]dt=\frac{37}{30}\,. \end{equation*}

Remark 10. For Example 19, if the force field is \(F(x, y) = xi + yj,\) then \(F\) is conservative force and in cases a and b, the work done is independent of the path i.e., \(W = f(1, 1)- f(0, 0) = 1\) where \(f(x, y) = \frac{x^{2}}{2} + \frac{y^{2}}{2} + c\) is a potential function for a force field \(F(x, y) = xi + yj.\)

4.2.4. Ampere\('\)s law [7]

The line integral of a magnetic field \(B\) around a closed path \(C\) is equal to the total current flowing through the area bounded by the contour \(C\) (Figure 13). This is expressed by the formula \begin{equation*} \oint_{C} B~.~dr= \mu_{0} I\,, \end{equation*} where \(\mu_{0}\) is the vacuum permeability constant, equal to \(4\pi \times 10^{-7} H/m \thickapprox 1.26\times 10^{-6} H/m.\)

Example 20. Let us develop the formula of the magnetic field in vacuum a distance \(r\) from the axis of a long straight wire carrying current \(I.\)

To find the field a distance \(r\) from the wire, we consider a loop of radius \(r,\) centered on the wire with its plane perpendicular to the wire with the current \(I\) (Figure 14). Since the field \(B\) has a constant magnitude and the field is tangent to the loop everywhere, the dot product of the vectors \(B\) and \(dr\) is just \(Bdr.\) Then we can write \begin{equation*} \oint_{C} B~.~dr=\oint_{C} Bdr=B\oint_{C} dr=B\int_{0}^{2\pi}dr=2\pi rB\,. \end{equation*} As a result, we have \(2\pi rB=\mu_{0} I \Longrightarrow B=\frac{\mu_{0} I}{2\pi r}\).

4.2.5. Faraday\('\)s law [7]

The electromotive force \(\epsilon\) induced around a closed loop \(C\) is equal to the rate of the change of magnetic flux passing through the loop (Figure 15). \begin{equation*} \epsilon=\oint_{C} E~.~dr=-\frac{d\psi}{dt}\,. \end{equation*}

Example 21. Let us evaluate the maximum electromotive force \(\epsilon\) and the electric field \(E\) induced in a finger ring of radius \(1cm\) when the passenger flies on an airplane in the magnetic field of the Earth with the velocity of \(900km/h.\)

According to Faraday\('\)s law, \begin{equation*} \epsilon=\oint_{C} E~.~dr=-\frac{d\psi}{ dt}\,. \end{equation*} As the conducting ring moves through the Earth magnetic field, there is a change in the magnetic flux \(\psi\), passing through the ring. Suppose that the magnetic field \(B\) is perpendicular to the plane of the ring. Then change in the flux for the time \(\vartriangle t\) is \begin{equation*} \vartriangle \psi=2rBx=2rBv\vartriangle t\,, \end{equation*} where \(x = v\vartriangle t, v\) is the velocity of the airplane and \(B\) is the magnetic field of the Earth. It follows from the last expression that \begin{equation*} \epsilon=-\frac{d\psi}{dt}=2rBv. \end{equation*} Substituting the given values \( v = 900km/h = 250m/s, r = 1cm = 0.01m\) and \(B = 5\times10^{-5} T\), we obtain the electromotive force \( \epsilon=2rBv=2 \times 0.01 \times 5\times 10^{-5} \times 250=0.00025V\). As it can be seen, it's safe for human.

We can find the electric field in the conducting ring by the formula \(\epsilon=\oint_{C} E~.~dr\).

By symmetry, the induced electric field will have a constant magnitude along the ring. Its direction will be tangential to the circle at every point. Hence, the line integral around the circle is

\begin{equation*} \epsilon=\oint_{C} E~.~dr=\oint_{C} Edrcos0=E\oint_{C} dr=2\pi rE\,. \end{equation*} Hence, the electric field strength is \begin{equation*} E=\frac{\epsilon}{2\pi r}=\frac{0.00025}{2\pi \times 0.01}=0.00398V/m. \end{equation*}

5. Summary

From this study, we acquire an idea regarding to line integral and its applications. Before we deal line integral, we need to have an idea regarding to space curves, and scalar and vector fields. As stated earlier, we deal line integral in either Cartesian plane or Complex plane. This integral has geometrical as well as physical applications.

Acknowledgments First of all, I would like to thank my colleagues at Debark University for giving me their genuine advices and constructive comments from the very beginning of this study till its end. Secondly, my thanks also go to department of Mathematics, Debark University, which helped me in copying available hard materials and printing available soft copy materials for this study. Finally, I am deeply indebted to my wife Manayesh Melkamu for her encouragement to do this study and I would like to say God bless her.

Conflicts of Interest:

''The author declares no conflict of interest.''

1. Introduction

An electrically conducting fluid moving in a magnetic field generates an electric current which induces a magnetic field and the magnetohydrodynamic force, known as Lorentz force, is built up in the flow [1]. Applications of magnetohydrodynamics flow can be found in jet printers, fusion reactors, and MHD generators. Most importantly is the study of heat and mass transfer in a magnetohydrodynamic flow which has practical applications in biosensors, aerosol generation and dispersion, and nuclear waste repository. Katagiri [2] studied the MHD Couette motion formation in a viscous incompressible fluid and found out that the velocity declines with increasing magnetic field strength. [3] presented an analysis of an unsteady MHD free convective HAMT in a boundary layer flow and found out that velocity profiles decrease with magnetic field, while concentration decreases with Schmidt number. Sheri and Modugula [4] analysed an unsteady MHD flow across an inclined plate and inferred that temperature profiles decrease with Prandtl number while velocity profiles increase with either the solutal Grashof number or thermal Grashof number. Sivaiah and Reddy [5] analysed HAMT of an unsteady MHD flow past a moving inclined porous plate. Flow velocity was found to rise with an increase in Magnetic field strength; as against the results from [2]. Their results showed that velocity profile increases with increasing solutal and thermal Grashof number; in agreement with the results from [2]. Also, velocity and concentration decrease with Schmidt number while temperature profile decreases with Prandtl number. Iva et al. [6] also supported the results of [2] across a rotating plane. Sreedhar and Reddy [7] considered the impact of chemical reaction in the presence of heat absorption and found out that velocity profiles decrease with both Prandtl number and magnetic field strength. Zafar et al. [8] analysed the effect of inclination angle on MHD flow. Hussain et al. [9] examined the magnetohydrodynamic flow of Maxwell nanofluid and deduced that flow velocity decreases as either magnetic field strength and/or inclination angle increases. The results also show that flow velocity increases as Maxwell parameter increases while flow temperature is enhanced with rising inclination angle.

Williamson fluid is a non-Newtonian fluid that exhibits shear thinning characteristics of non-Newtonian fluids. Williamson and Ouru [10,11] in 1929 experimentally introduced the Williamson fluid model which models a fluid whose viscosity reduces indefinitely as shear rate increases (meaning " an infinite viscosity when there is no fluid motion but zero viscosity as the shear rate tend to infinity" ). In a study by Khan et al. [12], an extensive investigation is conducted to unravel the thermophysical properties of MHD Williamson flow past a simultaneously rotating and stretching surface. Results indicated that velocity is boosted as values of rotation gets larger and increment in \(Pr\) inhibits temperature distribution. Yusuf and Mabood [13] examined chemical reaction on MHD Williamson fluid flow over an inclined permeable wall. The results indicate that both the magnetic strength and the Williamson fluid parameter have an adverse effect on the fluid velocity. Srinivasulu and Goud [14] explored the impact of Lorentz force on Williamson' s nanofluid. With a rise in magnetic strength, velocity profile diminishes but boosts the temperature and concentration profiles. The temperature and concentration profiles increase and velocity profile decreases with an increase in inclination angle. Li et al., [15] considers the heat generation and/or heat absorptions on MHD Williamson nanofluid flow.

Coriolis force is an inertia force that is generated in a rotating frame. It is the force responsible for the apparent curved trajectory of a linearly moving object in a rotating frame. The Coriolis force increases as the angular velocity of the rotating plane increases. Application of Coriolis force can be found in astrophysics, oceanography, and bioreactors [16,17]. The study of magnetohydrodynamic flow of Newtonian fluid over a rotating non-uniform surface was studied by [18]. It is found that simultaneously increasing the Coriolis force and Magnetic field strength leads to an increase in the flow temperature. Meanwhile, [19] extended the work of [18] by considering the non-Newtonian Casson fluid flow over a rotating non-uniform surface. The study showed that increasing Coriolis force increases the temperature profile and increases the primary velocity. A study of the flow of nanofluid flow over a rotating flat surface is conducted by [20]. It is recorded that the presence of Coriolis force has a significant impact on the arrangement of nanoparticles in the nanofluid.

Based on the available information, very little has not been done to figure out how Williamson fluid flows across an inclined plate. In this present study, a two-dimensional flow of Williamson flow past an inclined plate is considered. This study provides answers to the following questions; 1) what are the combined effects of Lorentz force and inclination angle on the magnetohydrodynamic flow of Williamson fluid over an inclined rotating plate? 2) what are the combined effects of Lorentz force and Coriolis force on the magnetohydrodynamic flow of Williamson fluid over an inclined rotating plate?

2. Governing equations

This study considers a steady boundary layer flow of a viscous, thermally and electrically-conducting Williamson fluid over an inclined porous plate that rotates at angular velocity \(\Omega.\) The plane is inclined at an angle \(\alpha\) and the flow configuration is shown in Figure 1.

The magnetic field is applied at an angle \(90^{o}\) to the direction of the flow with a constant magnetic field strength of \(B_{0}.\) The inclined surface is stretched linearly and the no-slip condition is upheld so that the velocity at the wall is the same as the velocity of the Williamson fluid layer closest to the wall. Hence, the velocity at the wall is given as \(u=ax.\) By including the porosity, Coriolis force and inclination angle in the formulations of [11], the system of governing equations comprising of the continuity equation, momentum equation, energy equation, and the species equation is given as

with the boundary conditions The quantities of industrial and engineering importance [4] are the coefficient of skin friction, Nusselt number and Sherwood number defined as \[ C_{f}=\frac{\nu}{a^{2}}\left(\frac{\partial u}{\partial y}\right)_{y=0},\;Nu=-\frac{x\left(\frac{\partial T}{\partial y}\right)_{y=0}}{\left(T_{w}-T_{\infty}\right)},\;Sh=-\frac{x\left(\frac{\partial C}{\partial y}\right)_{y=0}}{\left(C_{w}-C_{\infty}\right)}. \]

3. Methodology

By using the similarity variables \[ u=axf',\;v=-\left(a\nu\right)^{\frac{1}{2}}f,\;\theta=\frac{T-T_{\infty}}{T_{w}-T_{\infty}},\ \Phi=\frac{C-C_{\infty}}{C_{w}-C_{\infty}},\;\eta=y\left(\frac{a}{\nu}\right)^{\frac{1}{2}}, \] the system of governing equations (Eqs. (1)-(4)) is transformed to the dimensionless form as with the dimensionless boundary conditions where the dimensionless parameters are the solutal and thermal Grashof number, Rotation parameter, magnetic field parameter, porosity parameter, Schmidt number, Brownian parameter, thermophoretic parameter, Prandtl number, and the Williamson fluid parameter defined as \begin{align*} & Gr_{t}=\frac{g\beta\left(T_{w}-T_{\infty}\right)}{a^{2}x},\;Gr_{s}=\frac{g\beta^{*}\left(C_{w}-C_{\infty}\right)}{a^{2}x},\;K=\frac{2\Omega}{a},M=\frac{\sigma B_{0}^{2}}{a\rho},\;K_{c}=\frac{\nu}{a\rho},\\ & Sc=\frac{\nu}{D_{B}},\;N_{b}=\frac{\tau D_{B}}{\alpha},\;N_{t}=\frac{\tau D_{T}\left(T_{w}-T_{\infty}\right)}{\alpha T_{\infty}},\;Pr=\frac{\nu}{\alpha},\;\gamma=\Gamma\left(\frac{2a^{3}x^{2}}{\nu}\right)^{\frac{1}{2}}\,. \end{align*} Also, the dimensionless form of the local skin friction \(C_{f},\) the Nusselt number \(Nu,\)and the Sherwood number \(Sh\) are \[ R_{e}^{\frac{1}{2}}C_{f}=2\left(1+\frac{\gamma}{2}f''\left(0\right)\right)f''\left(0\right),\;R_{e}^{-\frac{1}{2}}Nu=-\theta'\left(0\right),\;Re^{-\frac{1}{2}}Sh=-\Phi\left(0\right). \] Eqs. (6)-(8) can be written as a system of first order ordinary differential equations by using the transformations \[ X_{1}=f,\ X_{2}=f',\ X_{3}=f'',\ X_{4}=\theta,\;X_{5}=\theta',\ X_{6}=\Phi,\ X_{7}=\Phi'. \] Hence, the system is with the boundary conditions To solve the system (10), the boundary conditions (11) and (12) need to be converted to initial values. This can be achieved by using shooting technique which requires the choice \[ X_{1}\left(0\right)=0,\ X_{2}\left(0\right)=1,\ X_{3}\left(0\right)=s_{1},\ X_{4}\left(0\right)=0,\;X_{5}\left(0\right)=s_{2},\ X_{6}\left(0\right)=1,\ X_{7}\left(0\right)=s_{3}. \] By making repeated arbitrary assumptions for \(s_{1},s_{2}\) and \(s_{3},\) the problem is solved until the three remaining boundary conditions \[ X_{2}\left(\infty\right)\to0,\;X_{4}=1,\ X_{6}\left(\infty\right)=0. \] are satisfied. The problem is solved numerically using the MATLAB bvp4c solver (for other methods of solution, see [11]). The results from the present work is validated against the results of [22] in Table (1) with the choice of parameter values as \[ Gr_{t}=Gr_{s}=K_{c}=K=0,\alpha=\pi/2,M=N_{b}=N_{t}=Pr=0 \]

Table 1. Results validation for \(Re^{\frac{1}{2}}C_{f}\).

