1. Introduction

Solid insulators are exposed to diverse environmental disorders during operation that weakens the dielectric strength and other properties of solid insulation media. An insulator is exposed to different environmental stresses like humidity and contamination during their service life. Mostly polymers contain carbon atoms. When an insulator degrades, carbon under its insulating surface makes it conducting that could lead to breakdown. The development of a conducting path over insulator surface is called tracking [1]. So, if somehow carbon is eliminated or reduced from its surface then tracking can be minimized. But due to removal of material, an erosion path is formed. The significant erosion may initiate insulation failure due to exposure of less resistant material to outdoor conditions. By adding appropriate inorganic filler in insulators sample, tracking phenomenon can be minimized and erosion resistance could be enhanced.

Most common test method used is the Inclined Plane Test and is considered the standard test. In literature, the behavior of the tracking resistance of a blend of silicone and Ethylene Propylene Diene Monomer(EPDM) is analyzed. The performance of insulators was identified in various ways and it was noticed earlier that due to tracking, failure of out-door insulation structure could occur. It is well known that the material performance depends on the characteristics of the insulating material (ingredients). Similarly, the enactment of silicone rubber mixtures having 0, 10, and 30% by weight of silica filler were examined by Inclined plane test via application of DC and AC. The results showed most superior erosion class for AC followed by -DC and most inferior for +DC voltage [2]. Inclined plane is a standard test method for the examination of surface tracking [3]. Discharges are formed due to continuous contaminant flow over the surface of insulator during this test, and these discharges cause Surface degradation [4]. American Society of Testing and Materials (ASTM) D2302 standard test is an easy and effective testing method used to assess surface tracking ability of polymeric insulators [5].

The insulators made of polymeric materials provides plentiful benefits for outdoor insulation systems because of their excellent dielectric properties like lighter weight, improved pollution resisting performance, less budget, and quick processing. Though the insulators could be tarnished by ecological stresses like moisture, heat, and pollutants that may lead to tracking and surface discharges and eventually causing flashover of insulators. Tracking, however, remains to be the utmost problem. Attention has also been paid on addition of Nano-scale silica fillers, and its effect on the erosion, tracking, arcing and flame resistance have also been studied. The experimental studies showed increased resistance to erosion, tracking, arcing and flame [6]. In another study Nano-BN particles were added to silicon rubber insulators with different concentrations of 0, 2.5 and 5% by weight. The results showed that with the increasing concentration thermal properties are improved and erosion and weight loss were decreased, which proves that resistance to erosion and tracking is improved [7].The authors presented the alternative gases such as R12 and R134 gases as an alternatives to SF6 gas in high voltage insulation [8, 9].

Though the above authors successfully described and evaluated the effect of filler on insulator surface via various testing techniques and also determined the track length but lack to provide the implementation of standard Inclined plane testing in HV laboratory. Therefore, our paper evaluates the effect of filler on Insulator surface using Inclined Plane test.

The rest of our paper is organized as follows. The design and testing of Specimen is elaborated in Section 2. The properties of proposed filler are described in Section 3. Section 4 is dedicated to Test setup and Results. Conclusion and future work is examined in Section 5.

2. Specimen design and test method

Insulator samples are prepared using Polyester Resin-C. The resin was poured in a steel box of dimensions 120x50x10 mm. Steel box was covered with silicone rubber from inner side. Filler added was 20% by weight of aluminum oxide and zinc oxide. 2-3 drops of hardener are added and specimen is prepared within 30 minutes. A total of 8 samples were prepared; 4 samples of insulator without any filler while 4 samples each of zinc oxide and aluminum oxide filler were prepared.

A pair of electrodes is fixed on the sample 50mm apart. Four filter paper layers are clamped on the insulator sample to evenly spread the contaminant over the surface of insulator. Contaminant solution was made by preparing the solution of NH4Cl in distilled water; small amount of wetting agent (Triton X-100) was also added to the solution. Solution prepared was 0.2% solution of NH4Cl that was fed on the insulator via a small pipe. Electrolyte flow rate is maintained at \(1.0ml/min\). Inclined plane test method [10, 11] is used in this research.

The sample of insulator is anchored over Teflon test bed at \(45^{o}\) angle as shown in Figure 1. The contaminant is made to flow over insulator surface.

1. A continuous test voltage is to be used for 6 hours.
2. Test voltage is increased every hour.

Sample passes the test if the track length is shorter than \(25mm\). If track doesn't appear after 6 hours, then extend the test to further 6 hours. The method used in this test was application of a constant voltage. Voltage used in this test was 33kV.

3. Properties of fillers used

The pertinent features of proposed filler are studied and are listed in Table 1.

Table 1. Properties of fillers.

Properties	Aluminum Oxide	Zinc Oxide
Surface energy ( \(mJm^{-2}\) )	800	2799
Work of  Adhesion ( \(mJm^{-2}\) )	362.2	677.52
Fracture toughness (\(kNm^{-3/2}\))	4500	842
Melting point (\(C^{o}\))	2050	1975

Surface energy is the sum of all inter molecular forces that are acting on the surface of the material, which is much greater in case of zinc oxide, compared to aluminum oxide. Contrarily, work of adhesion is the tendency of dissimilar particles to cling to one another that is also greater in case of zinc oxide and we can deduce that zinc oxide particles will have much stronger forces of attraction with polyester resin, compared to aluminum oxide. Fracture toughness describes the ability of a material containing a track to resist fracture. As this work is on the initial stages of the track, so this property is not taken into account in this research.

4. Test setup and Results

Test setup shown in Fig. 3 was used to apply high voltage to the insulator sample. The test bed was made of Teflon. Insulator was fixed at an angle of \(45^{o}\). Wet layers of filter paper were clamped over it and continuous flow of contaminant was maintained. The circuit diagram for the experimental setup is shown in the Figure 2.

The specimen that was without the filler couldn't pass the test as can be seen from Table 2, that the track length was more than \(25mm\) after 6 hours. In case of aluminum oxide, track was developed after 12 hours while in case of zinc oxide the track was developed after 18 hours and was very small as compared to that of aluminum oxide. The results of our test are summarized in the table below.

Table 2. Test Results.

Sr. No.	Track length in specimen without filler (After 6 hours)	Track length in specimen with aluminum oxide filler (After   12 hours)	Track length in specimen with Zinc oxide filler (After  18 hours)
1	\(48mm\)	\(13mm\)	\(3mm\)
2	\(46mm\)	\(11mm\)	\(5mm\)
3	\(50mm\)	\(9mm\)	\(2mm\)
4	\(45mm\)	\(10mm\)	\(2mm\)

The Inclined plane test for proposed filler is shown in Figure 4 and graphical comparison is shown in Figure 5.

Table 3. Average track length of each specimen and standard deviation.

Specimen	Average Track length (mm)	Standard deviation
Polyester resin without filler	47.25	2.21
Polyester resin with aluminum oxide filler	10.75	1.70
Polyester resin with zinc oxide filler	3.00	1.41

The results show that zinc oxide filler outperformed aluminum oxide filler. Track was observed using the Polari-scope. As we can see only a small area of specimen through the Polari scope, therefore we have taken the image of only the part where a track terminates. Images of all three types of specimen used are shown in the Figures 6. Figure 6(a) shows Polyester resin without filler, Figure 6(b) shows Polyester resin with filler of Aluminum oxide and Figure 6(c) shows Polyester resin with filler of Zinc oxide:

5. Conclusion and future work

From this research we can conclude that, by adding different fillers to the insulator specimens can greatly enhance their resistivity to erosion and track length. Zinc oxide showed much better results even after 18 hours of application of voltage. Whereas in aluminum oxide the track length was greater even after 12 hours. Conversely, specimen without any filler presented poor results, compared to aluminum oxide and zinc oxide. It is due to the fact that zinc oxide has the ability to disperse the electric field, it doesn't allow electric discharges to accumulate quickly and lead to breakdown, so the track length in case of zinc oxide is less. Tracking results into breakdown or flashover which could be fatal for the insulator and could affect the power system stability. Inclined plane test decides the life of insulating materials and can be used to check whether the insulating material is resistant against tracking or not. In near future, this work can be extended for determining the development of electric field and leakage current due to the applied voltage when the track starts to develop.

Competing Interests

The authors declare no competing interest.

