Probe of Radiant Flow on Temperature-Dependent Viscosity Models of Differential Type MHD Fluid

This paper numerically investigates the combined eﬀects of the radiation and MHD on the ﬂow of a viscoelastic Walters’ B liquid ﬂuid model past a porous plate with temperature-dependent variable viscosity. To study the eﬀects of variable viscosity on the ﬂuid model, the equations of continuity, momentum with magnetohydrodynamic term, and energy with radiation term have been expanded. To understand the phenomenon, Reynold’s model and Vogel’s model of variable viscosity are also incorporated. The dimensionless governing equations are two-dimensional coupled and highly nonlinear partial diﬀerential equations. The highly nonlinear PDEs are transferred into ODEs with the assistance of suitable transformations which are solved with the help of numerical techniques, namely, shooting technique coupled with Runge–Kutta method and BVP4c solution method for the numerical solutions of governing nonlinear problems. Viscosity is considered as a function of temperature. Skin friction coeﬃcient and Nusselt number are investigated through tables and graphs in the present probe. The behavior of emerging parameters on the velocity and temperature proﬁles is studied with the help of graphs. For Reynold’s model, we have shrinking stream lines and increasing three-dimensional graphs. c and Pr are reduced for both models.


Introduction
Non-Newtonian fluids have been a subject of great interest to researchers recently because of their various applications in industry and engineering. is is due to distinctive characteristics of such fluids in nature. In general, the mathematical problems in non-Newtonian fluids are more complicated because they are nonlinear and higher order than those in viscous fluids. Despite their complexities, scientists and engineers are engaged in non-Newtonian fluid dynamics. e analysis of boundary layer flow of viscous and non-Newtonian fluids has been the locus of extensive research by various scientists due to its importance in continuous casting, glass blowing, paper production, polymer extrusion, aerodynamic extrusion of plastic sheet, and several others. Rajagopal et al. [1] have focused their research towards non-Newtonian fluid flows due to stretching of a flat surface. As far as literature survey is concerned many researchers  have worked on MHD radiation effects of viscous fluids.
Effects of thermal diffusion and chemical reaction on MHD flow of a dusty viscoelastic fluid have been inspected by Prakash et al. [34]. Abdul Hakeem et al. [24] have found the effect of heat radiation in Walters' B fluid over a stretching sheet with nonuniform heat source/sink and elastic deformation. Recently, unsteady free convection flow in Walters' B fluid and heat transfer analysis have been presented by Khan et al. [24]. Wang and Ng [25] investigated a similar flow to the present study but for an electrically nonconducting fluid and outside a magnetic field.
Uddin et al. [26] studied MHD flow bounded by a nonlinearly stretching surface with radiation. Brownian motion and thermophoresis in magnetohydrodynamic (MHD) bioconvection flow of nanoliquid via nonlinear thermal radiation is addressed by Makinde and Animasaun [27]. e hydromagnetic pivot flows of an Oldroyd-B fluid in a porous medium was discussed by Khan et al. [28]. Khan et al. [29] studied the heat and mass transfer of viscoelastic MHD flow over a porous magnifying sheet with degeneration of energy and stress work. e flows of Walters' B fluid for numerical or applicable results for both steady and transitory at great length in a distinct range of geometries using broad scale of analytical or computational approaches have been studied [30][31][32]. Prakash et al. [33] inspected the effects of chemical reaction and thermal diffusion on the MHD flow of a dusty viscoelastic fluid. e effect of heat radiation in Walters' B fluid over a magnifying sheet with nonuniform elastic deformation and heat source was found by Abdul Hakeem et al. [34]. Khan et al. [24] represented the unsteady free convection flow in heat transfer analysis and Walters' B fluid. Under different pressure gradients the thermal effects of a dusty viscoelastic fluid on unsteady fluid between two parallel plates was studied by Madhurai and Kalpana [35]. e objective here is to study numerically the combined effects of the radiation and MHD on the flow of a viscoelastic fluid model past a porous plate with temperature dependent variable viscosity. e problem is divided into two different parts in which the first part explicates that the plate has greater temperature than fluids temperature. e second part describes that plate is insulated. To understand the phenomenon, Reynold's model and Vogel's model of variable viscosity with magnetohydrodynamic and radiation effects of viscoelastic Walters' B non-Newtonian fluid flow are incorporated. e shooting technique is habituated to attain the numerical solution of arising governing equations and solved with BVP4 software of Maple program. ree-dimensional and stream lines graphs were enlarged and reduced, respectively. e behavior of emerging parameters on the velocity and temperature profiles is studied with the help of graphs.

Mathematical Equations
e Cauchy stress tensor for Walters' B fluid is given by where pressure of the fluid is pand Here, N(τ) is distribution function with relaxation time τ.