\(\gamma\)	0	0.1	0.2	0.3
Ahmed and Akbar \cite{22}	1.33930	1.29801	1.26310	1.22276
present study	1.33013	1.29880	1.26384	1.22345

4. Analysis and discussion of results

The resulting system (10) is solved using the MATLAB bvp4c solver and the solutions are presented in graphs and tables. The default values for the parameters are chosen as follows; \[ Gr_{t}=Gr_{s}=1.0,\;Sc=0.62,\;M=2,\;Pr=4,\;\alpha=\pi/6,\;K=N_{b}=N_{t}=K_{c}=\gamma=0.1. \] The results are analysed and discussed in this section.

4.1. Analysis of results

Figures 2 - 4 show the impact of simultaneously increasing the magnetic field strength and inclination angle on Williamson fluid flow over a rotating surface. Figures 2 and 3 show that the velocity reduces as magnetic field strength increases and inclination angle increases to \(90^{o}\) while the flow temperature increases as the magnetic field strength and inclination angle increase simultaneously as shown in Figure 4. The combined effects of rotation and Prandtl number on the velocity are shown in Figures 5 and 6. It is revealed that both the primary and secondary velocity profiles increase with increasing rotation, meanwhile, both the primary and secondary velocity profiles decrease with increasing Prandtl number. Figures 7 and 8 show the variations of velocity profiles with the simultaneous increase in Williamson fluid parameter and Coriolis force. The flow velocity profiles decrease in all directions with increasing Williamson fluid parameter but increase in all directions with increasing Coriolis force.

Tables 2 and 3 demonstrate how the rotation parameter and magnetic field strength affect the local skin friction, heat transfer rate and mass transfer rate. As rotation increases, the local skin friction decreases at the rate of -0.8052, the Nusselt number decreases at the rate 0.06 and Sherwood number increases at the rate 0.0218. On the other hand, the local skin friction increases at the rate of 0.7191, the Nusselt number decreases at the rate -0.0492 and Sherwood number decreases at the rate -0.016 as magnetic field strength increases. With this, it is evident that the local skin friction can be increased by increasing magnetic field strength and decreasing rotation. In addition, the rate at which heat is transferred can be improved by increasing rotation and reducing magnetic field strength. Finally, rate of convective mass transfer can be boosted by increasing rotation and reducing magnetic field strength.

Table 2. Quantities of interest with rotation parameter.

\(K\)	skin friction	Nusselt number	Sherwood number
0	2.70423	1.19702	1.23699
0.1	2.62641	1.20272	1.23899
0.2	2.54775	1.20851	1.24104
0.3	2.46822	1.21440	1.24315
0.4	2.38778	1.22039	1.24532
0.5	2.30637	1.22649	1.24755
0.6	2.22394	1.23270	1.24986
0.7	2.14044	1.23902	1.25224
slope	-0.80520	0.06000	0.02180

Table 3. Quantities of interest with magnetic field strength.

\(M\)	skin friction	Nusselt number	Sherwood number
1	1.22976	1.30996	1.28113
2	2.14044	1.23902	1.25224
3	2.93308	1.18045	1.23132
4	3.65485	1.13012	1.21514
5	4.32955	1.08584	1.20209
6	4.97109	1.04631	1.19122
7	5.58845	1.01062	1.18198
slope	0.71910	-0.04920	-0.01600

4.2. Discussion of results

The presence of a magnetic field generates the Lorentz force. The Lorentz force generated with the presence of magnetic field acts in the opposite direction to fluid flow and thereby causes a reduction in flow velocity. Increasing inclination angle reduces flow velocities in all directions since more work is done by the fluid to climb. Simultaneously increasing the magnetic field strength and inclination angle causes more reduction on flow velocity profiles in all directions. Meanwhile, heat energy is generated in the system as the Lorentz force opposes the motion (due to magnetic field presence) and more heat energy is also generated as the fluid climbs the plate (due to an increase in inclination angle). Hence, the temperature profile increases as both magnetic field strength and angle of inclination increase.

As the surface rotates, Coriolis force is generated. Hence, increasing rotation leads to an increase in the Coriolis force. Also, the Prandtl number measures the ratio of momentum diffusivity to thermal diffusivity. Hence, increasing Prandtl number consequently means an increase in momentum diffusivity or a decrease in thermal diffusivity or both. It is revealed that velocity profiles increase with increasing rotation, meanwhile, both the primary and secondary velocity profiles decrease with increasing Prandtl number. The increase in velocity profiles as rotation increases is because more kinetic energy is added to the flow as rotation amplifies. A surge in Prandtl number consequently reduces thermal diffusivity while momentum diffusivity increases; this is the reason for the decrease in the velocity profiles as Prandtl number increases. It is worth mentioning that angular speed can be increased or decreased to adjust the flow velocities of Williamson fluid.

Williamson fluid possesses a thinner boundary layer compared to the Newtonian fluid but increasing the Williamson parameter increases the boundary layer. The flow approaches Newtonian flow and the velocity reduces in the process. Hence, the velocity profiles in all directions decrease with increasing Williamson fluid parameter.

5. Conclusion

This study captured the effects of magnetic field strength, rotation and inclination angle on the flow of Williamson fluid in a porous medium. The equations are formulated and solved numerically to produce graphs that illustrate the dynamics of Williamson fluid flow over a rotating inclined surface. The outcomes of this study are summarised below;

1. Velocity profiles reduce in all directions with increasing magnetic field strength, inclination angle and Williamson fluid parameter.
2. While temperature decreases with increasing magnetic field strength and inclination angle
3. Velocity profiles increase in all directions with increasing Coriolis force
4. Velocity profiles decrease in all directions with increasing Prandtl number.
5. The local skin friction and the heat transfer rate decrease with increasing rotation but mass transfer rate increases with increasing rotation.
6. The local skin friction rate increases with increasing Lorentz force but heat and mass transfer rates decrease with increasing Lorentz force.

Author Contributions:

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest:

''The authors declare no conflict of interest.''

1. Introduction

Neggers and Kim [1] introduced the notion of \( d\)-algebras and investigated the properties of them. Neggers et al. [2] introduced the concepts of d-ideals in d-algebra. Urge to deal with uncertainty by tools different from that of probability lead the way to fuzzy sets, rough sets and soft sets. Zadeh introduced fuzzy sets [3]. Akram and Dar [4] introduced the notions of fuzzy subalgebras and \( d\)-ideals in \( d\)-algebras and investigated some of their results. Al-Shehrie [5] introduced the notions of fuzzy dot \( d\)-ideals of \( d\)-algebras and some properties are investigated. Dejen [6] investigated product of fuzzy dot \( d\)-ideals and strong fuzzy relation and the corresponding strong fuzzy dot \( d\)-ideals. The triangular norms, \( t\)-norms, originated from the studies of probabilistic metric spaces in which triangular inequalities were extended using the theory of T-norms. Later, Hohle [7], Alsina et al. [8] introduced the \( t\)-norms into fuzzy set theory and suggested that the \( t\)-norms be used for the intersection of fuzzy sets. Since then, many other researchers presented various types of \( t\)-norms for particular purposes [9,10].

The author by using norms, investigated some properties of fuzzy algebraic structures [11,12,13,14,15]. In this paper, we introduce the notion of fuzzy subalgebras (as \( FST(X) \)) and fuzzy \(d\)-ideals (as \( FDIT(X) \)) of \(d\)-algebras \(X\) by using \( t \)-norm \( T\) and then we investigate different characterizations and several basic properties which are related to them. Next we define cartesian product and intersection of them and we obtain some new results about them. Finally we show that the image and pre-image of them are also \( FST(X) \) and \( FDIT(X) \) uner homomorphisms of \(d\)-algebras.

2. Preliminaries

The following definitions and preliminaries are required in the sequel of our work and hence presented in brief.

Definition 1. [1] A nonempty set \( X\) with a constant \( 0\) and a binary operation \( \ast\) is called a \( d\)-algebra, if it satisfies the following axioms:

1. \( x \ast x=0, \)
2. \( 0 \ast x=0, \)
3. if \( x \ast y=0 \) and \( y \ast x=0, \) then \( x=y, \) for all \( x,y \in X. \)

Definition 2.[2] Let \( S\) be a non-empty subset of a \( d\)-algebra \( X\), then \( S\) is called subalgebra of \( X\) if \( x \ast y \in S \), for all \(x,y \in S. \)

Definition 3.[2] Let \( X\) be a \( d \)-algebra and \( I\) be a subset of \( X\), then \( I\) is called \( d\)-ideal of \( X\) if it satisfies following conditions:

1. \( 0 \in I, \)
2. if \( x \ast y \in I \) and \( y \in I, \) then \( x \in I, \)
3. if \( x \in I \) and \( y \in X, \) then \( x \ast y \in I. \)

Definition 4.[1] A mapping \( f: X \to Y \) of \( d\)-algebras is called a homomorphism if \(f(x \ast y) = f(x) \ast f(y)\), for all \(x, y \in X. \)

Definition 5.[16] Let \(X\) be an arbitrary set. A fuzzy subset of \(X\), we mean a function from \(X\) into \( [0,1]\). The set of all fuzzy subsets of \(X\) is called the \([0,1]\)-power set of \(X\) and is denoted \( [0,1]^X.\) For a fixed \(s \in [0, 1],\) the set \(\mu_{s} = \lbrace x \in X : \mu(x) \geq s \rbrace\) is called an upper level of \(\mu.\)

Definition 6.[16] Let \(f: X \to Y\) be a mapping of sets and \(\mu\in [0,1]^X\) and \(\nu\in [0,1]^Y.\) Define \(f(\mu)\in[0,1]^Y\) and \(f^{-1}(\nu)\in[0,1]^X,\) defined by \begin{equation*} f(\mu)(y) = \left\{ \begin{array}{rl} \sup \{ \mu(x) \hspace{0.1cm}|\hspace{0.1cm} x\in G,f(x)=y\} &\text{if } f^{-1}(y)\neq\emptyset\\ 0 &\text{if } f^{-1}(y)=\emptyset \\ \end{array} \right. \end{equation*} for all \( y\in Y. \) Also \(f^{-1}(\nu)(x)=\nu(f(x))\) for all \( x\in X. \)

Definition 7. [17] A \(t\)-norm \(T\) is a function \(T : [0,1]\times [0,1] \to [0,1]\) having the following four properties:

1. \(T(x,1)=x\) (neutral element),
2. \(T(x,y)\leq T(x,z)\) if \(y\leq z\) (monotonicity),
3. \(T(x,y)= T(y,x)\) (commutativity),
4. \( T(x,T(y,z))=T(T(x,y),z)\) (associativity),

for all \(x,y,z \in [0,1].\)

We say that \(T\) is idempotent if for all \(x \in [0,1]\),\(T(x, x) =x.\)

Example 1. The basic \(t\)-norms are \(T_m(x,y) = min \{ x,y \}\),\(T_b(x,y) = max\{0, x+y- 1 \}\) and \(T_p(x, y) = xy \), which are called standard intersection, bounded sum and algebraic product respectively.

Lemma 1.[1] Let \(T\) be a \(t\)-norm. Then \[T(T(x,y),T(w,z))= T(T(x,w),T(y,z)),\] for all \(x,y,w,z\in [0,1].\)

3. Main results

Definition 8. Let \(\mu\) be a fuzzy subset in \(d\)-algebra \(X.\) Then \(\mu\) is called a fuzzy subalgebra of \( X \) under \(t\)-norm \(T\) iff \( \mu(x \ast y) \geq T(\mu(x),\mu(y))\) for all \(x,y\in X.\) Denote by \(FST(X)\), the set of all fuzzy subalgebras of \( X \) under \(t\)-norm \(T.\)

Example 2. Let \(X = \lbrace 0, 1, 2 \rbrace\) be a set given by the following Cayley table:

*	0	1	2
0	0	0	0
1	2	0	2
2	1	1	0

Then \( (X,\ast, 0) \) is a \(d\)-algebra. Define fuzzy subset \( \mu: (X,\ast, 0) \to [0,1]\) as \begin{equation*} \mu(x) = \left\{ \begin{array}{rl} 0.35 &\text{if } x=0,\\ 0.25 &\text{if } x \neq 0. \\ \end{array} \right. \end{equation*} \(T(a, b) =T_p(a,b)=ab \) for all a,b \(\in [0,1]\) then \( \mu \in FST(X). \)

In the following propositions we investigate relation between \( \mu \in FST(X)\) and subalgebras of \(X.\)

Proposition 1. Let \( \mu \in [0,1]^X\) and \(T\) be idempotent. Then \( \mu \in FST(X)\) if and only if the upper level \( \mu_{t} \) is either empty or a subalgebra of \(X \) for every \(t \in [0, 1].\)

Proof. Let \( \mu \in FST(X)\) and \( x,y \in \mu_{t}.\) Then \[ \mu( x \ast y) \geq T(\mu(x),\mu(y)) \geq T(t,t)=t.\] Thus \( x \ast y \in \mu_{t}\) and so \( \mu_{t} \) will be a subalgebra of \(X \) for every \(t \in [0, 1].\)

Conversely, let \( \mu_{t} \) is either empty or a subalgebra of \(X \) for every \(t \in [0, 1].\) Let \( t=T(\mu(x),\mu(y)) \) and \( x, y \in S. \) As \( \mu_{t} \) is a subalgebra of \(X \) so \( x \ast y \in \mu_{t} \) and thus \( \mu( x \ast y) \geq t=T(\mu(x),\mu(y)). \) Then \( \mu \in FST(X).\)

In the following proposition we prove that any subalgebra of a \( d\)-algebra \( X\) can be realized as a level subalgebra of some fuzzy subalgebra of \(X. \)

Proposition 2. Let \( A\) be a subalgebra of a \( d \)-algebra \( X\) and \( \mu \in [0,1]^S\) such that \begin{equation*} \mu(x) = \left\{ \begin{array}{rl} t &\text{if } x\in A \\ 0 &\text{if } x \notin A \\ \end{array} \right. \end{equation*} with \(t \in (0, 1).\) If \(T\) be idempotent, then \( \mu \in FST(X).\)

Proof. We know that \( A= \mu_{t}.\) Let \( x, y \in X \) and we investigate the following conditions;

1. If \( x, y \in A, \) then \( x \ast y \in A \) and so \[ \mu( x \ast y)=t \geq t =T(t,t)=T(\mu(x),\mu(x)).\]
2. If \( x \in A \) and \( y \notin A, \) then \( \mu(x)=t \) and \( \mu(y)=0 \) and so \[ \mu( x \ast y) \geq 0 =T(t,0)=T(\mu(x),\mu(y)).\]
3. If \( x \notin A \) and \( y \in A, \) then \( \mu(x)=0 \) and \( \mu(y)=t \) and so \[ \mu( x \ast y) \geq 0 =T(0,t)=T(\mu(x),\mu(y)).\]
4. If \( x \notin A \) and \( y \notin A, \) then \( \mu(x)=0 \) and \( \mu(y)=0 \) and so \[ \mu( x \ast y) \geq 0 =T(0,0)=T(\mu(x),\mu(y)).\] Thus from (1)-(4) we get that \( \mu \in FST(X).\)

Corollary 1. Let \( A\) be a subset of \(X. \) Then the characteristic function \( \chi_{A} \in FST(X) \) if and only if \( A\) is a subalgebra of \( X. \)

Now under some conditions we prove that \( \mu_{s}=\mu_{t} \) for every \(s,t \in [0, 1].\)

Proposition 3. Let \( \mu \in FST(X)\) and \(s,t \in [0, 1].\) If \( s < t, \) then \( \mu_{s}=\mu_{t} \) if and only if there is no \(x \in X\) such that \(s \leq \mu(x) < t.\)

Proof. Let \( s < t\) and \( \mu_{s}=\mu_{t}.\) If there exists \(x \in X \) such that \(s \leq \mu(x) < t,\) then \( x \in \mu_{s}\) but \( x \notin \mu_{t}\) which is contradicting the hypothesis.