1. Introduction

The roles and importance of thermal radiation are evident in solar power technology, nuclear plants, propulsion devices for aircraft, combustion chambers, glass production, furnace design, and space technology applications, such as comical flight aerodynamics rocket, space vehicles, propulsion systems, plasma physics in the flow structure of atomic plants, combustion processes, internal combustion engines, ship compressors, solar radiations and in chemical processes and space craft re-entry aerodynamics which operates at high temperatures. Also, there are various engineering and industrial applications of magnetohydrodynamic (MHD) fluid behavior such as in the design of cooling system with liquid metals, accelerators, MHD generators, nuclear reactor, pumps, flow meters, study of crystal growth, metal casting, liquid metal cooling blankets for fusion reactors and blood flow. Additionally, recent research works in the past few decades have shown that the study of flow of non-Newtonian materials is a topic of great interest amongst the recent workers in the study of fluid dynamics. The flow applications of non-Newtonian fluids such as blood transport in micro-circulatory system, printer ink, paints, liquid detergents, multi grade oil, polymers, sauce, mud, apple sauce are evident in polymer devolatisation and processing, bubble absorptions, fermentation, plastic foam processing, bubble columns, composite processing, wire and fiber coating, heat exchangers, extrusion process, chemical processing equipment, etc. Also, the analysis of stretched flow with heat transfer is very significant in controlling the quality of the end product in the afore-mentioned areas of applications. Such processes have great dependence on the stretching and cooling rates [1]. Consequently, in the past few years, research efforts have been directed towards the analysis of this very important phenomenon of wide areas of applications. Moreover, the promising significance of magnetohydrodynamics (MHD) fluid behavior in various engineering and industrial applications (such as in the design of cooling system with liquid metals, accelerators, MHD generators, nuclear reactor, pumps, flow meters, study of crystal growth, metal casting, liquid metal cooling blankets for fusion reactors and blood flow) still provokes the continuous studies and interests of researchers.

Additionally, the study of thermal radiation is important in solar power technology, nuclear plants, and propulsion devices for aircraft, combustion chambers, glass production and furnace design, and also in space technology applications, such as comical flight aerodynamics rocket, space vehicles, propulsion systems, plasma physics in the flow structure of atomic plants, combustion processes, internal combustion engines, ship compressors, solar radiations and in chemical processes and space craft re-entry aerodynamics which operates at high temperatures. Therefore, the influences of external factors such as magnetic field and thermal radiation on the flow and heat transfer problem of Newtonian and non-Newtonian fluid have been widely analyzed in recent times. In an early study, MHD fluid flow over a stretching surface was carried out by Anderson et al. [2, 3]. The effect of unsteadiness parameter on the film thickness has been studied [2] and the effect of magnetic field on the flow characteristics of the fluid were explored numerically [3]. Few years later, Chen [4] investigated the power-law fluid film flow of unsteady heat transfer stretching sheet while Dandapat et al. [5, 6] analyzed the effect of variable viscosity and thermo- capillarity on the heat transfer of liquid film flow over a stretching sheet. Meanwhile, Wang [7] developed an analytical solution for the momentum and heat transfer of liquid film flow over a stretching surface. Also, Chen [8] and Sajid et al. [9] investigated the flow characteristics of a non-Newtonian thin film over an unsteady stretching surface considering viscous dissipation using homotopy analysis and homotopy perturbation methods. After a year, Dandapat et al. [10] presented the analysis of two-dimensional liquid film flow over an unsteady stretching sheet while in the same year, effect of power-law index on unsteady stretching sheet was studied by Abbasbandy et al. [11]. Santra and Dandapat [12] numerically studied the flow of the liquid film over an unsteady horizontal stretching sheet. A numerical approach was also used by Sajid et al. [13] to analyze the micropolar film flow over an inclined plate, moving belt and vertical cylinder. A year later, Noor and Hashim [14]1 investigated the effect of magnetic field and thermocapillarity on an unsteady flow of a liquid film over a stretching sheet while Dandapat and Chakraborty [15]) and Dandapat and Singh [16] presented the thin film flow analysis over a non-linear stretching surface with the effect of transverse magnetic field. Heat transfer characteristics of the thin film flows considering the different channels have also been analyzed by Abdel-Rahman [17], Khan et al. [18], Liu et al. [19] and Vajaravelu et al. [20] Meanwhile, Liu and Megahad [21] used homotopy perturbation method to analyze thin film flow and heat transfer over an unsteady stretching sheet with internal heating and variable heat flux. Effect of thermal radiation and thermocapillarity on the heat transfer thin film flow over a stretching surface was examined by Aziz et al. [22]. In their study on the numerical simulation of Eyring-Powell flow and unsteady heat transfer of a laminar liquid film over a stretching sheet using finite difference method, Khader and Megahed [23] established that increasing the Prandtl number reduces the temperature field across the thin film. Vajravelu et al. [24] analyzed the convective heat transfer over a stretching surface with applied magnetic field while Pop and Na [25] studied the influence of magnetic field flow over a stretching permeable surface. In another study, Xu et al. [26] presented a series solutions of the unsteady three-dimensional MHD flow and heat transfer over an impulsively stretching plate. Nazar et al. [27] examined the hydro magnetic flow and heat transfer over a vertically stretched sheet. The effect of MHD stagnation point flow towards a stretching sheet was investigated by Ishak et al. [28] while the influence of thermal radiation on heat transfer in an electrically conducting fluid at stretching surface was explored by Emad [29].

Also, Reddy [30] studied the thermal radiation boundary layer flow of a nanofluid past a stretching sheet under applied magnetic field. Effects of thermal radiation on convective heat transfer in an electrically conducting fluid over a stretching surface with variable viscosity and uniform free stream was examined by Abo-Eldahab and Elgendy [31]. In a recent study, Gnaneswara Reddy [32] investigated the thermal radiation and chemical reaction effects on MHD mixed convective boundary layer slip flow in a porous medium with heat source and Ohmic heating. In another study, Gnaneswara Reddy [33] studied the influence of thermophoresis, Viscous Dissipation and Joule Heating on Steady MHD Flow over an Inclined Radiative Isothermal Permeable Surface. The effect of thermal radiation on magnetohydrodynamics flow was examined by Raptis et al. [34] while Seddeek [35] investigated the impacts of thermal radiation and variable viscosity on magnetohydrodynamics in free convection flow over a semi-infinite flat plate. In another study, Mehmood et al. [36] analyzed unsteady stretched flow of Maxwell fluid in presence of nonlinear thermal radiation and convective condition. Hayat et al. [37] addressed the effects of nonlinear thermal radiation and magnetohydrodynamics on viscoelastic nanofluid flow. Effects of nonlinear thermal radiation on stagnation point flow Farooq et al. [38]. Also, Shehzad et al. [39] presented a study o MHD three-dimensional flow of Jeffrey nanofluid with internal heat generation and thermal radiation. In a recent study, Lin et al. [40] examined the effect of MHD pseudo-plastic nanofluid flow and heat transfer film flow over a stretching sheet with internal heat generation. Numerically, Raju and Sandeep [41] studied heat and mass transfer in MHD non-Newtonian flow while Tawade et al. [42] presented the unsteady flow and heat transfer of thin film over a stretching surface in the presence of thermal radiation, internal heating in the presence of magnetic field. Heat and mass transfer of MHD flows through different channels have been analyzed [43, 44, 45, 46, 47, 48]. Makinde and Animasaun [49] investigated the effect of cross diffusion on MHD bioconvection flow over a horizontal surface. In another study, Makinde and Animasaun [50] presented the MHD nanofluid on bioconvection flow of a paraboloid revolution with nonlinear thermal radiation and chemical reaction while Sandeep [51], Reddy et al. [52] and Ali et al. [53] studied the heat transfer behaviour of MHD flows. Maity et al. [54] analyzed thermocapillary flow of a thin Nanoliquid film over an unsteady stretching sheet.

The above studies have been the consequent of the various industrial and engineering applications of non-Newtonian fluids. Among the classes of non-Newtonian fluids, Carreau fluid which its rheological expressions were first introduced by Carreau [55], is one of the non-Newtonian fluids that its model is substantial for gooey, high and low shear rates [56]. On account of this headway, it has profited in numerous innovative and assembling streams [56]. Owing to these applications, different studies have been carried out to explore the characteristics of Carreau liquid in flow under different conditions. Kumar et al. [40] applied Runge-Kutta and Newton's method to analyze the flow and heat transfer of electrically conducting liquid film flow of Carreau nanofluid over a stretching sheet by considering the aligned magnetic field in the presence of space and temperature dependent heat source/sink, viscous dissipation and thermal radiation. Hayat et al. [57] studied the influence of induced magnetic field and heat transfer on peristaltic transport of a Carreau fluid. Olajuwon [58] presented a study on MHD flow of Carreau liquid over vertical porous plate with thermal radiation. Hayat et al. [59] investigated the convectively heated flow of Carreau fluid while in the same year, Akbar et al. [60] analyzed the stagnation point flow of Carreau fluid. Also, Akbar [61] presented blood flow of Carreau fluid in a tapered artery with mixed convection. A year later, Mekheimer [62] investigated the unsteady flow of a carreau fluid through inclined catheterized arteries haveing a balloon with time-variant overlapping stenosis. Elmaboud et al. [63] developed series solution of a natural convection flow for a Carreau fluid in a vertical channel with peristalsis. Using a revised model, flow of Carreaunanoliquid in the presence of zero mass flux condition at the stretching sheet has been examined by Hashim and Khan [64]. The MHD flow of Carreau fluid with thermal radiation and cross diffusion effects was investigated by Machireddy and Naramgari [65]. Sulochana et al. [66] provided an analysis of magnetohydrodynamic stagnation-point flow of a Carreau nanofluid.