Physical Modeling of the Problem
e governing equations are For an incompressible fluid, (4) takes the form where v 0 > 0 is suction and v 0 < 0 represents blowing at the plate. Momentum equation under the effects of magnetohydrodynamics for the current problem is Define pressure as modified as Equations (8)-(10) with magnetohydrodynamic effects formed as Equation (12) can be written as e boundary conditions are As we have 3 rd -order equation (15), so we need another boundary condition. erefore, in free stream, We use the following conditions: and also take another assumption that Now, we are going to discuss the heat transfer in (6).
where q is heat flux. e radiation parameter is C p is specific heat and the boundary conditions for (21) are given in two parts as follows: is part gives conditions for constant wall temperature of the fluid: Case 2: is gives insulated wall of the fluid

Solution for Constant Wall
Temperature. e dimensionless parameters can be defined as where is the characteristic "length" and also Using the above relations, (15) and (23) become For simplicity, the bars are removed from (29)-(30) and get Mathematical Problems in Engineering Here, c is dimensionless length, M is MHD term coefficient, R is radiation, Pr is Prandtl number and λ is dimensionless quantity. e dimensionless boundary conditions are

Solution for Insulated Plate.
Here, we introduce nondimensional temperature parameter where θ b is bulk temperature. Eckert number is e boundary conditions for dimensionless flow are e skin friction and Nusselt number [7] are expressed as where C f is skin friction coefficient and Nu is Nusselt number. Also, and by using similarity transformation, we get Here, Re represents Reynold number.

Reynold's Model
e viscosity for this model is expressed as which can be solved by using Maclaurin's series as Using the value of η 0 in (31) and (32), we obtain

Vogel's Model
In this case, which implies the following.
e above equation can be written in the form Using (47) in (31) and (32), we get

Numerical Solution
For the purpose of numerical investigation, we have made comparison of our current article with three previous publications, which shows our results in this study are better than the previous literature [4,12,20] . e solution for (44) and (45) and (48) and (49) Now, we define new variables, By using new variables, we get

Mathematical Problems in Engineering
Velocity Profile

Mathematical Problems in Engineering
Along with boundary conditions, (53)

Solution for Vogel's Model. In this solution, (48) and (49) are
As previous case, with the same boundary conditions as in (53).

Graphical Results and Discussion
In graphical portray, Figure 1 explains the physical geometry of the problem. Figure 2 gives portray of  profile. Figure 6 draws the consequences of D � 0.3, 1.5, 2.5, 3.0 on temperature distribution for Reynold's model. Figure 7 limns the impact of G * � 0.1, 0.17, 0.25, 0.4 on temperature profile for Vogel's model. Figure 8 tells the influence of G � 0.3, 1.4, 2.8, 4.5 for Vogel's model on temperature profile. Figure 9 represents c � 0.1, 2.5, 4.5, 6.0 on Vogel's model for velocity outline. Figure 10 shows the effect of λ � 0.1, 0.4, 0.7, 1.0 on Vogel's model for temperature. Figure 11 depicts impact of N � 1.0, 3.0, 5.0, 7.0 on Vogel's model for velocity profile. Figure 12 shows the behavior of R � 0.2, 0.6, 1.4, 4.0 on Vogel's model for temperature portray. Figure 13 gives the effects of N � 0.1, 1.2, 2.3, 3.4 on velocity profile for Reynold's model. Figure 14 shows the impact of c � 0.1, 0.2, 0.3, 0.4 for temperature field of Reynold's model. Figure 15 depicts the influence of c and N � 0.1, 0.5, 1.0 for Reynold's model's Nusselt number. Figure 16 represents the impact for Vogel's model on D � 0.1, 0.25, 0.5 and c for skin friction. Figure 17 shows effects for Vogel's model on Nusselt number for c and   Table 1: e values of change in c for temperature of Reynold's model at the wall. Table 3: e values of change in R for temperature of Reynold's model at the wall.  Table 4: e values of change in λ for temperature of Reynold's model at the wall.  Table 6: e values of change in N for temperature of Reynold's model at the wall. Table 7: e values of change in c for temperature of Vogel's model at the wall.              Table 18 shows the behavior of N and on skin friction coefficient for Reynold's model. Table 19 indicates the consequences of G, D, E, G * , N, c, λ and Pr for skin friction coefficient of Vogel's model.

Concluding Remarks
In this inquisition, the numerical solution of Walters' B fluid model with MHD and radiation effects of both time dependent viscosity models has been discussed. Influences of these parameters are presented with the help of graphs and tables. Some important points of the study of this problem are the following: (1) A sensible growth is seen in the velocity portray as increase in G and the velocity curve decreases with the enlargement of c, D and N for both models (2) c andPr are decreases for Reynold's well as Vogel's models (3) e stream lines are sighted to shrink and the 3 − D graphs bended with the increase in c of Reynold's model (4) Skin friction curve increases with the increase in N, while Nusselt number graph decreases with the enlargement in λ Data Availability e data used to support the study are available from the corresponding author upon request.