Conversely, let there is no \(x \in X\) such that \(s \leq \mu(x) < t.\) As \( x \in \mu_{s}\) so \( x \in \mu_{t}\) then \( \mu_{s}\subseteq \mu_{t}. \) If \( x \in \mu_{t}\) then \( \mu(x) \geq t > s \) so \( x \in \mu_{s}\) then \( \mu_{t}\subseteq \mu_{s}.\) Therefore \( \mu_{s}=\mu_{t}.\)

Definition 9. \( \mu \in [0,1]^X\) is called fuzzy \( d\)-ideal of \( X\) under \(t\)-norm \(T\) if it satisfies the following inequalities:

1. \( \mu(0) \geq \mu(x), \)
2. \( \mu(x) \geq T(\mu(x \ast y),\mu(y)),\)
3. \( \mu(x \ast y) \geq T(\mu(x),\mu(y)), \) for all \( x,y \in X. \)

The set of all fuzzy \( d\)-ideals of \( X\) under \(t\)-norm \(T\) is denoted by \(FDIT(X)\).

Corollary 2. Let \(\mu \in FDIT(X). \) Then

1. \( \mu \in FST(X). \)
2. \( \mu(0) \geq \mu(x)\) and \( \mu(x) \geq T(T(\mu(x),\mu(y)),\mu(y))\) for all \( x,y \in X. \)

Example 3. Let \(X = \lbrace 0, 1, 2, 3 \rbrace\) be a set given by the following Cayley table:

*	0	1	2	3
0	0	0	0	0
1	1	0	0	1
2	2	2	0	0
3	3	3	3	0

Then \( (X,\ast, 0) \) is a \(d\)-algebra. Define fuzzy subset \( \mu: (X,\ast, 0) \to [0,1]\) as \begin{equation*} \mu(x) = \left\{ \begin{array}{rl} 0.65 &\text{if } x=0\\ 0.15 &\text{if } x \neq 0 \\ \end{array} \right. \end{equation*} \(T(a, b) =T_p(a,b) =ab \) for all a,b \(\in [0,1]\) then \( \mu \in FDIT(X). \)

Definition 10. Let \( \mu \in [0,1]^X\) and \( \nu \in [0,1]^Y.\) The cartesian product of \( \mu \) and \( \nu \) is denoted by \( \mu \times \nu: X \times Y \to [0,1] \) and is defined by \( (\mu \times \nu)(x,y)=T(\mu(x),\nu(y)) \) for all \( (x,y) \in X \times Y. \)

In the following propositions we investigate the properties of cartesian product \(FST(X)\) and \( FDIT(X). \)

Proposition 4. Let \( \mu \in FST(X)\) and \( \nu \in FST(Y).\) Then \( \mu \times \nu \in FST(X \times Y).\)

Proof. Let \( (x_{1},y_{1}), (x_{2},y_{2}) \in X \times Y. \) Then \begin{align*}(\mu \times \nu)((x_{1},y_{1}) \ast (x_{2},y_{2}))&=(\mu \times \nu)(x_{1} \ast x_{2},y_{1} \ast y_{2}) \\ &=T(\mu(x_{1} \ast x_{2}), \nu(y_{1} \ast y_{2})) \\ &\geq T(T(\mu(x_{1}),\mu(x_{2})),T(\nu(y_{1}),\nu(y_{2}))) \\ &=T(T(\mu(x_{1}),\nu(y_{1})),T(\mu(x_{2}),\nu(y_{2})))\hspace{0.1cm} \text{(Lemma 1)}\\ &=T((\mu \times \nu)(x_{1},y_{1}),(\mu \times \nu)(x_{2},y_{2})).\end{align*}Thus \[(\mu \times \nu)((x_{1},y_{1}) \ast (x_{2},y_{2})) \geq T((\mu \times \nu)(x_{1},y_{1}),(\mu \times \nu)(x_{2},y_{2})) \] and so \( \mu \times \nu \in FST(X \times Y).\)

Proposition 5. Let \( \mu \in FDIT(X)\) and \( \nu \in FDIT(X).\) Then \( \mu \times \nu \in FDIT(X \times Y).\)

Proof.

1. Let \( (x,y) \in X \times Y.\) Then \((\mu \times \nu)(0,0)=T(\mu(0),) \nu(0)) \geq T(\mu(x),) \nu(x)). \)
2. Let \( x_{i} \in X \) and \( y_{i} \in Y \) for \( i=1,2, \) then \begin{align*}(\mu \times \nu)(x_{1},y_{1})&=T(\mu(x_{1}),\nu(y_{1})) \\ &\geq T(T(\mu(x_{1} \ast x_{2}),\mu(x_{2})),T(\nu(y_{1} \ast y_{2}),\nu(y_{2})))\\ &=T(T(\mu(x_{1} \ast x_{2}),\nu(y_{1} \ast y_{2})),T(\mu(x_{2}),\nu(y_{2})))\hspace{0.1cm} \text{(Lemma 1)}\\ &=T((\mu \times \nu)(x_{1} \ast x_{2},y_{1} \ast y_{2}),(\mu \times \nu)(x_{2},y_{2}))\\ &=T((\mu \times \nu)(x_{1}, y_{1})\ast (x_{2} , y_{2}),(\mu \times \nu)(x_{2},y_{2})).\end{align*} Then \( (\mu \times \nu)(x_{1},y_{1}) \geq T((\mu \times \nu)(x_{1}, y_{1})\ast (x_{2} , y_{2}),(\mu \times \nu)(x_{2},y_{2})).\)
3. \begin{align*}(\mu \times \nu)((x_{1},y_{1}) \ast (x_{2},y_{2}))&=(\mu \times \nu)(x_{1} \ast x_{2},y_{1} \ast y_{2})\\ &=T(\mu(x_{1} \ast x_{2}), \nu(y_{1} \ast y_{2})) \\ &\geq T(T(\mu(x_{1}),\mu(x_{2})),T(\nu(y_{1}),\nu(y_{2})))\\ &=T(T(\mu(x_{1}),\nu(y_{1})),T(\mu(x_{2}),\nu(y_{2})))\hspace{0.1cm} \text{(Lemma 1)}\\ &=T((\mu \times \nu)(x_{1},y_{1}),(\mu \times \nu)(x_{2},y_{2})).\end{align*} Thus \[(\mu \times \nu)((x_{1},y_{1}) \ast (x_{2},y_{2})) \geq T((\mu \times \nu)(x_{1},y_{1}),(\mu \times \nu)(x_{2},y_{2})).\]

Therefore from (1) -(3) we get that \( \mu \times \nu \in FDIT(X \times Y).\)

Proposition 6. Let \( \mu \in [0,1]^X\) and \( \nu \in [0,1]^Y.\) If \( \mu \times \nu \in FDIT(X \times Y),\) then at least one of the following two statements must hold.

1. \( \mu(0) \geq \mu(x) \) for all \( x \in X.\)
2. \( \nu(0) \geq \nu(y) \) for all \( y \in Y.\)

Proof. Let none of the statements (1) and (2) holds, then we can find \( (x,y) \in X \times Y \) such that \( \mu(0) < \mu(x) \) and \( \nu(0) < \nu(y).\) Thus \[(\mu \times \nu)(x,y)= T(\mu(x),\nu(y)) > T(\mu(0),\nu(0))=(\mu \times \nu)(0,0)\] and it is contradiction with \( \mu \times \nu \in FDIT(X \times Y).\)

Proposition 7. Let \( \mu \in [0,1]^X\) and \( \nu \in [0,1]^Y.\) If \( \mu \times \nu \in FDIT(X \times Y)\) and \(T\) be idempotent, then we obtain the following statements:

1. If \( \mu(0) \geq \mu(x), \) then either \( \nu(0) \geq \mu(x) \) or \( \nu(0) \geq \nu(y) \) for all \( (x,y) \in X \times Y.\)
2. If \( \nu(0) \geq \nu(y), \) then either \( \mu(0) \geq \nu(y) \) or \( \mu(0) \geq \mu(x) \) for all \( (x,y) \in X \times Y.\)

Proof.

1. Let \( \mu(0) \geq \mu(x)\) and we have \( (x,y) \in X \times Y\) such that \( \nu(0) < \mu(x) \) and \( \nu(0) < \nu(y). \) Then \( \mu(0) \geq \mu(x) > \nu(0)\) and so \( \nu(0) = T(\mu(0),\nu(0)).\) Thus \[(\mu \times \nu)(x,y)= T(\mu(x),\nu(y)) > T(\nu(0),\nu(0))=\nu(0)=T(\mu(0),\nu(0))=(\mu \times \nu)(0,0)\] and it is contradiction with \( \mu \times \nu \in FDIT(X \times Y).\)
2. Let \( \nu(0) \geq \nu(y)\) such that for \( (x,y) \in X \times Y\) we have \( \mu(0) < \nu(y) \) and \( \mu(0) < \mu(x). \) So \( \nu(0) \geq \nu(y) > \mu(0)\) and \( \mu(0) = T(\mu(0),\nu(0)).\) Thus \[(\mu \times \nu)(x,y)= T(\mu(x),\nu(y)) > T(\mu(0),\mu(0))=\mu(0)=T(\mu(0),\nu(0))=(\mu \times \nu)(0,0)\]

and it is contradiction with \( \mu \times \nu \in FDIT(X \times Y).\)

Now we prove the converse of Proposition 5.

Proposition 8. If \( \mu \times \nu \in FDIT(X \times Y)\) and \(T\) be idempotent, then \( \mu \in FDIT(X) \) or \( \nu \in FDIT(Y).\)

Proof. We prove that \( \mu \in FDIT(X) \) and the proof \( \nu \in FDIT(Y)\) is similar. As Proposition 6 we have that

for all \( x \in X\) and from Proposition 7(1), we have either \( \nu(0) \geq \mu(x) \) or \( \nu(0) \geq \nu(y) \) for all \( (x,y) \in X \times Y\) thus \((\mu \times \nu)(x,0) = T(\mu(x),\nu(0)) =\mu(x).\) Let \( (x,y), (\acute{x},\acute{y}) \in X \times Y\) and as \( \mu \times \nu \in FDIT(X \times Y)\) so \[(\mu \times \nu)(x,y) \geq T((\mu \times \nu)((x,y) \ast (\acute{x},\acute{y})),(\mu \times \nu)(\acute{x},\acute{y})))=T((\mu \times \nu)((x \ast \acute{x},y \ast \acute{y})),(\mu \times \nu)(\acute{x},\acute{y})))\] thus \[(\mu \times \nu)(x,y) \geq T((\mu \times \nu)((x \ast \acute{x},y \ast \acute{y})),(\mu \times \nu)(\acute{x},\acute{y})))\] and by putting \( y= \acute{y}=0 \) we will have \[(\mu \times \nu)(x,0) \geq T((\mu \times \nu)((x \ast \acute{x},0 \ast 0)),(\mu \times \nu)(\acute{x},0)))\] and so Also since \( \mu \times \nu \in FDIT(X \times Y)\) so \[(\mu \times \nu)((x,y) \ast (\acute{x},\acute{y})) \geq T((\mu \times \nu)(x,y),(\mu \times \nu)(\acute{x},\acute{y}))\] thus \[(\mu \times \nu)(x \ast \acute{x},y \ast \acute{y}) \geq T((\mu \times \nu)(x,y),(\mu \times \nu)(\acute{x},\acute{y}))\] and by letting \( y= \acute{y}=0 \) we get that \[(\mu \times \nu)(x \ast \acute{x},0 \ast 0) \geq T((\mu \times \nu)(x,0),(\mu \times \nu)(\acute{x},0))\] which means that Thus from Eqs. (1)-(3), we have that \( \mu \in FDIT(X).\)

Definition 11. Let \( A: S \to [0,1]\) be a fuzzy set in a set \( S\). The strongest fuzzy relation on \( S\) under \(t\)-norm \(T\) is fuzzy relation on \( A\) with \(\mu_{A}: S \times S \to [0,1] \) given by \[\mu_{A}(x, y)= T(A(x), A(y))\] for all \(x, y \in S.\)

Proposition 9. Let \(T\) be idempotent. Then \[ A \in FDIT(X) \Longleftrightarrow \mu_{A} \in FDIT(X \times X).\]