Another non-Newtonian fluid is Casson fluid. Casson fluid is a non-Newtonian fluid first invented by Casson in 1959 [67]. It is a shear thinning liquid which is assumed to have an infinite viscosity at zero rate of shear, a yield stress below which no flow occurs, and a zero viscosity at an infinite rate of shear [68, 69]. If yield stress is greater than the shear stress then it acts as a solid, whereas if yield stress lesser than the shear stress is applied then the fluid would start to move. The fluid is based on the structure of liquid phase and interactive behaviour of solid of a two-phase suspension. It is able to capture complex rheological properties of a fluid, unlike other simplified models like the power law [70] and second, third or fourth-grade models [71]. Some examples of Casson fluid are Jelly, honey, tomato sauce and concentrated fruit juices. Human blood is also treated as a Casson fluid in the presence of several substances such as fibrinogen, globulin in aqueous base plasma, protein, and human red blood cells. Concentrated fluids like sauces, honey, juices, blood, and printing inks can be well described using this model. It has various applications in fibrinogen, cancer homeo-therapy, protein and red blood cells form a chain type structure. Due to these applications many researchers are concentrating characteristics of Casson fluid. Application of Casson fluid for flow between two rotating cylinders is performed in [72]. The effect of magnetohydrodynamic (MHD) Casson fluid flow in a lateral direction past linear stretching sheet was explained by Nadeem et al. [73].

Numerical methods such as Euler and Runge--Kutta methods are limited to solving initial value problems. With the aid of shooting method, the methods could be carried out iteratively to solve boundary value problems. However, these numerical methods are only useful for solving ordinary differential equations i.e. differential equations with a single independent variable. On the other hand, numerical methods such as finite difference method (FDM), finite element methods (FEM) and finite volume method (FVM) can be adopted solve differential equations with single and multiple independent variables as they have been used to different linear and non-linear differential equations in literatures. The numerical solution of FDM represents an efficient way of obtaining temperature profile for the steady heat transfer processes. The FDM can be used for solving any complex body by breaking the body into small domains. Also, choice of finer grids which requires high computing capability can remove approximation errors to larger extent. Hence, in this work, finite difference method. Therefore, in this study, finite difference method is applied to analyze the combined influences of thermal radiation, inclined magnetic field and temperature-dependent internal heat generation on unsteady two-dimensional flow and heat transfer analysis of dissipative Casson-Carreau nanofluid over a stretching sheet embedded in a porous medium are examined. Using kerosene as the base fluid embedded with the silver (Ag) and copper (Cu) nanoparticles, the effects of other pertinent parameters on flow and heat transfer characteristics of the nanofluids are investigated and discussed.

2. Problem Formulation

Consider an unsteady, two-dimensional boundary layer flow of an electrically conducting and heat generating Casson and Carreau nanofluids over a stretching sheet bounded by a thin liquid film of uniform thickness \(h(t)\) over a horizontal elastic sheet which emerges from a narrow slit at the origin of the cartesian coordinate system which is schematically represented in Figure 1. The sheet is stretched along the x-axis with stretching velocity \(U(x,t)\) and y--axis is normal to it. An inclined magnetic field. The effects of non-uniform heat source/sink, thermal radiation, viscous is applied to the stretching sheet at angle dissipation and volume fraction are taken into consideration.

Using the rheological equation for an isotropic and incompressible Casson fluid, reported by Casson [67], is or where \(\tau\) is the shear stress, \(\tau_{o}\) is the Casson yield stress, \(\mu\) is the dynamic viscosity, \(\dot{\sigma }\) is the shear rate, \(\pi= e_{ij}e_{ij}\) and \(e_{ij}\) is the \((i,j)\)th component of the deformation rate, \(\pi\) is the product of the component of deformation rate with itself, \(\pi_{c}\) is a critical value of this product based on the non-Newtonian model, \(\mu_{B}\) the is plastic dynamic viscosity of the non-Newtonian fluid and \(p_{y}\) is the yield stress of the fluid. The velocity as well as the temperature is functions of \(y,t\) only. The extra stress tensor for Carreau fluid is given as where, \(\bar{\tau }_{ij} \) is the extra tensor, \(\eta_{o}\) is the zero shear rate viscosity, \(\Gamma\) is the time constant, \(n\) is the power-law index and \(\bar{\dot{\gamma }}\) is defined as \[\bar{\dot{\gamma }}=\sqrt{\frac{1}{2} \sum _{i}\sum _{j}\bar{\dot{\gamma }}_{ij} \bar{\dot{\gamma }}_{ji} } =\sqrt{\frac{1}{2} } \Pi \] where \(\Pi \) is the second invariant strain tensor. Following the assumptions, the equations for continuity and motion for the flow analysis of Carreau and Casson fluids are where Assuming no slip condition, the appropriate boundary conditions are given as It should be stated at this juncture that the mathematical problem is implicitly formulated only for \(x \ge 0.\) In other to avoid the complications due to surface waves, a further assumption is made that the surface of the planar liquid film is smooth. Also, the influence of interfacial shear due to the quiescent atmosphere i.e. the effect of surface tension is assumed to be negligible. The viscous shear stress \(\tau =\mu \frac{\partial u}{\partial y} \) and the heat flux \(\dot{q''}=-k\left(\frac{\partial T}{\partial y} \right)\) vanish at the adiabatic free surface at \(y = h\).\\ It should be noted that The above boundary conditions are in line with the works of Kumar et al. [56].
The non-uniform heat generation/absorption \(q'''\) is taken as where the surface temperature \(T_{s}\) of the stretching sheet varies with respect to distance \(x\)-from the slit as And the stretching velocity varies with respect to \(x\) as On introducing the following stream functions And the similarity variables Substituting Equations (19) and (20) into Equations (5), (6), (13) and (14), we have a partially coupled third and second orders ordinary differential equation where And the boundary conditions become

Method of Solution

Equations (21) and (22) are systems of coupled non-linear ordinary differential equations which are to be solved by using the boundary conditions (24). The exact solution is not possible for this set of equations. The use and the accuracy of finite difference method for the analysis of nonlinear problems has earlier been pointed out by Han et al. (2005). Therefore, in this work, finite difference method is used to discretize the governing Equation (17) combined with the boundary conditions of Equation (19). The finite difference forms or schemes for each differential in the governing differential equations are given as
\(f''''=\frac{2f_{i+1} -9f_{i} +16f_{i-1} -14f_{i-2} +6f_{i-3} -f_{i-4} }{\left(\Delta \eta \right)^{4} } \)
\(f'''=\frac{3f_{i+1} -10f_{i} +12f_{i-1} -6f_{i-2} +f_{i-3} }{2\left(\Delta \eta \right)^{3} } \)
\(f''=\frac{f_{i+1} -2f_{i} -f_{i-1} }{\left(\Delta \eta \right)^{2} } \)
\(f'=\frac{f_{i+1} -f_{i-1} }{2\left(\Delta \eta \right)} =\frac{f_{i+1} -f_{i} }{\Delta \eta } =\frac{f_{i} -f_{i-1} }{\Delta \eta } \)
\(\theta ''=\frac{\theta _{i+1} -2\theta _{i} -\theta _{i-1} }{\left(\Delta \eta \right)^{2} } \)
\(\theta '=\frac{\theta _{i+1} -\theta _{i-1} }{2\left(\Delta \eta \right)} =\frac{\theta _{i+1} -\theta _{i} }{\Delta \eta } =\frac{\theta _{i} -\theta _{i-1} }{\Delta \eta } \)
Substituting the above finite difference schemes into Equations (21) and (22), one obtains an equivalent finite difference schemes as The boundary conditions are expressed in finite difference form as The Equations (25) and (26) along with the boundary conditions (27) are reduced into a block tri-diagonal system which are solved by block elimination method.

4. Results and discussion

For the computational domain, numerical solutions are computed and grid-independence study is made in order to obtain the results accurately. The necessary convergence of the results is achieved with the desired degree of accuracy. The results with the discussion are illustrated through the Figures 2-21 to substantiate the applicability of the present analysis. The influence of pertinent parameters such as magnetic field parameter, unsteadiness parameter, heat source/sink parameter, Eckert number, volume fraction of nanoparticles etc. on the flow and heat transfer of the thin film flow are investigated.