Proof. Let \( A \in FDIT(X). \)

1. Let \( x \in X \) then \(\mu_{A}(0,0)=T(A(0),A(0)) \geq T(A(x),A(x))=\mu_{A}(x,x).\)
2. Let \( (x_{1},x_{2}),(y_{1},y_{2}) \in X \times X. \) Then \begin{align*} \mu_{A}(x_{1},x_{2})&=T(A(x_{1}),A(x_{2}))\\&\geq T(T(A(x_{1} \ast y_{1}),A(y_{1})),T(A(x_{2} \ast y_{2}),A(y_{2}))) \\ &=T(T(A(x_{1} \ast y_{1}),A(x_{2} \ast y_{2})),T(A(y_{1}),A(y_{2}))) \hspace{0.9cm} \text{(Lemma 1)}\\ &=T(\mu_{A}(x_{1} \ast y_{1},x_{2} \ast y_{2}),\mu_{A}(y_{1},y_{2}))\\ &=T(\mu_{A}((x_{1},x_{2}) \ast (y_{1},y_{2}) ),\mu_{A}(y_{1},y_{2})).\end{align*} Thus \[\mu_{A}(x_{1},x_{2}) \geq T(\mu_{A}((x_{1},x_{2}) \ast (y_{1},y_{2}) ),\mu_{A}(y_{1},y_{2})).\]
3. Let \( (x_{1},x_{2}),(y_{1},y_{2}) \in X \times X. \) Then \begin{align*}\mu_{A}((x_{1},x_{2}) \ast (y_{1},y_{2}))&=\mu_{A}(x_{1} \ast y_{1},x_{2} \ast y _{2})\\ &=T(A(x_{1} \ast y_{1}),A(x_{2} \ast y_{2})) \\ &\geq T(T(A(x_{1}),A(y_{1})),T(A(x_{2}),A(y_{2})))\hspace{0.9cm} \text{(Lemma 1)}\\ &=T(T(A(x_{1}),A(x_{2})),T(A(y_{1}),A(y_{2})))\\ &=T(\mu_{A}(x_{1},x_{2}),\mu_{A}(y_{1},y_{2})).\end{align*} So \[\mu_{A}((x_{1},x_{2}) \ast (y_{1},y_{2})) \geq T(\mu_{A}(x_{1},x_{2}),\mu_{A}(y_{1},y_{2})).\]

Then (1)-(3) give us \( \mu_{A} \in FDIT(X \times X). \)

Conversely, suppose that \( \mu_{A} \in FDIT(X \times X).\)

1. Let \( x \in X \) then \[A(0)=T(A(0),A(0))=\mu_{A}(0,0) \geq \mu_{A}(x,x)=A(x)\] and \[A(0) \geq A(x) . \] So \( \mu_{A}(x,0)=T(A(x),A(0))=A(x). \)
2. Let \((x_{1},x_{2}),(y_{1},y_{2}) \in X \times X\), then \[\mu_{A}(x_{1},x_{2}) \geq T(\mu_{A}(x_{1},x_{2}) \ast (y_{1},y_{2})),\mu_{A}(y_{1},y_{2})))=T(\mu_{A})(x_{1}) \ast y_{1},(x_{2} \ast (y_{2}),\mu_{A}(y_{1},y_{2})).\] If we let \( x_{2}=y_{2}=0 \) , then \[\mu_{A}(x_{1},0) \geq T(\mu_{A}(x_{1} \ast {y_{1},0} \ast 0),\mu_{A}(y_{1},0)).\] Thus \( A(x_{1}) \geq T(A(x_{1} \ast y_{1}),A(y_{1})).\)
3. Let \((x_{1},x_{2}),(y_{1},y_{2}) \in X \times X,\) then \[\mu_{A}( (x_{1},x_{2}) \ast (y_{1},y_{2})) \geq T(\mu_{A}(x_{1},x_{2}),\mu_{A}(y_{1},y_{2}))\] and \[\mu_{A}(x_{1} \ast y_{1},x_{2} \ast y_{2}) \geq T(\mu_{A}(x_{1},x_{2}),\mu_{A}(y_{1},y_{2})).\] By letting \( x_{2}=y_{2}=0 \), we get that \[\mu_{A}(x_{1} \ast y_{1},0) \ast 0) \geq T(\mu_{A}(x_{1},0),\mu_{A}(y_{1},0))\] and thus \( A(x_{1} \ast y_{1}) \geq T(A(x_{1}),A(y_{1})).\)

Now, from (1)-(3), we have \(A \in FDIT(X).\)

Definition 12. Let \( \mu \in [0,1]^X\) and \( \nu \in [0,1]^X.\) The intersection of \( \mu \) and \( \nu \) is denoted by \( \mu \cap \nu: X \to [0,1] \) and is defined by \( (\mu \cap \nu)(x)=T(\mu(x),\nu(x)) \) for all \( x \in X.\)

In the following propositions we investigate the intersection of two \( \mu, \nu \in FST(X)\) and \( \mu, \nu \in FDIT(X).\)

Proposition 10. If \( \mu, \nu \in FST(X),\) then \( \mu \cap \nu \in FST(X).\)

Proof. Let \( x,y \in X.\) Then \begin{align*} (\mu \cap \nu)(x \ast y)&=T(\mu(x \ast y),\nu(x \ast y))\\ & \geq T(T(\mu(x),\mu(y)),T(\nu(x),\nu(y)))\\ &=T(T(\mu(x),\nu(x)),T(\mu(y),\nu(y)))\\ &=T((\mu \cap \nu)(x),(\mu \cap \nu)(y)).\end{align*} Thus \( (\mu \cap \nu)(x \ast y) \geq T((\mu \cap \nu)(x),(\mu \cap \nu)(y)) \) and so \( \mu \cap \nu \in FST(X).\)

Proposition 11. If \( \mu, \nu \in FDIT(X),\) then \( \mu \cap \nu \in FDIT(X).\)

Proof. Let \( x,y \in X.\) Then

1. \[(\mu \cap \nu)(0)=T(\mu(0),\nu(0)) \geq T(\mu(x),\nu(x))=(\mu \cap \nu)(x).\]
2. \begin{align*}(\mu \cap \nu)(x)&=T(\mu(x),\nu(x)) \\ &\geq T(T(\mu(x \ast y),\mu(y)),T(\nu(x \ast y),\nu(y)))\\ &=T(T(\mu(x \ast y),\nu(x \ast y)),T(\mu(y),\nu(y)))\\ &=T((\mu \cap \nu)(x \ast y),(\mu \cap \nu)(y))\end{align*} and thus \[(\mu \cap \nu)(x) \geq T((\mu \cap \nu)(x \ast y),(\mu \cap \nu)(y)).\]
3. \begin{align*}(\mu \cap \nu)(x \ast y)&=T(\mu(x \ast y),\nu(x \ast y))\\ &\geq T(T(\mu(x),\mu(y)),T(\nu(x),\nu(y)))\\ &=T(T(\mu(x),\nu(x)),T(\mu(y),\nu(y)))\\ &=T((\mu \cap \nu)(x),(\mu \cap \nu)(y)).\end{align*} Then \[ (\mu \cap \nu)(x \ast y) \geq T((\mu \cap \nu)(x),(\mu \cap \nu)(y)).\]

Now from (1)-(3), we get \( \mu \cap \nu \in FDIT(X).\)

In the following propositions we consider \(FST(X)\) and \(FDIT(X)\) under homomorphisms of \(d\)-algebras.

Proposition 12. If \( \mu \in FST(X)\) and \(f: X \to Y\) be a homomorphism of \(d\)-algebras, then \( f(\mu) \in FST(Y).\)

Proof. Let \( y_{1},y_{2} \in Y \) and \( x_{1},x_{2} \in X \) such that \( f(x_{1})=y_{1} \) and \( f(x_{2})=y_{2}. \) Then \begin{align*} f(\mu)(y_{1} \ast y_{2})&=\sup \{ \mu(x_{1} \ast x_{2}) \hspace{0.1cm}|\hspace{0.1cm} x_{1},x_{2} \in X,f(x_{1})=y_{1},f(x_{2})=y_{2}\}\\ &\geq \sup \{ T(\mu(x_{1}),\mu(x_{2})) \hspace{0.1cm}|\hspace{0.1cm} x_{1},x_{2} \in X,f(x_{1})=y_{1},f(x_{2})=y_{2}\}\\ &=T(\sup \{ \mu(x_{1}) \hspace{0.1cm}|\hspace{0.1cm} x_{1} \in X,f(x_{1})=y_{1}\},\sup \{ \mu(x_{2}) \hspace{0.1cm}|\hspace{0.1cm} x_{2} \in X,f(x_{2})=y_{2}\})\\ &=T(f(\mu)(y_{1}),f(\mu)(y_{2})).\end{align*} Thus \[f(\mu)(y_{1} \ast y_{2}) \geq T(f(\mu)(y_{1}),f(\mu)(y_{2})) \] and then \( f(\mu) \in FST(Y).\)

Proposition 13. If \( \nu \in FST(Y)\) and \(f: X \to Y\) be a homomorphism of \(d\)-algebras, then \( f^{-1}(\nu) \in FST(X).\)

Proof. Let \( x_{1},x_{2} \in X .\) Then \begin{align*} f^{-1}(\nu)(x_{1} \ast x_{2})&=\nu(f(x_{1} \ast x_{2}))\\ &=\nu(f(x_{1}) \ast f(x_{2}))\\ &\geq T(\nu(f(x_{1})),\nu(f(x_{2})))\\ &=T(f^{-1}(\nu)(x_{1}),f^{-1}(\nu)(x_{2})).\end{align*} Thus \[ f^{-1}(\nu)(x_{1} \ast x_{2}) \geq T(f^{-1}(\nu)(x_{1}),f^{-1}(\nu)(x_{2}))\] then \( f^{-1}(\nu) \in FST(X).\)

Proposition 14. If \( \mu \in FDIT(X)\) and \(f: X \to Y\) be a homomorphism of \(d\)-algebras, then \( f(\mu) \in FDIT(Y).\)

Proof.

1. Let \( x \in X \) and \( y \in Y \) with \( f(x)=y. \) Now \[f(\mu)(0)=\sup \{ \mu(0) \hspace{0.1cm}|\hspace{0.1cm} 0 \in X,f(0)=0 \rbrace \geq \sup \{ \mu(x) \hspace{0.1cm}|\hspace{0.1cm} x \in X,f(x)=y \rbrace =f(\mu)(y).\]
2. Let \( x,x_{1} \in X \) such that \( f(x)=y, f(x_{1})=y_{1}. \) Now \begin{align*} f(\mu)(y)&=\sup \{ \mu(x) \hspace{0.1cm}|\hspace{0.1cm} x \in X,f(x)=y \rbrace \\ &\geq \sup \{ T(\mu(x \ast x_{1}),\mu(x_{1})) \hspace{0.1cm}|\hspace{0.1cm} x,x_{1} \in X,f(x)=y,f(x_{1})=y_{1} \rbrace \\ &=T(\sup \{ \mu(x \ast x_{1}) \hspace{0.1cm}|\hspace{0.1cm} x,x_{1} \in X,f(x)=y,f(x_{1})=y_{1} \rbrace,\sup \{ \mu(x_{1}) \hspace{0.1cm}|\hspace{0.1cm} x_{1} \in X,f(x_{1})=y_{1} \rbrace)\\ &=T(\sup \{ \mu(x \ast x_{1}) \hspace{0.1cm}|\hspace{0.1cm} x,x_{1} \in X,f(x \ast x_{1})=y \ast y_{1} \rbrace,\sup \{ \mu(x_{1}) \hspace{0.1cm}|\hspace{0.1cm} x_{1} \in X,f(x_{1})=y_{1} \rbrace \\ &=T(f(\mu)(y \ast y_{1}),f(\mu)(y_{1})).\end{align*} Therefore \[f(\mu)(y) \geq T(f(\mu)(y \ast y_{1}),f(\mu)(y_{1})).\]
3. Let \( y_{1},y_{2} \in Y \) and \( x_{1},x_{2} \in X \) such that \( f(x_{1})=y_{1} \) and \( f(x_{2})=y_{2}. \) Then \begin{align*} f(\mu)(y_{1} \ast y_{2})&=\sup \{ \mu(x_{1} \ast x_{2}) \hspace{0.1cm}|\hspace{0.1cm} x_{1},x_{2} \in X,f(x_{1})=y_{1},f(x_{2})=y_{2}\}\\ &\geq \sup \{ T(\mu(x_{1}),\mu(x_{2}) \hspace{0.1cm}|\hspace{0.1cm} x_{1},x_{2} \in X,f(x_{1})=y_{1},f(x_{2})=y_{2}\}\\ &=T(\sup \{ \mu(x_{1}) \hspace{0.1cm}|\hspace{0.1cm} x_{1} \in X,f(x_{1})=y_{1}\},\sup \{ \mu(x_{2}) \hspace{0.1cm}|\hspace{0.1cm} x_{2} \in X,f(x_{2})=y_{2}\})\\ &=T(f(\mu)(y_{1}),f(\mu)(y_{2})).\end{align*}

Thus from (1)-(3), we have that \( f(\mu) \in FDIT(Y).\)

Proposition 15. If \( \nu \in FDIT(Y)\) and \(f: X \to Y\) be a homomorphism of \(d\)-algebras, then \( f^{-1}(\nu) \in FDIT(X).\)

Proof.