Figures 2 and 3 depicts the effects of Casson parameter on velocity and temperature profiles Casson nanofluid, respectively. It is obvious from the figure that Casson the parameter has influence on axial velocity. From Figure 6, the magnitude of velocity near the plate for Casson nanofluid parameter decreases for increasing value of the Casson parameter, while temperature increases for increase in Casson fluid parameter as shown in Figure 7. Physically, increasing values of Casson parameter develop the viscous forces. These forces have a tendency to decline the thermal boundary layer. Figures 4 and 5 depict the effect of thermal radiation parameter on the velocity and temperature profiles. From the figure, it is shown that an increase in radiation parameter causes the velocity of the fluid to increase, while the temperature profiles increases with increasing radiation parameter values. This is because, increases in thermal radiation causes the thermal boundary layer of fluid to increase. Generally, increasing radiation parameter values enhances the temperature near the boundary. Effects of other pertinent parameters such as magnetic field parameter, unsteadiness parameter, heat source/sink parameter, Eckert number, volume fraction of nanoparticles etc. on the flow and heat transfer of the thin film flow are investigated. Figures 6 and 7 show the effects of magnetic field (Ha) on the velocity and temperature fields, respectively. It is revealed that there is a diminution in the velocity field and enhancement in the temperature field occur for increasing values of the Hartmann number Ha. This confirms the general physical behavior of the magnetic field that say that the fluid velocity depreciates for improved values of Ha. According to the physical point, Ha represents the ratio of electromagnetic force to the viscous force so large Ha implies that the Lorentz force increases, which is drag-like force that produces more resistance to transport phenomena due to which fluid velocity reduces. Consequently, the boundary layer thickness is a decreasing function of Ha. i.e. presence of magnetic field slows fluid motion at boundary layer and hence retards the velocity field. It should be noted that the magnetic field tends to make the boundary layer thinner, thereby increasing the wall friction. It is seen through Figure 7 that the temperature profile \(\theta(\eta)\) enhances increasing the Hartmann number Ha. Practically, the Lorentz force has a resistive nature which opposes motion of the fluid and as a result heat is produced which increases thermal boundary layer thickness and fluid temperature. The magnetic field tends to make the boundary layer thinner, thereby increasing the wall friction.

The effects of unsteadiness parameter on velocity and temperature profiles are shown in Figures 8 and 9, respectively. It is observed that increasing values of S increases the velocity field while decreases the temperature field. This is because as the rate of heat loss by the thin film increases as the value of unsteadiness parameter increases. Figures 10 and 11 depict the effects of Weissenberg number (\(We\)) on the velocity and temperature profiles. It is shown from the figures that the velocity increases for increasing values of \(We\) and opposite trend was observed in temperature field. The observed trends in the velocity and temperature fields are due to the fact that a higher value of \(We\) will reduce the viscosity forces of the Carreau fluid. Increasing the Weissenberg number reduces the magnitude of the~fluid velocity for~shear thinning~fluid~while it arises for the~shear thickening fluid. The influence of aligned angle on velocity and temperature profiles is presented in Figures 12 and 13. From the figures, it is shown that as the value of aligned parameter increases, the velocity field increases while temperature field decreases.

Figures 14 and 15 demonstrated the effect of power law index on velocity and temperature fields. As the power index is increased, it was observed that the velocity profile increases while the temperature profile decreases. This is because, increasing value of the power law index, thickens the liquid film associated with an increase of the thermal boundary layer. An increase in the momentum boundary layer thickness and a decrease in thermal boundary layer thickness is observed for the increasing values of the power law index including~shear thinning~to~shear thickening~fluids. Also, it should be pointed out that an increase in Weissenberg number correspond a decrease in the local skin~friction~coefficient and the magnitude of the local Nusselt number s decreases when the Weissenberg number increases. The effects of nanoparticles volume fraction on the velocity and temperature profiles are depicted in Figures 16 and 17, respectively. The result shows that as the solid volume fraction of the film increases both the velocity and temperature field increases. This is because as the nanoparticle volume increases, more collision occurs between nanoparticles and particles with the boundary surface of the plate and consequently the resulting friction enhances the thermal conductivity of the flow and gives rise to increase the temperature within the fluid near the boundary region.

Figures 18 and 19 depict the influence of non-uniform heat source/sink parameter on the temperature field. It is revealed that increasing the non-uniform heat source/sink parameter enhances the temperature fields. It is observed in the analysis that the temperature and thermal boundary layer thickness is depressed by increasing the Prandtl number Pr. The effect of Eckert number on temperature profile is shown in Figure 20. It was established that as the values of Eckert number increases, the values of the temperature distributions in the fluid increases. This is because as Ec increases, heat energy is saved in the liquid due to the frictional heating.

The effect of nanoparticle volume fraction \(\mathrm{\phi}\) on the film thickness of the nanofluid is shown in Figure 21. It is evident from the figure that the film thickness is enhanced as the values of \(\mathrm{\phi}\) is increased. It can be inferred from Equation (15) that if nanoparticle volume fraction \(\mathrm{\phi}\) is increased, the nanofluid viscosity will increased as there exist a direct relationship or proportion between the two parameters. As a result, the increasing viscosity resists the fluid motion along the stretching direction leading to the slowdown of the film thinning process [54].

5. Conclusion

In this paper, combined influences of thermal radiation, inclined magnetic field and temperature-dependent internal heat generation on unsteady two-dimensional flow and heat transfer analysis of dissipative Casson-Carreau nanofluid over a stretching sheet embedded in a porous medium have been investigated examined numerically with the aid of finite difference method. Using kerosene as the base fluid embedded with the silver (Ag) and copper (Cu) nanoparticles, the effects of other pertinent parameters on flow and heat transfer characteristics of the nanofluids are investigated and discussed. From the results, it was established temperature field and the thermal boundary layers of Ag-kerosene nanofluid are highly effective when compared with the Cu-kerosene nanofluid. Thermal and momentum boundary layers of Cu-kerosene and Ag-kerosene nanofluids are not uniform. Heat transfer rate is enhanced by increasing in power-law index and unsteadiness parameter. Skin friction coefficient and local Nusselt number can be reduced by magnetic field parameter and they can be enhanced by increasing in aligned angle. Friction factor is depreciated and the rate of heat transfer increases by increasing the Weissenberg number. This analysis can help in expanding the understanding of the thermo-fluidic behaviour of the Carreau nanofluid over a stretching sheet. Also, the present study has numerous applications involving heat transfer and other applications such as chemical sensors, biological applications, glass, solar energy transformation, electronics, petrochemical products, light-weight, heat-insulating and refractory fiberboard and metallic ceramics etc.

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

Fuzzy set generalize classical sets by use of a membership function such that each element is assigned a number in the real unit interval [0,1], which measures its grade of membership in the set. The theory of fuzzy sets was proposed by Zadeh in 1965 [1]. Since then, the theory was used in a wide range of domains in which information is incomplete or imprecise, such as such as management science, medical science, social science, financial science, environment science and bioinformatics [2]. In 1993, Gau et al. [3] presented the concept of vague set theory as a generalization of fuzzy set theory, which allow a separation of evidence for membership (grade of membership) and evidence against membership (negation of membership). They used a subinterval of [0,1] to replace the value of an element in a set. That is, a vague set is characterized by two functions. Namely, a truth-membership function \(t_{v}(x)\) and false-membership function \(f_{v}(x)\) are used to describe the boundaries of the membership degree.

Graph theory is a very useful and well developed branch of discrete mathematics, and it also is an important tool for modeling many types of relations and processes in biological, physical, social and information systems. Realizing the importance of graph theory and inspiring of Zadeh's fuzzy relations [4], Kauffman [5] proposed the definition of fuzzy graph in 1973. Then Rosenfeld [6] proposed another elaborated definition of fuzzy graph in 1975. Since then, there was a vast research on fuzzy graph [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 19]. Inspired by fuzzy graph, in 2009, Ramakrishna [20] introduced the concept of vague graphs and studied some important properties. After that, Samata et al. [21] analysed the concepts of vague graphs and its strength. Rashmanlou et al. [22] introduced the notion of vague h-morphism on vague graphs and regular vague graphs, and they investigated some properties of an edge regular vague graph [23]. At the same time, they introduced some connectivity concepts in the vague graphs [24].

The Dominator coloring of a graph was proposed by Gera et al [25] in 2006. In the same paper, they showed that dominator chromatic number is NP-complete. After that, they studied the bounds and realization of the dominator chromatic number in terms of chromatic number and domination number [26] and the dominator colorings in bipartite graphs [27]. Recently, several researchers have theoretically investigated the dominator coloring number of Claw-free graph [28], Certain Cartesian Products [29], trees [30] and more [31, 32]. Motivated by dominator chromatic number, Kazemi [33] studied the new concept of a total dominator chromatic number of a graph. And they showed that total dominator chromatic number is NP-complete. A survey of total dominator chromatic number in graphs can also be found in [34, 35].

Borzooei et al. [36] in their work introduced the concepts of special kinds of dominating sets in vague graph. Kumar et al. [37] discuss the new concepts of coloring in vague graphs with application. In this paper, we introduce the concept of the dominator coloring and total dominator coloring of a vague graph and establish mathematical modelling for these problems.

2. Preliminaries

A vague set \(A\) in an ordinary finite non-empty set \(X\) is a pair \((t_{A}, f_{A})\), where \(t_{A} : X \rightarrowtail [0, 1]\), \(f_{A} : X \rightarrowtail [0, 1]\), and \(0 \leq t_{A}(x) + f_{A}(x) \leq 1\) for each element \(x \in X\). Note that the truth-membership \(t_{A}(x)\) is considered as the lower bound on grade of membership of \(x\) derived from the evidence for \(x \in X\) and the false-membership \(f_{A}(x)\) is the lower bound on negation of membership of \(x\) derived from the evidence against \(x \in X\). The grade of membership for \(x\) is characterized by the interval \([t_{A}(x), 1 - f_{A}(x)]\) not a crisp value. And if \(t_{A}(x) = 1 - f_{A}(x)\) for all \(x \in X\), the vague set degrades to a fuzzy set. In this paper, we denote by \(P_{n}, \ C_{n}, \ K_{n}\) the path, cycle and complete graph on \(n\) vertices, respectively. The complete bipartite graph with part size \(m,\ n\) is denoted by \(K_{m,n}\) and the ladder graph is the Cartesion product of \(P_{2}\) and \(C_{n}\), denoted by \(P_{2} \Box C_{n}\).

Definition 1. Let \(G = (V,E)\) be a graph. A pair \(G' = (A,B)\) is called a vague graph on \(G\) where \(A = (t_{A}, f_{A})\) is a vague set on \(V\) and \(B = (t_{B}, f_{B})\) is a vague set on \(E\) such that \(t_{B}(uv) \leq min\{ t_{A}(u), t_{A}(v) \}\), \(f_{B}(uv) \geq max \{f_{A}(u), f_{A}(v) \}\) for each \(uv \in E\).

Definition 2. For a vague graph \(G = (A, B)\), an edge \(uv\) is called a strong edge if \(t_{B}(uv) = min \{ t_{A}(u), t_{A}(v) \}\), \(f_{B}(uv) = max \{ f_{A}(u), f_{A}(v) \}\). Let \(N(u) = \{v|uv \ is \ a \ strong \ edge \ in \ G \}\) and \(N[u] = N(u) \cup \{u\}\). We say \(u\) dominates all vertices in \(N(u)\) and totally dominates all vertices in \(N[u]\).

Definition 3. Dominator coloring of a vague graph \(G\) is a coloring of the vertices of \(G\) such that every vertex dominates all vertices of at least one other class. The dominator chromatic number \(\chi^d(G)\) of \(G\) is the minimum number of colors among all dominator colorings of \(G\).

Definition 4. Total dominator coloring of a vague graph \(G\) is a coloring of the vertices of \(G\) such that every vertex totally dominates all vertices of at least one other class. The total dominator chromatic number \(\chi^d_{t}(G)\) of \(G\) is the minimum number of colors among all total dominator colorings of \(G\).

3. Dominator coloring problems

Let \([k] = \{1, 2, \dots , k \}\). Let \(V_{c} \subseteq V\) denotes set of vertices with assigned color \(c\). Further, let decision variables \(x_{i,c}\) be defined as \[ x_{i,c} = \begin{cases} 1, & i \in V_{c} \\ 0, & i \in V_{c} \end{cases} \] For a vague graph \(G\) and an integer \(k\), let \(E^s\) be the set of all strong edges of \(G\). We propose integer linear programming \((ILP)\) formulations (called Dominator Coloring ILP and Total Dominator Coloring ILP, respectively), for the dominator coloring problem and total dominator coloring problem as follows:

Dominator coloring ILP

Theorem 5. Conditions \((1)-(7)\) defined for the graph \(G\) are satisfied if and only if \(G\) admits a dominator coloring with \(k\) colors.

Proof. \(\Rightarrow\) Condition \((1)\) ensures that each vertex is assigned with exactly one color. Condition \((2)\) ensures that each color should be used. Conditions \((3)\) and \((4)\) ensure that every vertex dominates all vertices of at least one other class. Condition \((5)\) ensures that the assignment is a proper coloring. Conditions \((6)\) and \((7)\) ensure that each variable is boolean. Therefore, if each condition is satisfied, then \(G\) admits a dominator coloring with \(k\) colors. \(\Leftarrow\) By the definition of the dominator coloring, it is clear that Conditions \((1)-(7)\) defined for the graph \(G\) are satisfied.

Total dominator coloring ILP

Theorem 6. Conditions \((8)-(14)\) defined for the graph \(G\) are satisfied if and only if \(G\) admits a total dominator coloring with \(k\) colors.

Proof. Condition \((8)\) ensures that each vertex is assigned with exactly one color. Condition \((9)\) ensures that each color should be used. Conditions \((10)\) and \((11)\) ensure that every vertex totally dominates all vertices of at least one other class. Condition \((12)\) ensures that the assignment is a proper coloring. Conditions \((13)\) and \((14)\) ensure that each variable is boolean. Therefore, if each condition is satisfied, then \(G\) admits a total dominator coloring with \(k\) colors. \(\Leftarrow\) By the definition of the total dominator coloring, it is clear that Conditions \((1)-(7)\) defined for the graph \(G\) are satisfied.

Example 1. Let \(G\) be a vague graph depicted in Figure 1. Then the set of strong edges is \(\{(u_{1}u_{2}), (u_{2}v_{2}), (u_{2}u_{4}),(u_{2}u_{3}), (u_{3}u_{4}), (u_{4}v_{3}), (u_{4}u_{5}), (u_{5}v_{4}), (u_{5}u_{6}), (u_{6}u_{7}),(v_{1}v_{2}), (v_{2}v_{3}), (v_{3}v_{4}), (v_{4}v_{5})\}\).

Example 2. Let \(G\) be a vague graph depicted in Figure 2. Then by solving the instance from Dominator Coloring ILP, we obtain \(\gamma^d(G) = 6\). A dominator coloring \(f\) with \(6\) colors is \(f(u_{1}) = 1, \ f(u_{2}) = 2, \ f(u_{3}) = 4, \ f(u_{4}) = 1, f(u_{5}) = 4, f(u_{6}) = 6, \ f(u_{7}) = 1, f(v_{1}) = 3, \ f(v_{2}) = 1, \ f(v_{3}) = 4, \ f(v_{4}) = 5, \ f(v_{5}) = 1\) which is presented in Figure 2.

Example 3. Let \(G\) be a vague graph depicted in Figure 3. Then by solving the instance from Total Dominator Coloring ILP, we obtain \(\gamma^d_{t}(G) = 7\). A dominator coloring \(f\) with \(7\) colors is \(f(u_{1}) =7, \ f(u_{2}) = 1, f(u_{3}) = 7, f(u_{4}) = 4, \ f(u_{5}) = 6, f(u_{6}) = 5, f(u_{7}) = 7, \ f(v_{1}) = 7, \ f(v_{2}) = 2, f(v_{3}) = 7, \ f(v_{4}) = 3, f(v_{5}) = 7\) which is presented in Figure 3.

4. Dominator coloring number of some classes of vague graphs

Definition 7. For a vague graph \(G\), we define an underlying graph of \(G\), denoted by \(\widetilde{G}\), with \(V(\widetilde{G}) = V (G)\), and \(xy \in E(\widetilde{G})\) if and only if \(x \in N(y)\) in \(G\).

By the definition of Dominator coloring, we have

Proposition 8. For any vague graph \(G\), \(\chi^d(G) = \chi^d(\widetilde{G})\) and \(\chi^t_{d}(G) = \chi^t_{d}(\widetilde{G})\).

Proposition 9.(see [28]) For any vague graph \(G\) with \(\widetilde{G} \cong K_{n},\) we have \(\chi_{d}(G) = \chi^t_{d}(G) = n\).

Proof. By the definition, we have \(\chi^t_{d}(G) \geq \chi_{d}(G) \geq \chi(G) = n\). Let \(V(G) = \{v_{1}, v_{2}, \ldots, v_{n}\}\). We consider a function \(f : V(G) \rightarrowtail \{1, 2, \ldots , n\} \) with \(f(v_{i}) = i\) for any \(i\), then we have \(f\) is a total dominator coloring of \(G\) with \(n\) colors. Therefore, we have \(\chi^t_{d}(G) \leq n\) and so the desired result holds.

Proof. The following results are straightforward:

Proposition 10(see [28]) For any vague graph \(G\) with \(\widetilde{G} \cong K_{m,n}\), we have \(\chi^d(G) = \chi^t_{d}(G) = 2\).

Proposition 11.(see [28]) For any vague graph \(G\) with \(\widetilde{G} \cong C_{n} (n \geq 3)\), we have \(\chi^d(G) = 3\) for \(n \equiv 3\) (mod 6) and \(\chi_{d}(G) = 2\) otherwise.

Proposition 12.(see [35]) For any vague graph \(G\) with \(\widetilde{G} \cong P_{n} \ or \ C_{n} (n \geq 3)\), we have \(\chi^t_{d}(G) = \bigg[ \frac{n}{2}\bigg] + \bigg[ \frac{n}{4}\bigg] - \bigg[ \frac{n}{4}\bigg]\).