1. Let \( x \in X. \) Then \[ f^{-1}(\nu)(0)=\nu(f(0)) \geq \nu(f(x)= f^{-1}(\nu)(x).\]
2. Let \( x,x_{1} \in X. \) As \begin{align*}f^{-1}(\nu)(x)&=\nu(f(x))\\& \geq T(\nu(f(x) \ast f(x_{1})),\nu(f(x)))\\&=T(\nu(f(x \ast x_{1})),\nu(f(x)))\\&=T(f^{-1}(\nu)(x \ast x_{1}),f^{-1}(\nu)(x)).\end{align*} So \[f^{-1}(\nu)(x) \geq T(f^{-1}(\nu)(x \ast x_{1}),f^{-1}(\nu)(x)).\]
3. Let \( x_{1},x_{2} \in X .\) Then \begin{align*} f^{-1}(\nu)(x_{1} \ast x_{2})&=\nu(f(x_{1} \ast x_{2}))\\ &=\nu(f(x_{1}) \ast f(x_{2}))\\& \geq T(\nu(f(x_{1})),\nu(f(x_{2})))\\&=T(f^{-1}(\nu)(x_{1}),f^{-1}(\nu)(x_{2})).\end{align*}Then \[ f^{-1}(\nu)(x_{1} \ast x_{2}) \geq T(f^{-1}(\nu)(x_{1}),f^{-1}(\nu)(x_{2})).\]

Therefore, from (1)-(3,) we have \( f^{-1}(\nu) \in FDIT(X).\)

Acknowledgments

We would like to thank the referees for carefully reading the manuscript and making several helpful comments to increase the quality of the paper.

Conflicts of Interest:

''The author declares no conflict of interest.''

1. Introduction and Preliminaries

Africa is the store house of about 70% of the world's resources out of which 80% residents in rural areas. But despite this enviable position, the continent merely controls five percent of the enormous resources. Smarter outsiders are the ones reaping the benefits of African people especially the rural dwellers bedevilled by bad politics or poor governance of the belly and unbridled bitterness.

Water is the most basic natural resource. More than 97% of the earth's water is saline ocean water. Another largely unavailable reservoir of water is the 2% of the earth's water frozen in polar ice caps and glaciers. Of the remaining 1% of' the earth's water, more than half (0.6% of the total supply) is contained in groundwater [1]. Water is one of the most common substances known, being a good solvent for many materials but rarely exists in its pure form in nature. Natural sources of water include rain, spring, well, river, lake and sea. Rain water is the purest form of natural water because it is formed as a result of the condensation of water vapor in the atmosphere and can be classified as a natural form of distilled water. The water from the other sources especially river, lake and sea, contains a lot of impurities which include dissolved gases, mineral salts, bacteria and organic remains [2].

The United Nations as part of its Millennium Development Goals (MDGs) stipulates the proportion of the population without sustainable access to safe drinking water be reduced by half by 2015. As a result, the developed nations had 91% to 100% increase in drinking water coverage and the lowest level of coverage was in sub-Saharan Africa, South Asia and East Asia from less than 50% to 75% coverage [3].

In Nigeria, there are signs of major water crises that are getting worse and will continue to deteriorate if not checked by corrective action. The continuously intensifying scarcity of water resources is a crucial problem [4]. The primary reason for the water scarcity being experienced in Nigeria could be traced to growing disparity between the decreasing effective supply and increasing demand for water. The causes of this obvious imbalance between urban demand and water supply manifests in the continuous inability of supply quantities to meet demand. This could be traced to the population growth, higher living standards, increased irrigation, urbanization coupled with the impacts of climate change [5]. Nigeria still lacks the human and financial resources to provide adequate water and sanitation infrastructure and services for the total populace. Surveys carried out in major parts of Nigeria, showed obvious gaps and challenges in meeting the Millennium Development Goals (MDG) for water supplies and services [6]. Rapid population growth has not been accompanied by an increase in the delivery of essential urban services such as water supply, sewerage and sanitation, and collection and disposal of solid wastes. It is estimated that currently only about 50% of the urban and 20% of the semi- urban population have access to reliable water supply of acceptable quality (i.e. something better than a traditional source). Overall effective urban water supply coverage may be as low as 30% of the total population due to poor maintenance and unreliability of supplies. Rural coverage is estimated at 35% [7].

It is currently estimated that about 65% of the rural population do not have access to safe and reliable water supply and adequate sanitation facilities. Access to inadequate water supply and means of safe disposal of excreta, could lead to a significant effect on health from bacteria diseases and virus infections, loss of educational participation and attainment, affect economic productivity, and dignity in rural areas [8].

2. The study area

The study location is presented in Figure 1. There is a maze of numerous rivers, and lakes in and around Olorunda and Egbedore local government areas with very prominent rivers like Apala and Osun River. The study area enjoys luxuriant vegetation with high forest zone. The evident all year round green tree vegetation is an indication of availability of adequate soil moisture that provide underground water to the trees and other vegetation types in the dry season. On the other hand, the observed vegetation pattern signifies enhanced groundwater infiltration potential across the study area with the exception of rock outcrops and heavily populated urban areas.

3. Methodology

This study was done through surveys which included one on one interviews, notation of hydrologic features, demographic survey and data collection on occurrence of water borne diseases from nearby hospitals. Water demand, water usage and sanitation investigations were done through the use of questionnaires and personal interviews granted by inhabitants of the village as well as site visitation. The data on water related diseases were collected at the only closest primary health care centre to the study area. The data on rainfall was collected from metrological station, Oshogbo. Demographic and population data were obtained from the National Population Commission office. GIS approach was used to depict the groundwater recharge potential of the study area.

4. Results and discussion

4.1. Water resources

The interview conducted with individuals during preliminary survey confirmed the level of water scarcity. There were few wells, some with pumps to serve individual houses and others were drawn with rubber pails and buckets. No stream or river flows through the community but there is a pond where they fetch water (Figure 2). There are also hand pump boreholes. This results in waste of productive hours. This made the community to settle for the use of naturally impounded untreated surface water with low quality which is about 600m distance from the centre of the village (Figure 2). This water source is however not reliable because it does not meet the demand of the community during dry season.

4.2. Analysis of water supply promoters

Table 1 gives the water supply situation in the community. Columns 1 give the 2006 Population figure as provided by the National Population Commission NPC. The third column is the water demand of 40 lpcd. The sum of these gives the values shown in column 3 of Table 1. Column 4 and 5 are output from field survey.

Table 1. Water supply capacity in Oyo state.

1	2	3							4		5
Pop.	Estimated water	Source of Water Supply in Number										Capacities
	demand l/d	Motorized	Hand Pump Borehole		River		RWH					1/d
		Borehole							Others		Total Installed	Currents Outputs
		OP NOP	OP	NOP	OP	NOP	OP	NOP	OP	NOP
4508	180320	01	1	4	1				4	1	182500	28000

Surface source of water under column 4 is in terms of location only. The installed capacities in column 5 were obtained from the number of existing facilities. Motorized boreholes were assumed to produce 100,000 l/d, hand pump 13,000 l/d and the installed capacities, current production and distribution patterns of the surface schemes were obtained from the promoters and verified by site visits. Column 5 is thus a product of all installed facilities as against the current output. The Table shows that if all the facilities are working at full installed capacity, there will be no shortfall. At the current production of 28,000 l/d, the shortfall is up to 152,320 l/d. In simple terms 15.53% of the water required is available even with the installed facilities.

Table 2 presents the summary of water supply promoters and the type of facility they promote. A summary of the current status of these facilities is also indicated in the table. It shows that 3 of the boreholes are currently not operational; this is due in part to the attitudes of the community and non integrative approach of providing infrastructure for the populace. Other facilities as indicated in the table refer to protected hand dug wells. Table 3 gives a comparative involvement of the promoters in the community. There is a difference between the number of facilities operated by a promoter and the quantity of water produced and subsequently the impact on the citizenry. The percentage of facilities promoted by the various agencies is presented both in terms of the total number of facilities and the total quantity of water provided. This is to ensure that a wrong impression is not given concerning the direct effect of the agencies on the study area. The Table shows that while the Local Government is responsible for about 41.67% of the facilities in the Study area, it is responsible for 46% of the available safe water supply. Other promoters are the politicians, Donors (8.3%) and community 8.33% and others 41.67%. Figures 3 and Figure 4 are from result of Table 2 showing water supply by different levels of promoters.

Table 2. Summary of water supply promoters in Oyo state.

Promoter	Motorized BH		Hand Pump BH		Surface Schemes		Others		Total
	OP*	NOP**	OP	NOP	OP	NOP	OP	NOP
COMMUNITY	0	0	0	0	1		4	1	6
DONOR	0	0	0	1					1
FGN	0	0	0	0					0
JOINT	0	0	0	0					0
LGA	0	0	1	4					5
STATE	0	0	0	0		0			0
Total	0	1	1	4	1	0	4	1	12

\(^{*}\)OP Operational, \(^{**}\)NOP, Not Operational, MBH \(=\) Motorised Borehole, HBH \(=\) Hand pump Borehole, Others (Protected hand-dug wells mostly)

Table 3. Comparative performance of water supply promoters in Oyo state.

S/No	Item	% water Supply Facilities Promoted
		% By Number	% By Quantity
1	Facilities by Federal Government	0	0
2	Facilities by State Government	0	0
3	Facilities by Local Governments	41.67	83.3
4	Facilities by Communities	8.33	2.7
5	Facilities by Donors	8.33	7.2
6	Facilities by Others	41.67	6.8
	Total	100	100

Table 4 gives the water supply services demand and supply in the community. The first two columns are from Table 1. Column 3 gives the theoretical availability of water against demand which is defined as the total available safe water divided by total water demand. This was simply calculated by dividing the current product in Table 1 with the demand in column 3. Column 4 is the area of coverage achieved by the available water. This is defined simply as the ratio of the sum of people who get their safe water from this source. Supply coverage is about 30%. Column 5 is a direct output of the household survey. This is the actual number of respondents with access to safe water. Access to safe water is generally is poor and is about 18%.

Table 4. Water supply services demand and supply for Oyo state.

1	2	3	4	5
Population	Water  Demand 1/d	Theoretical % of Water Supply vs Demand	% of Water  Supply Coverage	% Access to safe Water
4508	180320	43	20	10

Table 5 gives a summary of the water supply situation in the study area. This table was prepared like Tables 4.4 and 4.5. The installed capacity is 182,500.00 l/d while the current production is 28,000.00 l/d. It shows that capacity utilization is about 15.3%. It also shows that if the facilities are working at full capacity, the water supply will be surplus over demand. Water supply coverage is poor and this is responsible for the reason why the rural people result to taking unsafe water Figure 2.

Table 5. Water supply capacity and supply services in Dagbolu.

Total LGA  Population	Estimated Urban  Water demand l/d	Available l/d		Theoretical % of  Water Supply vs  Demand		% of Water Supply Coverage	% Access  to safe  Water
		Total  installed	Current  Outpot
4508	180320	182500	28000	15.53		20	10

Distance and Time to access water

Now we present the average distance and average time taken to get water.

Average time travelled to get water and return \(=\frac{Time\ \ travelled\ \ (minutes)}{No.\ \ of\ \ people}\)

\(=\frac{30+60+40+30+30+60+30+30+180+30+60+120+60+120+50+180+60+40+90+120+90+60+120+90+60+180+240+180++120+60+120+180+120}{30}\) \(=\frac{2770}{30}=\) 92 minutes \(=\) 1 hour 30 minutes.

Average distance to trek to fetch water \(=\frac{Distance \ \ travelled(m)}{No.\ \ of \ \ people }\)

\(=\frac{100+100+100+280+500+82+300+80+250+200+180+80+240+240+500+150+100+120+60+60+180+5+250+230+100+120+5+10+250}{30}\)

\(=\frac{4870}{30}=160m\).

It takes an average of one hour and thirty minutes for the community people to get water and return to their houses and an average of 162m to trek to fetch water. This shows that many of the people in the community go through stress in order to have water for use.

4.3. Water Haulage

Figure 5 shows haulers with adult male, adult women, male children and female children being 6%, 27%, 18% and 49% respectively. The adult women and female children mostly go to the source to fetch water for the household. This accounts for the poor performance of rural ladies in Senior Secondary School certificate Examination (Figure 6 and 7).

4.4. Ground water penitential

The result of the groundwater potential zones for the study area is shown in Figure 8. The groundwater potential map is generated through the reclassified, weighted and overlaid of some factors (maps) such as; lineament density, hydraulic conductivity, land use, rainfall intensity, soil texture (sand, clay and silt) and permeability using geographic information system techniques. The groundwater recharge potential zone is categorized into three classes from fair to very high. From the final map, the locations with very good covers 18% o, good covers 80% and fair covers 2% of the study area. The areas with very good and good groundwater recharge potential zones fall on the locations such as Ofatedo, Gbonmi, Oke oro, Awosuru, Gbodofon, and Dagbolu in the study area.

4.5. Sanitation

Survey shows that approximately 2.1% of respondent households have access to improved sanitation facilities and inversely, 97.9% uses unimproved sanitation facilities in the area. The proportions of households using the various forms of improved sanitation facilities in the State are given in Table 6. The table indicates that simple pit latrines (covered) are the most prominent sanitation facilities in most of the area.

Table 6. Proportion of Households (%) Using the Various Improved Sanitation Facilities in Dagbolu community.

Sampled population	Latrine	Water closet	Hand wash	Septic system	Public server	Sullage disposal	Storm water	Total facilities
350	1.2	0.9	0.0	0.0	0.0	0.0	0.0	2.1

4.6. Sanitation

Findings from the study indications are that out of those who have access to improved toilet facilities in the area, 57.0% utilize simple pit latrines (covered), water closet (43%). Most of the respondents have access to simple pit latrines because it is the most affordable facility to them; some of the pit latrines are also available in some public places like schools and religious houses. The simple pit latrines in the areas, where available within the households, are shared by members of the household.

4.7. Promoters of sanitation facilities in Dagbolu community

There are no promoters from the State, Local governments, as well as Communities nor donor agencies. The Abandoned Latrine in a church is shown in Figure 9.

4.8. Diseases

The result of the household survey shows that the most common water-related diseases in area are malaria, diarrhoea, typhoid fever, dysentery, and cholera. Their frequencies are 81.2 %, 8.41 %, 3.40 %, 3.22 %, and 2.76 % respectively (Table 7 and Figure 10). The least common water-related diseases in the area are streptococci, trachoma, Guinea worm, schistosomiasis, and Ring worm. Their frequencies are (0.02%), (0.04%), 0.11%), (0.32%), and (0.39%) respectively. Malaria in particular has been associated with the tropics where weather conditions are favorable to mosquitoes.