The Cartesian product \(G \Box H\) of two graphs \(G\) and \(H\) is a graph with \(V(G) \times V (H)\) and two vertices \((g_{1}, h_{1})\) and \((g_{2}, h_{2})\) are adjacent if and only if either \(g_{1} = g_{2}\) and \((h_{1}, h_{2}) \in E(H)\), or \(h_{1} = h_{2}\) and \((g_{1}, g_{2}) \in E(G)\). Let \(V(C_{n}) = \{1, 2, 3, \dots n\}\), \(E(C_{n}) = \{i(i + 1)\}, \ 1n|i = 1, 2, \ldots n - 1\}\) and \(V (P_{2}) = \{1, 2\}, \ E(P_{2}) = \{(12)\}\). Let \(u_{i,j}\) be a vertex of \(P_{2} \Box C_{n}\) where \(i = 1, 2, \ j = 1, 2, \ldots n\). We have the following result:

Proposition 12. For any vague graph \(G\) with \(\widetilde{G} \cong P_{2} \Box C_{n}\) with \(n \geq 6\), \[ \gamma^d_{t}(G) \leq \begin{cases} \frac{2n}{3} + 2, & n \equiv 0 \ \ \ \ (mod \ 6) \\ \frac{2n}{3} + 4, & n \equiv 1,2 \ (mod \ 6) \\ \frac{2n}{3} + 3, & n \equiv 3 \ \ \ \ (mod \ 6) \\ \frac{2n}{3} + 2, & n \equiv 4,5 \ (mod \ 6) \\ \end{cases} \]

Proof. We use two lines of numbers to denote a total dominator coloring of \(P_{2} \Box C_{n}\). The total dominator coloring can be represented as a \(2 \times n\) array as follows: \[ f(P_{2} \Box C_{n}) = \begin{cases} f(u_{1,1}) \ f(u_{1,2}) \ldots f(u_{1,n-1}) \ f(u_{1,n})
f(u_{2,1}) \ f(u_{2,2}) \ldots f(u_{2,n-1}) \ f(u_{2,n})
\end{cases} \] If \(i = 1\) and \(j \equiv 1, 3\) (mod 6), let \(f(u_{i,j}) = 1\).
If \(i = 1\) and \(j \equiv 2, 4\) (mod 6), let \(f(u_{i,j}) = 2\).
If \(i = 1\) and \(j \equiv 5\) (mod 6), let \(f(u_{i,j}) = 4 \frac{j}{6} + 5\).
If \(i = 1\) and \(j \equiv 0\) (mod 6), let \(f(u_{i,j}) = 4\frac{j}{6} + 6\).
If \(i = 2\) and \(j \equiv 4, 5\) (mod 6), let \(f(u_{i,j}) = 1\).
If \(i = 2\) and \(j \equiv 0, 5\) (mod 6), let \(f(u_{i,j}) = 2\).
If \(i = 2\) and \(j \equiv 2\) (mod 6), let \(f(u_{i,j}) = 4\frac{j}{6} + 3\).
If \(i = 2\) and \(j \equiv 3\) (mod 6), let \(f(u_{i,j}) = 4\frac{j}{6} + 4\).
We will consider the following cases:
Case 1. \(n \equiv 0\) (mod 6).
Obviously, \(f\) is total dominator coloring with desired number of colors. For example, let \(n = 12\), we have \[ f(P_{2} \Box C_{12}) = \begin{cases} 1 \ 2 \ 1 \ 2 \ 5 \ 6 \ 1 \ 2 \ 1 \ 2 \ 9 \ 10 \\ 2 \ 3 \ 4 \ 1 \ 2 \ 1 \ 2 \ 7 \ 8 \ 1 \ 2 \ 1 \end{cases} \] Case 2. \(n \equiv 1, 2, 4, 5\) (mod 6).
Let \(h(x) = f(x)\) for any \(x \in V (P_{2} \Box C_{n}) \backslash \{u_{1,n}, u_{2,n}\}\), \(h(u_{1,n}) = 2 \times \frac{n}{3}, \ h(u_{2,n}) = 2 \times \frac{n}{3} + 4\). Obviously, \(h\) is total dominator coloring with desired number of colors. For example, let \(n = 13\), we have \[ f(P_{2} \Box C_{13}) = \begin{cases} 1 \ 2 \ 1 \ 2 \ 5 \ 6 \ 1 \ 2 \ 1 \ 2 \ 9 \ 10 \ 11 \\ 2 \ 3 \ 4 \ 1 \ 2 \ 1 \ 2 \ 7 \ 8 \ 1 \ 2 \ 1 \ 12 \end{cases} \] Let \(n = 14\), we have \[ f(P_{2} \Box C_{14}) = \begin{cases} 1 \ 2 \ 1 \ 2 \ 5 \ 6 \ 1 \ 2 \ 1 \ 2 \ 9 \ 10 \ 1 \ 11 \\ 2 \ 3 \ 4 \ 1 \ 2 \ 1 \ 2 \ 7 \ 8 \ 1 \ 2 \ 1 \ 2 \ 12 \end{cases} \] Let \(n = 16\), we have \[ f(P_{2} \Box C_{16}) = \begin{cases} 1 \ 2 \ 1 \ 2 \ 5 \ 6 \ 1 \ 2 \ 1 \ 2 \ 9 \ 1 0 \ 1 \ 2 \ 1 \ 13 \\ 2 \ 3 \ 4 \ 1 \ 2 \ 1 \ 2 \ 7 \ 8 \ 1 \ 2 \ 1 \ 2 \ 11 \ 12 \ 14 \end{cases} \] Let \(n = 17\), we have \[ f(P_{2} \Box C_{17}) = \begin{cases} 1 \ 2 \ 1 \ 2 \ 5 \ 6 \ 1 \ 2 \ 1 \ \ 2 \ 9 \ 10 \ 1 \ 2 \ 1 \ 2 \ 13 \\ 2 \ 3 \ 4 \ 1 \ 2 \ 1 \ 2 \ 7 \ 8 \ 1 \ 2 \ 1 \ 2 \ 11 \ 12 \ 1 \ 14 \end{cases} \] Case 3. \(n \equiv 3\) (mod 6).
Let \(h(x) = f(x)\) for any \(x \in V (P_{2} \Box C_{n}) \backslash \{u_{1,n}\}, \ h(u_{1,n}) = \frac{2n}{3}+ 3\). Obviously, \(h\) is total dominator coloring with desired number of colors. As an example, let \(n = 15\), we have \[ f(P_{2} \Box C_{15}) = \begin{cases} 1 \ 2 \ 1 \ 2 \ 5 \ 6 \ 1 \ 2 \ 1 \ 2 \ 9 \ 10 \ 1 \ 2 \ 13 \\ 2 \ 3 \ 4 \ 1 \ 2 \ 1 \ 2 \ 7 \ 8 \ 1 \ 2 \ 1 \ 2 \ 11 \ 12 \end{cases} \] Now the proof is complete.

5. Conclusion

Fuzzy graph theory has substantial applications for real-world life in different domains, such as in the fields of biological science, neural networks, decision making, physics and chemistry. At present, the graph coloring problem can be applied in sequencing, timetabling, scheduling, electronic bandwidth allocation, computer register allocation and printed circuit board testing. Also the domination is also one of the fundamental concepts in graph theory and it has been wide used to distributed computing, biological networks, resource allocation and social networks. In this paper, motivated with the combination of fuzzy graph theory, graph coloring and graph domination, we introduce the concept of the dominator coloring and total dominator coloring of a vague graph and establish mathematical modelling for these problems.

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

Three phase SAASMs are workhorse of all industries due to its versatility and robustness. However, such machines cause limitations on exceeding that cause premature failure of the stator, rotor, and shaft. Main short-comings in SAAMs includes: (a) stator winding fault failures, (b) shaft eccentricity failure, and (c) rotor faults. Damages in SAASMs are due to following reasons:

• Due to defective bearings, rotor starts vibration and results in damage of machine.
• Turns of stator winding are insulated from each other by di-electric medium, when insulation failure or short circuit occurs and machine stops working. This type of fault results in crack or damage to rotor bars where it touches end rings [1, 2].

Numerous literature reports and reviews exist on SAASMs, for example Zhang et al. discussed fault detection and diagnosis system that can detect different type of faults in [3]. Stator current and time are used as inputs to system and direct torque control technology (DTC) is used as technical control in drive system. Time is factor that plays an important role for both detection of fault and selection of most appropriate corrective action in accordance with type of fault [4]. Benefits of DTC include: (a) relatively simple construction, and (b) very good flow and torque control performance. There is no transformation in power modulation block and no current control loops. Although discussed technology has some drawbacks such as: (a) variable switching frequency, (b) operating at low speed, and (c) high torque ripples. Vijayakumar et al. presented simulation of most common faults in three-phase squirrel cage machines with finite element method in [5]. Finite elements software is used to graphically record: (a) electric and magnetic fields, and (b) waveform of flux density distribution in air gap and electromagnet core. Analysis and data processing of aforementioned software is done using MATLAB. Moreover, broken rotor winding and inter-turn short circuit stator faults are studied in [1]6}. Results of this study show that broken rotor bars and inter-turn short circuit affects flux density in air gap. Table 1 illustrates the various stresses of the machine with respect to stator.

Table 1. Various stresses effects on SAASMs

		Stator	Rotor
Electrical Stresses		Dielectric aging	Shear
		Corona	Loose lamination
		Transients	Thermal overload
		Tracking
Mechanical Stresses		Rotor strikes	Casting vibration
		Flying objects	Loose lamination
			Incorrect shaft\core fit
			Geometry
			Material deviations
			Part breakage
Thermal Stresses		Thermal aging	Thermal overload
		Voltage variation	Thermal unbalance
		No proper Ventilation	Excessive rotor losses
			Sparking
Environmental Stresses		Moisture	Contamination
		Chemical	Temperature
		Damaged parts
Dynamic Stresses			Vibration
			Rotor rub
			Centrifugal force
			Over speeding

The abbreviation used in this paper are explained in Table 2. Important parameters that were discussed in the selected research papers are listed below in the comparative tabular analysis. In Table 3, '✔' justifies the presence of feature while '✖' justifies that feature is not discussed in the referred study.

Table 2. Abbreviations

Abbreviation	Full-form	Abbreviation	Full-form
BB	Broken Bars	BF	Bearing Faults
DE	Dynamic Eccentricity	SCS	Stator Current Sensor
SF	Shaft Faults	RCS	Rotor Current Sensor
BR	Broken Rings	SST	Stator shorted Turns
VCS	Vibration and Current Sensors	ECS	Eccentricity Current Sensor
SFM	Stray Flux Measurements	EVS	Eccentricity Vibration Sensor
ACS	Air Coil Sensor	BRB	Broken Rotor Bars
BVS	Bearing vibration sensor	BCS	Bearing Current Sensor

Table 3. Summary of some generic state of art surveys

Papers	BB	SST	BVS	BF	BCS	ACS	BR	SCS	DE	RCS	BRB	EVS	SF	ECS	VCS	SFM
[2]	✔	✔	✖	✔	✖	✖	✖	✔	✖	✔	✖	✔	✔	✖	✖	✔
[3]	✔	✖	✖	✔	✔	✖	✖	✖	✖	✔	✔	✖	✔	✔	✖	✖
[6]	✖	✔	✔	✖	✖	✖	✖	✔	✔	✖	✖	✔	✖	✖	✔	✖
[7]	✔	✖	✖	✔	✖	✖	✖	✔	✖	✔	✔	✖	✖	✔	✖	✖
[8, 9]	✔	✔	✖	✔	✖	✖	✔	✔	✖	✔	✖	✖	✖	✖	✔	✔
[10]	✖	✔	✖	✔	✔	✖	✔	✔	✖	✔	✖	✖	✖	✔	✔	✔
[11]	✖	✔	✖	✖	✔	✖	✖	✔	✖	✔	✖	✖	✖	✔	✖	✔
[12]	✔	✖	✖	✖	✔	✖	✔	✔	✖	✔	✔	✔	✔	✖	✖	✖
[13]	✖	✖	✖	✖	✔	✖	✖	✔	✔	✔	✔	✖	✖	✔	✖	✔
[14]	✖	✔	✖	✖	✖	✔	✔	✔	✔	✔	✖	✖	✔	✔	✖	✖
Our Survey	✔	✔	✔	✔	✔	✔	✔	✔	✔	✔	✔	✔	✔	✔	✔	✔

2. SAASMs rotor faults

Three phase IMs are extensively employed in industrial and domestic applications and is helpful for practical implementation of SG due to its efficiency, durability, and robustness. Successful and safe operation of electric machines is important for maximum economy and high efficiency in industry. Monitoring of machine parameters in industry is very important to recognize and diagnose failures of electrical machine. IM performance is affected by the defects, namely: (a) electrical failures, (b) mechanical failures, and (c) environmental disturbances. Electrical failures consist of (a) asymmetric faults, (b) under voltage fault, (c) overload, and (d) fault earth leakage. Rotor winding failure, failure of stator winding and bearings are mechanical faults. Moisture, contamination, temperatures, and external vibrations also affects motor performance and considered as environmental disturbances.

Statistics of IM failures and faults due to rotor are 10% [1] and most common cause of rotor failure is broken rotor bars, as shown in Figure 1. Causes of broken rotor bar fault of squirrel cage IM are thermal and mechanical. Thermal stresses arise due to design and construction, while mechanical stresses arise because of magnetic force and vibrations.

Manufacturing defects like weak joints at ends of bar, internal stresses (thermal and electromagnetic) and external constraints (frequent and overloading starts) are major causes of rotor faults. Other reasons of rotor failure are thermal overload and unbalanced mechanical stresses due to lose slots and bearing failure, magnetic stresses incurred by electromagnetic forces, dynamic loads generates thermal stresses, centrifugal forces and residual stresses are due to vibrations. These causes play pivotal part because their symptoms are detected by conventional measurement systems and affect rotating characteristics of motor [3].

2.1. SAASMs rotor fault diagnosis

Rotor bar is considered as most severe faults in SAASMs. Normally, these types of fault occur in heavy motors which are used in industries to drive large loads having higher inertia. Generally, these motors are very expensive and needed to maintain with no downtime. There are several causes for rotor bar breakage including centrifugal forces, electromagnetic forces, vibrations, thermal stress, electromagnetic noise, mechanical stress, fatigue parts, environmental stress, or often bearing faults. One important factor causing rotor bar is temperature gradient appearing in rotor cages and high current forces due to which current is interrupted through bars. Normally, this type of fault starts from the junction between short-circuit end rings and bars. Rotors of IM are of two types: (a) Cast, and (b) Fabricated used in squirrel-cage rotors. Cast rotors are used in high power rating machines up to 3000-kW and it's very difficult to repair after cracks or bar breakage. In case of cracked bar, current is increased by 50% of rated current in the bars. Fabricated cages are used for special application machines and for higher powers. Broken bar and cracked end-ring faults share only (5-10)% of induction machine faults. Detection of these faults can be done by different methods, including: (a) broken rotor bar detection-current signature analysis, (b) broken-rotor-bar-induced stator current harmonics, (c) Broken-rotor-bar-detection - fault severity analysis, (d) fault severity index, and (e) broken rotor bar detection - practical considerations. For example, MCSA method is used widely to find the end- rings fault and breakage of rotor bars. Failures of rotor bars are totally analyzed from current analysis drawn by the machine. Rotating field theory states that when any rotor rotates at infinite inertia and at a constant speed, it generates asymmetry component of \(f*(1-2s)\) in stator current spectrum. These symmetries are mostly aligned around fundamental frequency referred as side band components given by equation below: \begin{equation} f_{sh}=(1 \pm 2s) \end{equation} where, \(s\) is slip. To reduce these drawbacks, current drawn analysis of machine during transient operations is proposed by some authors also known as transient motor current signature analysis (TMCSA). In this aspect, methods based upon stator start-up current are recently introduced. Figure 2 and 3 present speed ripple effect on stator and rotor current of SAASMs due to rotor asymmetry and saturation respectively. Table 4 illustrates diagnostic techniques for different fault types and Table 5 illustrates effects of these aforementioned current components.

Table 4. Diagnostic techniques in term of adopted signal processing techniques and input signal

Type of Faults	Input Signal	Signal Processing	References
Rotor	Stator current	Frequency analysis	[5,12]
Rotor	Stator current	Signal injection	[8]
Stator	Flux	Frequency analysis	[15,16]
Stator	Stator current	Frequency analysis	[5]

Table 5. Effect of unbalanced voltages

Current Component	Estimation of \(n_{r}\)	References
\((1-2s) f\)	\( \frac{I_{l}}{I} = \frac{sin \gamma}{2P(2 \pi-\gamma)}\)	[17]
	\(\gamma = \frac{2 \pi}{N_{r}/P}n_{r} \)
\((1 \pm 2s) f\)	\(n_{r} = \frac{2N_{r}}{10-\frac{I_{db}}{20}+P}\)	[18]
\((1 \pm 2s) f\)	\( \frac{I^{'}_{l}}{I} = \frac{I_{l}+ I_{r}}{I}  \simeq \frac{n_{r}}{N_{r}}\)	[19]

2.2. SAASMs stator faults

Stator faults account for 38% of total faults of IM causing motor failure, and stator winding insulation is one of main sources of motor drive system failure. Percentage of motor drive system failures due to the stator winding insulation is about 26% [1]5}. Failure of electrical insulation system of electrical machine can be catastrophic and can result in injuries to personnel pose adverse effects on SG and also disturb economy due to shut-down processes.

2.3. Stator insulation failure

Stator insulation failure can occur due to different modes and patterns of failure, such as: (a) phase-to-ground failure mode, (b) turn-to-turn, (c) coil-to-coil, (d) phase-to-phase, (e) short-circuits, and (f) open-circuits. Phase to ground failure mode is most severe out of all failure modes. To find reason behind occurrence of failure mode, we need to analyze its pattern that is whether it has occurred in a single phase, is symmetrical or asymmetrical, includes or excludes grounding [1]5}. Figure 4 illustrates aforementioned failure modes.

2.3.1. Turn-to-turn failure mode

Turn-to-turn failure mode is one in which two or more turns of coil become short-circuited. Short-circuit compels current in turns to become much higher than normal operating current. Such a high current can cause temperature of stator winding to increase to levels where it can cause severe damage to insulation of winding. Majority of insulation failures starts with turn-to turn failure mode, and result in more serious insulation problems.