Table 7. Statistics of water-related diseases in Dagbolu.

Cholera	Diarrhea	Dysentery	G. Worm	Hepatitis B	Malaria  Fever	Oncho cerciasis
134	409	157	5	29	3946	66
Others	R. Worm	Scabies	Schisto somiasis	Strepto cocci	Trachoma	Typhoid  Fever
26	19	21	15	1	2	164

Each value was computed as the proportion of households where any member suffered the water-related disease in 2019. *Note that a respondent may pick more than one disease.

Table 8 reveals that access to adequate supply of water would help in reducing the expenditure of time and energy spent in fetching water, increase water consumption, reduce time spent in queuing or seeking alternative sources and water-borne diseases. The economic loss due to water related diseases in a small popupaltiion of 4,508 is \(\aleph\) 29,964,000 per annum, but productive fore has been said to be 30 %, then the loss in the community is \(\aleph\) 8,989,200. If we have to extrapolate to the current Nigeria population of 208,000,000:00, the yearly loss for the unproductive downtime due to water related disease is N 414,763,442,768:00 excluding cost of treatment. This shows that insufficient water supply in rural areas amount to double loss. The patients are not working and are spending.

Table 8. Summary of the prevalence of water related diseases.

Population	Total  incidence  of diseases	Total Productive  days for the population	Down  unproductive days1*	Population  Expected  Income2*	Down  unproductive  days ost3*
4508	4994	64152	19976	96,228,000	29,964,000

5. Conclusion and recommendation

5.1. Conclusion

Water unavailability among other challenges such as poverty, energy shortage, food insecurity (hunger and malnutrition) and climate change facing developing countries; seems to be one of the most important, which cut across all other challenges (UN DESA, 2016). Water shortage also lead to issue of social menace as dropped out secondary girls will have nothing to do but give in to prostitution when they cannot compete with their male counterparts in labour market. Rural dwellers are not lacking in resources but they are bereft of ideas to harness their resources. Improve water management through adequate policies will bring about development to the underdeveloped countries.

5.2. Recommendation

The government should ensure sustainable management of water resources and improving water supply and sanitation to the poor majority. This can be by aggressive public enlightenment; and effective legislative policies that are well enforced. They should improve management of water and water utilities to generate secure revenue streams to increase financial resources. They should support better integration of water supply and sanitation issues into integrated water resources management. These are interrelated. Reduced supply will increase sanitation problems. This will increase water borne diseases, injure health and subsequently increase the overall cost of water supply. There is the need for the Federal, State and local Governments to further increase budgetary allocation for hygiene and sanitation in the Nation. Apart from increase budgetary allocations, the fund should also be released as at when due There is also the need to introduce simple, adaptable low cost toilet facilities to rural areas and small towns that constitute about 60 % of our population and the government should encourage strong private sector participation in the sanitation programmes of the state so as to exploit the extensive potential for financial and technical resources from this sector. It is time for us to use adaptive technology to enhance our rural developments and remove poverty in our DNA. We should stop being busy but productive

Author Contributions:

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest:

''The authors declare no conflict of interest.''

1. Introduction

The textile industry is indeed one of the world's most significant sectors. This sector employs people without any extraordinary skill, which plays a crucial role in employment in developing countries, namely, Bangladesh, Vietnam, Pakistan, Sri Lanka, and India. Hence, it plays a significant role in developing the value of the Gross Domestic Product (GDP) of these countries [1]. Moreover, due to the rising demand from the population, textile goods have increased, and textile mills and their wastewater have risen proportionately, leading the planet to have a big pollution problem. Since the textile industry represents two-thirds of the total demand for dyes, approximately 10-15% of the used dyes are discharged into the wastewater during the dyeing phase [2]. Therefore, T the textile industries are one of the world's leading causes of various pollution. Besides, more than 1,000,000 synthetic dyes are generated worldwide with an annual production of approximately \(7 \times 105\) metric tons [3,4], and these dyes are widely used in the textile, paper, pharmaceutical, food, and cosmetics industries [5]. However, the textile industries are the largest consumers of dyes [6]. The World Bank estimates that approximately 20% of global industrial water pollution comes from the treatment of wastewater and dyeing of textiles. Thus, the textile industries are second to agriculture practices as the biggest polluting agents for freshwater bodies.

The effluents contain heavy metals, trace metals, coloring agents, and some toxic elements. The effluent discharged into the rivers goes far away and is used by people for their day-to-day activities, and irrigation [7]. Consequently, the physicochemical parameters of water such as pH, BOD, COD, TDS, DO get degraded owing to the polluted water. It will contaminate the food chain and ecosystem. These make water very toxic and harmful for humans, crops, and aquatic livings. This causes severe diseases such as cancer, damage of infants' brains, body shrinkage on human beings, reduces soil fertility, and damages crops [8].

These days, textiles are dyed with aromatic and heterocyclic dyes. Dyes are more challenging to degrade in textile effluent because various chemical and physical materials are found in textile waste. The unfixed dyes in textile effluents were considered to be in massive amounts. These unfixed dyes with used water are mostly dumped into various nearby water sources, which are known as textile wastewater, or effluent [9]. As industrial effluents or wastewater are dumped directly into sewage systems without treatment from most factories, the sewage systems are directed into canals, which discharge their contents into rivers and lagoons. The result of this is the contamination of surface water, which has a consequent impact on human wellbeing. Industrial effluents have been reported to contaminate water, soil, and air, resulting in high disease burdens and, ultimately, a shorter life span in developed countries [10]. The environmental hazard due to wastewater is alarming for developing countries like Bangladesh. Many international investors are now aware of environmental pollution and looking for whether textile factories have ETP or not for the safe emission of effluents. Hence, the industries should be aware of the effect of polluted effluent in the environment and the threat it has on human life. Before dumping dyeing wastewater into our surroundings, these dyes must be eliminated from industrial effluents. The study was done to offer a complete overview of dye categorization, environmental impacts, their detrimental consequences, and the different techniques for removing dyes from textile dyeing effluent, followed by the findings of numerous investigations.

2. Water consumption and discharge of the textile effluent into Aquatic ecosystem

Textile industries significantly contribute to freshwater pollution, especially cotton and cotton-based fabric, the main culprits. A 2017 study estimated that 79 billion cubic meters of water were used by the apparel industry alone in 2015; this amount would be sufficient to replenish 32 million Olympic-sized swimming pools. By 2030, the number is projected to grow by 50%. As the earth's water resources are running low, this is a staggering volume [11].

Furthermore, the entire demand for clothes is expected to climb by 63% from 62 million tons now to 102 million tons in 2030 if the world's population expands to 8.5 billion people as forecasted by 2030 [12]. In other words, the apparel market is growing at an unprecedented pace. It takes roughly three thousand liters of water to manufacture only one cotton shirt. About 93 billion cubic meters of water are used annually for textiles production (including cotton cultivation), accounting for 4% of the total global freshwater extraction [13].

Nearly 4,560 textile factories may be found in Bangladesh. There are currently 500-700 wet processing plants in operation, and the quantity is increasing daily [14]. In Bangladesh, it was anticipated that around 1,700 wet processing units were devoted to textile washing, dyeing, and finishing, and textile factories consume an estimated 250-300 liters of water for each kilogram of cloth produced. Which is the equivalent of two people's daily water use [15]. Some simple activities, such as sizing, use less water, while others, such as dyeing many washes and rinsing, need a large number of consecutive operations. The amount of water used depends on the type of material being treated and the final finish. The dyeing process and numerous activities and energy production, namely, steam generation, cooling water, sanitation, all, use water.

However, annually, the textile industry dumps massive amounts of dyeing effluent into our waterways. It has been calculated that a single factory may utilize two hundred tons of freshwater for each ton of colored cloth. This dye-based wastewater is dumped into Bangladesh's neighboring rivers, mostly in untreated conditions, gradually expanding into the sea. It leads to the use of highly poisonous chrome, which indicates a drastic increase in these countries' diseases [13].

2.1. Textile Dyes

A dye is a coloring substance used to add color to various substances or modify something's color. Dyes contain chromophores and auxochromes that are accountable for their substantiveness and color. Textile dyes are mostly synthetic chemical compounds with an aromatic structure, and these synthetic dyes are always responsible for decreasing light penetration and disrupting photosynthesis in the aquatic ecosystem. In other words, we can say, 'Dyes are detrimental for aquatic species.' Dyes that are employed by the textile industry presently, for the most part, are synthetic. The majority of them are extracted from two sources: coal tar and fossil fuel intermediates. Powders, granules, pastes, or solvent dispersions are sold as most of these dyes. These latest dyes are frequently formulated to satisfy newer technology's needs, new fabric styles, detergents, advancements in dyeing equipment and address the significant environmental problems raised by certain current dyes. The fact that nearly all goods are subject to seasonal demand and variations is also another significant consideration. In order to satisfy these all modern and unique technological specifications, industrial textile dyes must expand. Since the textile industry's product quality is being frequently changing, the trend of using these dyes is also changing rapidly, ranging from durable, flexible synthetic fibers to high-cost cellulosic fibers [16].

2.2. Types of dyes and their brief properties

Dependent on numerous parameters, dyes can be categorized. We can, nonetheless, have a look at the four most noticeable ones in Figure 1.

2.2.1. Direct Dyes

As they have a good affinity, we may add these dyes directly to the fabric, called direct dyes. This dye is predominantly a sodium salt of sulphonic acid or carboxylic acid, with azo as their leading chromophoric group. They are soluble in water and are anionic. It is common to color cellulosic fibers; protein fibers are also used. The fastness properties are average; the wet fastness is predominantly low, which means this kind of direct dye produces a wide range of wastewater during the dyeing process. Thus, the after-treatment process is required for enhanced consistency. It is generally used for dyeing viscose, and cotton-related goods [17,18].

2.2.2. Reactive Dyes

By forming a covalent bond, halogen-containing reactive groups are present in reactive dyes and become part of the fiber structure. This dye is often used to color cotton fabrics. It is soluble in water and anionic in nature and shows strong wash fastness, which means less responsible for producing wastewater in volume than other dyes. In addition, this dye has a covalent bond, helping it lock into the fabric more firmly. Generally, it is used for dyeing cotton goods; it is also possible to dye protein, and polyamide-based products [17,18].

2.2.3. Basic Dyes

They are primarily organic-based salts. For color production, this dye structure contains cationic charges responsible for that. That is why they are called cationic dyes occasionally. This dye is easily soluble in methylated spirit and alcohol, but not water. It is mainly used for acrylic, and jute-related goods [17,18].

2.2.4. Vat Dyes

In their composition, these dyes typically consist of a keto group and are rendered water-soluble by vatting. Vat dyes are very similar to sulfur dyes in their application procedure. They are used primarily to dye denim or jeans. However, they are natural substances for coloring and are not soluble in water. For this reason, to turn them into a water-soluble form, the vatting process is required, and alkaline conditions are necessary for vatting. It is generally used for dyeing cotton-based goods [17,18].

2.2.5. Disperse Dyes

Dyeing thermoplastic hydrophobic fabrics with these dyes are done, as well as these dyes are not easily water-soluble. The dispersing agent is essential for water dispersion to be produced. They have little attraction for any fiber. Additionally, these synthetic fabric dyes are primarily replaced by azo, anthraquinone, or diphenylamine compounds. It is mainly used for dyeing acetate, triacetate, nylon, acrylic, polyester-related goods [17,18].

2.2.6. Acid Dyes

Acid dyes are mainly carboxylic, or sulphuric acid salts, highly soluble in water and anionic, and they form ionic bonds primarily, but van-der-Waals and H-bonds are formed as well. In addition, acid dyes are applied in an acidic bath; they can dye polyamide and protein fibers, as this dye is substantially effective on these fibers. It is primarily used for dyeing protein fibers and polyamide-based products. Dyeing thermoplastic hydrophobic fabrics with these dyes are done, and these dyes are water-soluble.

2.2.7. Azoic Dyes

These were mainly mono or bi-azo water-insoluble coloring substances that need coupling components to produce colors. Unlike other dyes, the most remarkable aspect was that they were not ready-made colors. Furthermore, their structure possesses an insoluble azo group and is not water-soluble; their preparation needs two baths, such as a developing bath and an impregnation bath; diazonium and coupling compounds affect their color. As a result, there is diversity in their application to textile industries. Cotton, nylon, and polyester-related materials are commonly used with these dyes [17].

2.2.8. Mordant Dyes

These dyes, also known as chrome dyes because these dyes primarily consist of inorganic chromium, possess no affinity towards textile materials. Instead, chemical binding agents, called mordants, assist them in sticking to the fiber. It is mainly used for dyeing natural protein fibers, nylon, and modacrylic fibers [17].

2.2.9. Sulfur Dyes

The sulfur dye used to produce black and brown cotton fabrics is analogous to vat dyes. They contain a disulfide linkage in their structure. However, they are not soluble in water, so they need reducing agents to make them soluble. Furthermore, it must be used in an alkaline medium, and oxidation is essential for the production of color. Generally, it finds particular applications in the dyeing of silk, paper, and certain types of leather and is used in limited quantities in the dyeing of cellulosic materials [17].

2.2.10. Anthraquinone Dyes

There are two significant classes of textile dyes: anthraquinone dyes and azo dyes. In terms of color range, these dyes cover nearly all visible spectrum. Moreover, as anthraquinone-based dyes possess glued aromatic structures, they resist degradation for a long time [17].

3. Textile operations

Pretreatment, dyeing/printing, finishing, and other technologies are included in textile printing and dyeing. Various methods are employed in the pretreatment stage, such as de-sizing, scouring, washing, and more. The primary goal of dyeing is to dissolve the dye in water, which will then be imparted to the cloth to make a colored cloth, given specific conditions. Printing is a type of dyeing known as "localized dyeing," which refers to dyeing that has been restricted to a specific area of the fabric. For the most part, the reactions involved in this dyeing are identical to those involved in dyeing. As opposed to printing, which uses a thick paste of colorant, dyeing employs liquid colorants. Finishing procedures are applied to both natural and synthetic textiles. As a result, various finishing agents are employed to soften, cross-link, and waterproof the completed cloth. Every step of the finishing process causes water pollution. Before dying or printing should have been done the following processes: singeing, mercerizing, base reduction, and others. Bleaching is a critical step in the textile dyeing process, and Bleaching using sodium hypochlorite or sodium chlorite is the most popular method. Also, chlorine dioxide, a strong oxidant, is corrosive and poisonous. The usual quantity of bleaching effluent is between 10 and 200 mg/L. Figure 2 depicts a typical printing and dyeing procedure.