2.3.2. Coil-to-coil failure mode

In coil-to coil failure mode, two or more coils from the same phase get short-circuited, and this can be a result of turn-to-turn fault. Coils are made of several materials such as copper, plastic, steel, brass and aluminum and each having different thermal expansion coefficient. As coil cools and heats in normal operation, their thermal expansion coefficient work against each other and effectively try to tear the coil apart. Coil failure may also occur as a result of shorted winding in either primary or secondary coil. Mechanical damage should also be other failure mode to coil itself. Coils with obvious mechanical damage like cracked and broken connectors should be replaced. Except thermal cycling and damage, it may cause, there is no wear-out factor in the coil itself. Boots and wires in coil do age that result in reduced insulation with passage of time. The majority of factors related to premature induction coils failures are: (a) Operation related factors, (b) Coil material related factors, (c) design, fabrication and maintenance related factors, and (d) Accessories and tooling. Operation related factors are: (a) localized coil areas with high current density concentration, (b) inappropriate frequency, (c) improper cooling and coil overheating, (d) quenching media, (e) coil arcing, and (f) thermal expansion. Coil material related factors are: (a) copper grade selection, (b) stress fatigue cracking, (c) copper work hardening, and (d) copper erosion and pitting. Improper design, fabrication and maintenance can also cause coil-to-coil failures. Taking into consideration the aforementioned factors properly can lessen chances of coil-to-coil failures. Last group pointing out the factors related to tooling and accessories are: (a) magnetic flux concentrators, (b) quenching devices, (c) coil self-relocation, and (d) improper bus tightness.

2.3.3. Phase-to-phase failure mode

In phase-to-phase failure, two or more phases become short-circuited. These failure modes can subsequently lead to phase-to-ground failure due to which machine is severely damaged. Probability of occurrence of an open-circuit fault is very small. Similar to short circuit, an open-circuit initiates asymmetry in machine, thus causing machine to malfunction.

Apart from investigating the failure mode and pattern of the machine, other factors such as the cleanliness of the machine, existence of foreign particles, amount of moisture, and the rotor condition should be taken into account. Furthermore, the operating condition under which the motor failed should be compared to the normal operating condition of the motor. The schedule of machine maintenance should also be considered. The various factors that can cause the winding insulation to deteriorate include thermal, mechanical, and electrical stresses.

2.4. Thermal stresses

Winding insulation has to undergo thermal aging referred as thermal stresses. Rise in temperature aggravates thermal aging process of winding insulation and causes it to deteriorate even further, thus reducing its lifetime. It is observed that an increase of \(10^{o}\) in the temperature causes 50% decrease in the insulation's life [4].

In order to mitigate the effects of thermal aging, either operate motor at lower temperatures or use higher insulation class. Thermal overloading is another class of thermal stress posing detrimental effect on winding insulation. Variations in voltage, obstructions in ventilation, and imbalances in phase voltages, overloading, cycling and surrounding temperature are main causes of thermal overloading. During start-up, current is usually 3 to 5 times larger than normal operating current due to which there is rise in winding temperature. Therefore, it should be ensured that motor is not repeatedly started. An increase in load can also be a contributing factor in problem of thermal stress. Hence, load on motor that will drive must be kept in mind while designing insulation system and it must have much greater rating than normal working temperature insulation system. It is observed that if there is a voltage imbalance of 3.5% per phase, it can cause a 25% increase in temperature of phase with highest current.

To overcome effects of thermal overloading, proper ventilation of motor should be ensured so that heat produced can be dissipated without causing winding temperature to increase. If this is not possible then a better insulation with a higher rating should be chosen for stator winding.

2.5. Mechanical stresses

Mechanical stress causes damage to insulation when rotor strikes stator, or due to coil movement. Force exerted on winding coils is directly proportional to square of current in motor and attains its maximum value at motor start-up. Due to this force, coils begin to vibrate and move, causing serious damage to conductor and insulation of coil. Misalignment between rotor and stator, bearing faults and shaft deflection can cause rotor to strike stator. This occurs mostly during start-up, but it can also happen when motor is operating at maximum speed.

2.6. Electrical stresses

Electrical Stresses in motor occur due to various factors such as: (a) issues with dielectric material, (b) transient voltages, (c) effects of corona, and (d) tracking. Voltage stresses along with material used for dielectric in turn-turn, phase-phase, and phase-ground insulation greatly affect lifetime of winding insulation. Therefore, to achieve required insulation lifetime and smooth operation of motor, material for dielectric has to be chosen carefully. Effects of tracking occur only above voltage level of 600V. Transient voltages can either cause damage to winding or even complete turn-turn or turn-ground failures. Transient voltages occur due to various faults such as: (a) line-line, (b) line-ground or multi-phase line-ground faults, (c) failure of power system insulation, (d) repetitive striking, and (e) rapid bus transfer. These transients can also occur when lightning strikes, circuit breakers open and close or when switching capacitors are employed for power factor improvement. Permanent voltage transients are produced when variable frequency drives are operating, especially during starting and stopping of drives [4].

3. Fault detection schemes in SAASMs

3.1. Fault detection schemes in IM

Predictive methods are implemented in induction machines to overcome the accidental breakout. Techniques which are frequently used for the fault diagnosis in machines are (a) Speed oscillations (b) circuit analysis (c) partial discharge and (d) vibration analysis. Due to high requirement for fault diagnosis in heavy industries Electrical signature analysis (ESA) caught huge attraction of researchers to this matter. Motor current signature analysis (MCSA), vibration analysis, Instantaneous power signature analysis (IPSA) and Extended Park's vector approach (EPVA) techniques are also frequently employed by the different consultants or dedicated groups to gain better results for fault tracing in industries to enhance the efficiency of IMs. Any kind of fault in IMs can disturb its symmetry which results in to change the flux interaction between rotor and stator. Furthermore, this flux changing phenomena results in change of voltages, machine vibration, electromagnetic torque, and stator currents.

3.2. Fault detection scheme in SM

Different fault detection schemes are presented in literature for smooth operation of synchronous machines (SMs). These schemes provide helps regarding: (a) Ground faults detection, (b) Eccentricity faults detection, (c) Inter-turn faults detection, and (d) Phase-fault detection.

3.2.1. Ground fault detection

Ground-fault-detection technique for synchronous machines is presented in [16]. This technique is appropriate for synchronous machines with static excitation system and excitation field winding fed by rectifiers via excitation transformer. The said technique contributes in a way that it can detect both ac and dc side ground faults in excitation without having requirement of conventional power injection sources.

3.2.2. Eccentricity fault detection

In [19, 20], standard short circuit test is proposed for the salient pole synchronous machine to monitor the static eccentricity (SE) and dynamic eccentricity faults. The above-mentioned method possess the advantage that it based on offline current signature analysis (OFCSA) in which machine disassembly does not requires. In this technique, SE fault diagnosis is gained by residual estimation method for monitoring where DE is observed by manipulation of SE.

3.2.3. Inter-turn fault detection

Faulty machine can shut-down overall system or operation in an industry, health monitoring of machine is attracting a lot of attention recently. Due to robustness and advantage in power density in wind power generation, Permanent magnet synchronous machine (PMSM) is extensively employed due to its high robustness level and better efficiency. On-line machine monitoring fault is quite good due to fact that it is uneasy to dismantle machine once installed and cause the machine to displace. Proposed method is on-line stator winding inter-turn fault detection using electromagnetic field signature and the proposed method assists in operating movable electrical machine.

3.2.4. Phase Fault Detection

A new strategy for phase fault detection of five phase permanent magnet synchronous reluctance motor (PMa-SynRM) is discussed in [20]. This fault detection technique is developed using novel decomposition process of sequential components of a five-phase electrical machine. An advance developed technique symmetrical component analysis (SCA) is implemented to trace out the phase faults in five phase or poly phase machines.

4. Conclusions and Future Work

Our work discussed in detail the several faults of synchronous and asynchronous machines. Induction machine is one of the most important machines playing a vital role in almost every industry due to its robustness and rigid nature. There are some common issues incurring in SAASMs as discussed in details in this paper. We investigated various rotor faults and stator faults occurring frequently in the SAASMs and affecting the efficiency of the machines resulting in a huge loss of power and uneconomical phenomenon. We also discussed several stresses that are thermal stress, mechanical stress, electrical stress and eccentricity faults and their diagnosis are also enlisted whereas fault detection techniques used in IMs and asynchronous machines are also presented. In near future, we will investigate the opening of stator phase, broken end rings, bowed shaft, broken rotor bars, offbeat rotor and bearing damage. To critically analyze the health of machine by any method, a huge information is required. For that reason, the modelling and simulation of induction motor under several conditions will be carried out in the MATLAB/SIMULINK software incorporating the stator intern-turn and broken bar conditions.

Acknowledgments

We would thankful to Prof. Dr. Muhammad Ali for providing valuable comments, suggestions and reviews for our final manuscript.

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.