Table 1. Unfixed dye percentage for distinct forms of dye and applications [16].

Polyester	Disperse	8-20
Wool, Viscose, and Cotton	Sulfur dyes	30-40
	Reactive dyes	20-50
	Azoic dyes	5-10
	Vat dyes	5-20
	Direct dyes	5-20
	Pigment	1

After the dyeing process, water used in the whole process, extracted from textile products, is named wastewater or effluents. We will find different chemical or physical substances and a large proportion of unfixed dyes from those extracted water. These dyes are the main culprits for wastewater formation because it is not relatively easy to extenuate unfixed synthetic dyes from wastewater than other chemicals. Therefore, we need to know the percentage of unfixed dyes. It varies depending on the dyes and raw goods where these dyes are generally applied. From Table 1, we will know which types of dyes are more responsible for textile wastewater formation, depending on their dye wash-out percentage after dyeing.

4. Characterization of textile wastewater

Textile mills could be divided into two categories based on waste and wastewater generation: wet processing and dry processing. The wastewater generated by the textile industry includes cleaning wastewater, process wastewater, noncontact cooling wastewater, and steam water. In dry processing units, generally solid waste is generated, and in wet processing units, most textile wastewater is generated. Raw textile dyeing effluent can be characterized by BOD, COD, TDS, color, total suspended solids (TSS), heavy metals, etc. The textile wastewater exhibits a wide range of pH (2-14), COD (50 ppm - 18000 ppm), TDS (50 ppm - 6000 ppm), and very strong color [20]. Textile wastewater is challenging to treat because of its variations in some of the factors, namely, pH, color, temperature, TDS (Total dissolved solids), Total suspended solids (TSS), and dissolved solids (DO) (see Table 2) [21].

Table 2. Characterization of textile wastewater [21] .

Category	BOD (ppm)	COD (ppm)	pH	Temp.
High	500	1500	10	28
Average	270	970	9	28
Low	100	460	10	31

4.1. Main characteristics of textile dye effluents

4.1.1. Color

Color present in the dyes is responsible for inhibiting the self-purification potential of dye effluent by reducing the photochemical synthesis of oxygen and disturbing the ecosystem. Furthermore, when dispersed in water, wastewater containing dye causes very harmful effects on aquatic life as the color present in dye effluent absorbs sunlight required mainly for aquatic plants. Which in return impacts less dissolved oxygen in wastewater, which is ultimately harmful to the aquatic animals. Hence, color is one of the critical characteristics of textile wastewater that must be treated before discharge from industries.

4.1.2. Dissolved solids

Dissolved solids are another essential characteristic of dye effluent. High TDS dye effluents have a lot of disadvantages, such as high TDS leading to disturbing the surface water quality of water. Therefore, high TDS water is not suitable for use as raw water for industry and is not suitable for irrigation.

4.1.3. Chlorine

The chlorine present in effluent wastewater is harmful to the water bodies. Chlorine is a toxic gas that irritates the skin, eyes, and respiratory system. The presence of chlorine in effluent reduces the dissolved oxygen content in water. Chlorine reacts with another compound to form complex chlorine salt.

4.1.4. Organic materials

Organic pollutants include pesticides, fertilizers, hydrocarbons, phenols, detergents, oils, greases, pharmaceuticals, proteins, carbohydrates, etc. Toxic organic pollutants cause several environmental problems. The presence of organic pollutants is detected by analysis of BOD and COD. Most valuable oxygen is consumed to decompose the organic pollutants in textile dye wastewater.

4.1.5. Toxic metals

Most of the metals are biodegradable, highly toxic, and carcinogenic. Thus removal of metals is a serious problem. Metals create adverse effects on treatment processes. Metals react with other compounds in dye in effluent and form complex metal salt, which is difficult to remove. Most of the toxic metals are chromium (Cr), cadmium (Cd), nickel (Ni), zinc (Zn), copper (Cu), lead (Pb), ferrous (Fe), etc. [4]

5. Effect of wastewater on algae, fish, river water and human

The wastewater disposal exacerbates major pollution issues by some factories under unregulated and unsuitable circumstances. Undoubtedly, the urgency of prevention and treatment of pollution is crucial for human life. If a textile mill expels wastewater into the surrounding area without any treatment, it would significantly affect existing water bodies and the adjacent area's soil. In addition, strong COD and BOD5 values, the accumulation of particulate matter and sediments, and oils in the effluent allow dissolved oxygen to become reduced, which has a detrimental impact on the marine ecological environment [22].

5.1. Impact textile dyes on aquatic environment

Textile dyes create a variety of environmental pollution and health risks. Due to their high thermal and photo-stability, dyes can last for a long time in the environment. The absorption and reflection of sunlight entering the water is the primary environmental problem with dyes. Besides, since cotton use has gradually grown over the last century [23], cotton fibers are mostly dyed using azo dyes, one of the major classes of synthetic dyes used in the industry [24]. Azo dye is impossible to degrade due to the ongoing conventional treatment methods. They are distinguished by the presence in the middle of the nitrogen-nitrogen bond (-N=N-) and are thus highly electron-deficient [25]. Such azo dyes are complicated and have been found to exhibit carcinogenic signs of reductive cleavage. These dyes can change the soil's physical and chemical properties, degrade the water's body, and cause disruption to the flora and fauna of the environment. It has been found that the poisonous nature of dyes leads soil microorganisms to die, which in turn influences agricultural productivity [26]. The presence of a minimal quantity of azo dye in water (< 1ppm) is prominent [27]. This affects aesthetic merit, transparency, and water-gas solubility. Reducing light penetration by water lowers photosynthetic activity, induces oxygen depletion, and deregulation of aquatic biota's biological cycles [28]. Most azo dyes are often too toxic to the environment and mutagens, which means that they can have severe chronic effects on animals, depending on the exposure period and concentration of azo dye [29].

The absorption and reflection of sunlight by dyes is the main reason for light absorption decrease and the photosynthetic activity of the algae that affects the food chain. Large quantities of textile dyes in water bodies hinder the re-oxygenation of the receiving water and the sunlight, thus affecting ecological development in aquatic life and the process of photosynthesis of aquatic plants [30]. The polluting effects of dyes on the aquatic ecosystem can be toxic due to their long-term existence in the environment accumulation in sediments, particularly in fish or other aquatic species, decomposition of contaminants in carcinogenic or mutagenic compounds. Several dyes and their products are carcinogenic, mutagenic, and life-threatening. The existence of relatively tiny amounts of dyes of water is highly noticeable. It has a significant effect on the consistency and transparency of water sources, such as waterways, rivers, and others, resulting in harm to the aquatic environment [16].

5.2. Impact on human

1,4-diamine benzene is an aromatic amine whose source azo dyes can induce skin irritation, contact dermatitis, chemosis, lacrimation, exopthamlmosis, lifelong blindness, rhabdomyolysis, severe tubular necrosis, vomiting, gastritis, hypertension, vertigo. Upon ingestion, oedema of the face, throat, pharynx, tongue, and larynx along with respiratory distress [31]. Aromatic amines can be stimulated by water, which facilitates their absorption through the skin and some other exposed areas, such as the mouth. Absorption by ingestion is quicker and possibly riskier since more dye can be ingested during a shorter time frame. Water-soluble azo dyes become risky after being metabolized by liver enzymes. All those things happen when we, humans, get exposed to wastewater [29]. Due to the unavailability of fresh canal water and subsoil water, farmers have to irrigate their fields with polluted wastewater that flows through their villages. The impact on wastewater-irrigated crops can be seen through symptoms such as plaque in the villagers' teeth, knee pain, and grey hair [32].

5.3. Impact on fish

Gambusia affinis, a comparative toxicological analysis of textile effluent in freshwater fish, has shown significant mortality reductions and cytotoxic effects on red blood cells (RBCs). As well as a decrease in their numbers and a percentage variance in their form (poikilocytosis) and size [33]. Another research on Mastacembelus armatus, a protensive edible freshwater fish subjected to textile effluent, has contributed to alterations in the ionic regulation of tissues such as the liver, kidneys, and muscles. The sodium and chloride ions concentration and increasing the potassium, calcium, and magnesium ions concentration [34]. The textile industry's effluent on teleost fish Poecilia recticula triggers abnormal behavior, including excessive swimming, hyper-excitation, rapid opercular activity, and thick mucus. Histopathological alters include the enlargement of the main gill bar and the detaching of the secondary gill bar. The disintegration of intestinal villi and penetration of haemocytes into the lumen has also been observed. The research was also performed on the effect of Textile-Dyeing Industry Effluent on certain Freshwater Fish Hematological Parameters Oreochromis mossambicus [35]. Hyperemia, necrosis, and degeneration are the significant histological changes found in the liver [36]. Effects of the textile dye industry effluent have also been seen on the nutritional value of the freshwater crab Spiralothelphusa hydrodrome, a crucial personal food supply in southern India, contributing to a loss of nutritional content, namely, lipids, carbohydrate, and protein [37]. Similarly, as for Catla, dye effluent strongly affects feed absorption and food conversion rates.

5.4. Impact on algae

The presence of increased usage of dyes in the water bodies influences several algae parameters, such as growing protein content, pigment content, and other nutrient content. Different dyes have various possible effects on algae. When it comes to measuring pollution in aquatic environments, algae are more than half percent susceptible to pollutants than organisms commonly utilized in toxicological testing [16]. When dye concentration in water is increased, it inhibits the growth of Spirulina platensis and reduces its nutritional content. Ramazol Red Brilliant dye even affects the chain in the water bodies. It thus creates an environmental imbalance [38] The use of indigo dye is also capable of significantly lowering growth and biomass production, as well as changing the morphological structure of freshwater microalgae S. Quadricauda [39].

6. An overview of the environment pollution condition of the textile industries in Banladesh perspective

Textile wet processing can be categorized as Knit dyeing units, Woven dyeing units, Denim units, Printing units [40].

6.1. Knit dyeing units

Knit dyeing factory has been one of Bangladesh and South Asia's major environmentally harmful textile industries. These are most active in the manufacture of export-focused knit garments. Various forms of textile items need different treatment methods. The manufacturing technology and their treatment and testing of various fabrics are very distinct. This is because knit fabrics are softer and need to be treated gently, while other fabrics are comparatively rigid and can receive more challenging treatment options. For this cause, knitted fabrics are dyeing in a winch-type dyeing machine in which treatment occurs at a high M:L ratio of 1:150-200 means adding about 150 to 200 liters of water to dye one kg of knitted products. It was observed that a knitting factory with a production capacity of 10 tonnes consumes about 100 to 150 Million of water per hour, taking into account all the factors. All the water mentioned above, however, is not equally dangerous. Some of these are very seriously contaminated, whereas others are slightly polluted. On average, 50 percent of the water is polluted and needs to be cleaned, and the remainder of the water should be directly drained or only slightly treated. Therefore, a general guideline for knit dyeing would be that a wastewater treatment plant of 40-60 M/hrs' treatment capability can be needed for a 10-ton dyeing capacity factory [40].

6.2. Printing unit

Printing-related contaminants include dissolved solids, chemicals, foam, color, and metals, and during the washing-off phases, large amounts of water are absorbed in general. Dyes containing metals, objectionable surfactants, air pollutants, water from the cleaning of the printing blanket, residual print paste, waste paste from containers, screens, and tanks. Urea use is the primary environmental contamination of textile printing as this raises the nitrogen in the effluent. Like denim, the textile printing industry's volume of effluent is minimal but heavily contaminated [40].

6.3. Woven dyeing units

Woven dyed fabric gets dyed differently from knitted fabrics in a completely different way. In contrast to the knit dyeing method, the amount of wastewater produced from a woven dyeing factory is quite limited. The attributes of woven dyeing plants, in addition to this, differ from those of knit fabric dyeing plants. To maximize the strength of warp yarns, sizing is conducted before weaving. Starch is a key part of the sizing process. The wet process begins with desizing to extract the starch and other sizing substances from the cloth; otherwise, it would not be perfect for subsequent steps and dyeing. Therefore, a desizing unit heavily contaminates the emission. Starch and other sizing products, unfixed dyes, inadequate washing of dyes, machine cleaning wastes before start-up, shut-down and color and style changes, salts, alkalis, etc., are contaminants of woven dyeing units. Typically, woven dyeing is conducted at such a low M: L ratio that may be as little as 1:5 (for consistent dyeing), so the volume of wastewater is quite limited, but the degree of effluent contamination is vast [40].

We already got the idea about all the different processing units in the wet process through the dyeing flowchart and the above brief description. To get a minimum idea, we will now look at probable wastewater pollutants, quantity, and nature found before and after the dyeing process (see Table 3).

Table 3. The toxic subtances found in the traditional textile dyeing and finishing plant [40].

Processing Unit	Probable Wastewater  Pollutants	Wastewater Quantity	Wastewater Nature
Sizing	Starch, waxes,  carboxymethyl cellulose, polyvinyl alcohol.	Minimal volume	High BOD and COD
Desizing	Starch, waxes, carboxymethyl cellulose,  polyvinyl alcohol, desizing, dissolved solids fats and waxes	Very small volume;	High BOD
(30Scouring	NaOH, Waxes, grease Na, CO3, Na2O2, and SiO2 fragments of cloth.	Small volume; Strongly alkaline; Dark color	Strongly a1kaline,
BOD (30Bleaching	NaOCl, Cl2, NaOH, H2O2, Acid etc.	Small volume	Alkaline constitutes

Increasing levels of biochemical compounds and heavy metals in water due to textile industry effluent have resulted in a significant hazard to the aquatic system and public health. There are different forms of toxicity present in textile wastewater. In order to control them, the Department of Environment (DoE) has listed the most detrimental to the environment are the Biochemical Oxygen Demand (BOD), Chemical Oxygen Demand (COD), Total Suspended Solids (TSS), Total Dissolved Solids (TDS), pH, oil and grease, color and temperature. To some point, the polluting parameters of other industries can differ. The concentration of pollutants in effluents varies significantly with different textile industries. Standard wastewater parameters are given in Table 4.

Table 4. Standard wastewater parameters [41].

Parameters on Quality  of water	Unit	Standard value of  discharging into the river
pH	---	6.5-9
BOD	ppm	<50
COD	ppm	<200
TSS	ppm	<150
TDS	ppm	<2100
Wastewater Flow	L/Kg of fabric    processing	100
Color	Co-pt unit	<200
Temperature	\(^{o}C\)	<30

Heavy metals constituted a concern to public health and organs at the same period as well [42]. The river serves as an essential supply of water for household and commercial usage and a mode of transportation, and it helps to dilute non-biodegradable contaminants in the environment. The presence of a large concentration of pollutants, on the other hand, reduces its capacity to purify. As a result, the dilution and reoxygenation potential is significantly decreased during the dry season, resulting in a more significant presence of BOD and COD and a low presence of dissolved oxygen (DO). During floods, such contaminated water comes into touch with humans and harms the general public's health [43]. More specifically, to figure out the reason for aquatic environment pollution, we collected some data from different textile industries situated in Bangladesh that do not have any effluent treatment plant (ETP) in their factory for processing the unfixed dyed water. We endeavor to come to a firm decision by comparing our collected value with government-given standard value to understand water conditions.

Comparing Tables 5 and 6 values with the standard value of textile effluent given DoE, shows a considerable discrepancy in each parameter's values. We took only six textile factories as our research source of collecting required values and found each of them as a leading polluter and destroyer of the water ecosystem in that specific area. To carry out our research work more precisely and easily, we chose these factories situated on the same riverbank named Shitalakkha. More than 100 of them like these six factories on the same riverbank, and most of them do not have any ETP system of their own. They all prefer to release their wastewater directly into the river without minimal treatment, which occurs consistently.

Table 5. Factories that do not have an ETP plant.

Name of the factory	BOD ppm	COD ppm	TSS ppm	TDS ppm	Color Co-pt	Temp. oC	pH
Ehsan Knitwear    Ltd.	300	445	2200	75	ND*	ND	9.3
Sadmusa Knit Ltd.	450	1060	3600	90	Dark	60	9.1
Northwest Textile    Ltd.	325	1000	3500	100	Dark	35	11
Purobi Knit Ltd.	640	1200	----	1000	Dark	ND	10
Brothers Denim    Ltd.	850	2150	----	350	<1000	35	9
Blue Denim	640	1312	3633	305	1380	----	11

Table 6. Factories that have ETP plant.

Name of the factory	BOD5 ppm	COD ppm	TSS ppm	TDS ppm	Color Co-pt	pH
KDS Textile Mills    Ltd.	25-45	60-150	<50	1800-2000	100-150	7-8
4H Group	35-48	70-120	<60	1900-2100	90-120	6.5-8
Karnafuli Polyester Dyeing Ltd.	30-45	60-150	45-55	1700-2000	120-160	7-8.5

It can be seen that how much responsible for damaging the aquatic environment a single factory that does not have any ETP plant than a factory that have

7. Techniques for treatment of textile dyeing effluent

Numerous methods have been employed to find a cost-effective and reliable way to treat textile dyeing wastewater, including physical, chemical, biological, integrated treatment processes, and some other technologies (see Figure 3). Specific treatment with one of these three methods has been ineffective in extracting color and other effluents from textile wastewater. While some dyes are difficult to degrade, namely hydrolyzed reactive dyes, and some acidic dyes are not readily taken up by active sludge; thus, they are not processed. A mixture of different effluent treatment methods can eliminate over 85 percent of unnecessary matter. The resultant effluent is typically strong in color. Complementary treatment is required to eliminate color and, if possible, residual impurities. The textile industry has been condemned as the world's worst aquatic environment polluter. It requires enormous quantities of chemicals and water at any point in the textile manufacturing and finishing process. Water is required to convey the chemicals to the fabric and wash them at the beginning and end of each phase. It is packed with chemical additives and then discharged as wastewater, which pollutes the environment. Water contamination is still a significant concern in most countries. The textile industry explores an economical solution to decolor the approximately 200 billion liters of colored effluent emitted annually. Countries, governments, and companies spend billions in cash on pollution-reduction studies and the development of effluent treatment plants. The environmental concern for industrial water pollution has contributed to substantial bans on all industrial practices that pollute the environment [44]. Governments have proposed regulations restricting the volume and form of waste that may be dumped because of the adverse effects on the environment and people's wellbeing resulting from the leakage of effluent from the Textile Dye Industry [32].

7.1. Chemical treatment process

An oxidative method consisting of Fenton reagent, Ozone, Photochemicals, Sodium hypochlorite, Cucurbituril, and Electrochemical destruction oxidative methods are the most widely employed chemical decoloration processes (see Table 8). This is primarily attributed to the ease of its application.

Table 7. The merits and drawbacks of chemical dye removal techniques.

Technique	Merits	Drawbacks	References
Fenton's reagent	1. Insoluble and soluble dyes can be effectively decolored. 2. Economical.	1. Sludge production. 2. Excessively costly.	[47]
Ozonation	1. Gases are used. 2. Wastewater and sludge volumes are not be increased.	1. A short half-life (20 minutes)	[48]
Oxidation with NaOCl	1. Initiates and accelerates the breaking of azo bonds.	1. Aromatic amines release.	[48]
Photochemical  oxidation	1. Doesn't generate sludge. 2. Low cost.	1. By-products formation.	[49]
Electrochemical destruction	1. Breakdown substances are not harmful.	1. Electricity and operation  costs are high.	[47,48]
Coagulation- flocculation	1. Easy and cost effective. 2. With a short detention period and inexpensive  capital expenditures. 3. Excellent removal efficiency.	1. Sludge production is high. 2. Difficulties with handling and disposal. 3. Chemicals for pH  adjustment are expensive. 4. Difficulties with dewatering and sludge handling.	[50]

7.2. Physicals treatments process

According to the data in Table 9, the most commonly utilized physical techniques include adsorption techniques, ion exchange, activated carbon, and membrane filtration methods (Electrodialysis, Nanofiltration, and Reverse osmosis). Adsorption techniques are quickly gaining popularity to handle aqueous effluent due to their effectiveness in eliminating too solid and commercially viable contaminants for conventional approaches. The physical process consists of activated carbon peat, wood chips, fly ash and coal combination silica gel, membrane filtration, ion exchange electro-kinetic coagulation, and other materials such as natural clay and agro-waste materials.

7.3. Biological treatment process

Biological treatment is one of the environmentally sustainable and pollution-free methods using different possible organisms. The biological method consists of decoloration of white-rot fungi, various microbial cultures, adsorption of living/dead microbial biomass, and an acromial textile dye bioremediation [49]. Also, as opposed to other physical and chemical procedures, biological treatment is frequently the most cost-effective option, as illustrated in Table 9.

Table 8. The merits and drawbacks of removal techniques for dye from dyeing effluent [51] .

\textbf{Technique}	\textbf{Merits}	\textbf{Drawbacks}
Adsorption	1. High adsorption capacity for all dyes, 2. Low cost	1. Need to dispose of adsorbents. 2. Low surface area for some adsorbents.
Activated carbon	1. Removes wide varieties of dyes	1. Very expensive, ineffective against disperse and vat dyes
Non-conventional adsorbents (agricultural and industrial byproducts)	1. Effective adsorbent, inexpensive, widely available, operation is easy, process design is simple	1. Transfer of pollutants from liquid phase to solid matrix (adsorbent) not selective
Membrane filtration	1. Removes all dye types, quick method and requires less space	1. Concentrated sludge production, membrane fouling, high cost and incapable to treat large volume
Ion exchange	1. Regeneration possibility 2. The adsorbent is not lost	1. Not effective for   all types of dyes
Nano-filtration	1. Separation of low molecular weight organic compounds and of divalent ions	1. High operation costs
Reverse osmosis	1. Removal of mineral salts, dyes and chemical reagents	1. High pressure needed

Table 9. The merits and drawbacks of biological dye removal techniques.

Organism (procedure)	Merits	Drawbacks	References
Bacteria (aerobic)	1. Azo and anthraquinone dyes can be decolored. 2. Biogas production.	1. Rate of decolorization that are too low. 2. Particular oxygen catalytic enzymes are needed. 3 Additional carbon and energy sources are needed.	[46]
Bacteria (anaerobic)	1. Effective for large-scale use.   2. For sludge treatment system, it is occurred in at a pH of 7. 3. Helps both obligatory  and facultative bacteria  to degrade azo dyes.	1. Toxic chemical compound production. 2. Post-treatment is needed. 3. Immobilization and recovery of redox mediator presents a challenge.	[47]
Fungi	1. Anthraquinone and  indigo-based dyes decolorize at a faster pace.	1. Azo dyes have a very low rate of decolorization. 2. A specialised bioreactor and an additional supply of carbon are required. 3. Acidic pH is needed. 4. Chemical and dye mixtures in textile wastewater inhibiting.	[48]

8. Techniques for removal of dyes from dyeing effluent

In an ETP, there are several ways to combine the tasks and processes:

1. A properly planned biological treatment plant, which usually includes screening, equalization, pH control, aeration, and settling, can effectively fulfill BOD, pH, TSS, oil, and grease demands. However, microorganisms can be harmful to the compounds in industrial effluent, so pretreatment may be essential. In addition, most dyes are complex chemicals that are impossible for microbes to degrade, so minimal color removal occurs typically.
2. A Physicochemical treatment plant, usually requiring filtering, equalization, pH monitoring, chemical storage tanks, mixing unit, flocculation unit, settling unit, and dewatering of sludge, is another choice. Depending on the procedures used, this procedure will destroy much of the color. Minimizing BOD and COD will be challenging to meet effluent requirements, and it is not feasible to eliminate TDS.
3. Physicochemical treatment will most commonly be associated with biological treatment. Screening, equalization, pH monitoring, storage facilities, mixing, flocculation, primary settling, aeration, and secondary settling are common components of such a plant. Physical treatment often comes before the biological units of treatment. Using a variety of treatments can usually decrease the levels of contaminants to below the standards of discharge.
4. The reed bed that can be used with a settling tank or other treatment processes is another type of biological treatment. It offers a natural effluent treatment technique that is also lower in capital, service, and maintenance. Reed beds can lead to a decrease in color, a decrease in COD, an increase in dissolved oxygen, and a reduction in heavy metals, but they work better with some pretreatment [41].

Latest works on textile dye removal techniques are presented in Table 10.

Table 10. Latest works on textile dye removal techniques.

Types of dye	Adsorbent used	pH & Temp.	Isotherms followed	References
Methyl Red	Adsorption by Guargum  Powder	pH 4.2 & \(34^{o}\)C	Langmuir model	[52]
Amido black-10 B	Nano photo catalyst	---	--------	[53]
Synthetic dye	Adsorption by sago waste	pH 4 & \(34^{o}\)C	Langmuir model	[54]
Acid blue 92,  Direct red 23,  & Direct red 81	Polymeric Adsorbent (poly amino primary secondary amine)	pH 12	Langmuir isotherm	[55]
Reactive red 120	Nano filtration poly etherimide  membrane			[56]
Acid black 210 & acid red 357	Activated carbon prepared  from leather shaving wastes	pH 2	Langmuir and  BET models	[57]
Residual Reactive blue 49	A coagulant and a flocculant	pH 7 \& \(60^{o}\)C	-----------	[58]
Reactive red	Belpatra Bark charcoal  adsorbent	\(50^{o}\)C \& pH 3	Langmuir, Freundlich and Temkin adsorption	[59]

9. Factors influencing the elimination of dyes

1. The adsorbent's physical and chemical properties, namely, composition of chemical, area of surface, and size of pore.
2. The adsorbate's physical and chemical properties, such as molecular polarity and chemical content.
3. The adsorbate concentration in the liquid phase (solution).
4. The liquid phase's properties, such as pH and temperature.
5. The system's residence time.
6. Incubation time, temperature, pH, agitation rate, and carbon, nitrogen, inorganic salts, and phosphorus sources in growth medium are all critical factors in dye decolorization in biological treatment [60].

10. Conclusion

One of the world's significant sectors is the garment and textile dyeing industry. The economy creates jobs and plays a significant role in many countries, such as Bangladesh, India, Pakistan, and Sri Lanka. Widely used in the textile industry, many synthetic dyes such as azo dye, vat dye, reactive dye, disperse dye, etc. The textile dyeing industries generate large quantities of effluent and solid waste ingredients daily. A complex combination of toxic substances, organic and inorganic, releases all these factories into the water bodies. Most pollutants are dissolved solids (DS), suspended solids (SS), colored compounds, higher BOD5, COD, and lightly alkaline, heavy metals. Most of the effluent parameters surpass the limits defined by DoE. The higher COD, TDS, TSS values indicate stronger effluent toxicity. The aquatic environment's physical, chemical, and biological natures differ due to the non-stop modification of alkalinity, odor, noise, temperature, pH, etc. Textile dyeing waste retains concentrations of metals capable of harming water, soil, and human health. Due to untreated effluents' discharge, the study found that the rivers' ecological balance in South Asia has decreased. In addition, the textile dyeing effluent seriously affects crop production in the surrounding agricultural fields. The study indicated an ETP in a few sectors, but it is not yet enough. The study revealed that some traditional treatment techniques, including flocculation, coagulation, adsorption, ozonation, and advanced oxidation procedure, are used to operate ETPs. However, a single treatment process cannot eliminate both toxic organic and inorganic pollutants from the effluent, and a series of treatment units must therefore be established.

Author Contributions:

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest:

''The authors declare no conflict of interest.''