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

Authors: Azad Hussain; Sobia Akbar; Lubna Sarwar; Sohail Nadeem
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

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.γ and Pr are reduced for both models.


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## Body

## 1. Introduction

Non-Newtonian fluids have been a subject of great interest to researchers recently because of their various applications in industry and engineering. This 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. The 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 [2–23] 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]. The 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. The 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 [30–32]. Prakash et al. [33] inspected the effects of chemical reaction and thermal diffusion on the MHD flow of a dusty viscoelastic fluid. The 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]. The 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. The problem is divided into two different parts in which the first part explicates that the plate has greater temperature than fluids temperature. The 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. The shooting technique is habituated to attain the numerical solution of arising governing equations and solved with BVP4 software of Maple program. Three-dimensional and stream lines graphs were enlarged and reduced, respectively. The behavior of emerging parameters on the velocity and temperature profiles is studied with the help of graphs.


## 2. Mathematical Equations

The Cauchy stress tensor for Walters’ B fluid is given by(1)S=−pI+2η0e−2k0δeδt,where pressure of the fluid is pand(2)η0=∫0∞Nτdτ,k0=∫0∞τNτdτ.Here,Nτ is distribution function with relaxation time τ.


## 3. Physical Modeling of the Problem

The current problem gives the flow of a Walters’ B fluid flow past an infinite spongy plate.The temperature and velocity fields are(3)θ=θy,v=vyi+wyk.The governing equations are(4)∂ρ∂t+divρv=0,(5)ρdvdt=divS+ρb,(6)ρdedt=S.L−divq+ρr.For an incompressible fluid, (4) takes the form(7)divv=0,u=−v0=Constant,where v0 > 0 is suction and v0<0 represents blowing at the plate. Momentum equation under the effects of magnetohydrodynamics for the current problem is(8)∂P∂z=ρv0dwdx+dη0dydwdy+η0d2wdy2+k0v0d3wdy3−σB02w,(9)∂P∂y=2k0ddydwdy2,(10)∂P∂x=0.Define pressure as modified as(11)P^=P−2k0dwdy2.Equations (8)–(10) with magnetohydrodynamic effects formed as(12)∂P^∂z=ρv0dwdy+dμ0dydwdy+η0d2wdy2+k0v0d3wdy3−σB02w,(13)∂P^∂y=0,(14)∂P^∂x=0.Equation (12) can be written as (15)ρv0dwdy+dμ0dydwdy+η0d2wdy2+k0v0d3wdy3−σB02w=L1,where(16)∂P^∂z=L1.The boundary conditions are(17)w0=0,wy⟶W∞,asy⟶∞.As we have 3rd-order equation (15), so we need another boundary condition. Therefore, in free stream,(18)Syz|y⟶∞=η0d2wdy2+k0v0d3wdy3y⟶∞=0.We use the following conditions:(19)dwdy=0,asy⟶∞,and also take another assumption that(20)L1=0.Now, we are going to discuss the heat transfer in (6).(21)q=−kgradθ,where q is heat flux. The radiation parameter is(22)qr=−4σ∗3K∗∂θ4∂x.Then(23)kd2θdy2+ρCpv0dθdy+η0dwdy+k0v0dwdyd2wdy2−1ρCp∂qr∂y=0.Cp is specific heat and the boundary conditions for (21) are given in two parts as follows:Part 1:This part gives conditions for constant wall temperature of the fluid:(24)θ0=θ0,θx⟶0∞,asx⟶∞.Case 2:This gives insulated wall of the fluid(25)dθdy|y⟶0=0,θy⟶θ∞,asy⟶∞.


### 3.1. Solution for Constant Wall Temperature

The dimensionless parameters can be defined as(26)Y¯=YL,w¯=wW∞,θ¯=θ−θ∞θ0−θ∞,where(27)L=k0v0η0∗,is the characteristic “length” and also(28)η0¯=η0η0∗.Using the above relations, (15) and (23) become(29)d3w¯dy¯3+η0d2w¯dy¯2+γdw¯dy¯+dη0dy¯dw¯dy¯−γMw¯=L2,(30)1+43Rd2θ¯dy¯2+γPrdθ¯dy¯+λη0dw¯dy¯2+λdw¯dy¯d2w¯dy¯2=0.For simplicity, the bars are removed from (29)–(30) and get(31)d3wdy3+η0d2wdy2+γdwdy+dη0dydwdy−γMw=L2,(32)1+43Rd2θdy2+γPrdθdy+λη0dwdy2+λdwdyd2wdy2=0,where(33)γ=ρk0W∞2η0∗2,M=σB02K0ρη02,R=4σ∗θ∞3KK∗,Pr=η0∗Ck,λ=W∞2η0kθ0−θ∞.Here,γ is dimensionless length, M is MHD term coefficient, R is radiation, Pr is Prandtl number and λ is dimensionless quantity. The dimensionless boundary conditions are(34)w0=0,w⟶1asy⟶∞,dwdy⟶0asy⟶∞,θ0=1,θ⟶1asy⟶∞.


### 3.2. Solution for Insulated Plate

Here, we introduce nondimensional temperature parameter(35)θ∗=θ−θ∞θb−θ∞,where θb is bulk temperature. Eckert number is(36)E∗=W∞2cθb−θ∞.The boundary conditions for dimensionless flow are(37)dθdy|y⟶0=0,θy⟶θ∞asy⟶∞.The skin friction and Nusselt number [7] are expressed as(38)Cf=τw1/2ρW2,Nu=yqwkTw−T∞,where(39)Nu=−θ′0.Cf is skin friction coefficient and Nu is Nusselt number. Also,(40)τw=η0dwdy+k0v0d2wdy2y⟶0,qw=−k∂θ∂yy⟶0,and by using similarity transformation, we get(41)12CfRe=1−Nθw′0+w″0,Nu=−θ′0.Here,Re represents Reynold number.


## 3.1. Solution for Constant Wall Temperature

The dimensionless parameters can be defined as(26)Y¯=YL,w¯=wW∞,θ¯=θ−θ∞θ0−θ∞,where(27)L=k0v0η0∗,is the characteristic “length” and also(28)η0¯=η0η0∗.Using the above relations, (15) and (23) become(29)d3w¯dy¯3+η0d2w¯dy¯2+γdw¯dy¯+dη0dy¯dw¯dy¯−γMw¯=L2,(30)1+43Rd2θ¯dy¯2+γPrdθ¯dy¯+λη0dw¯dy¯2+λdw¯dy¯d2w¯dy¯2=0.For simplicity, the bars are removed from (29)–(30) and get(31)d3wdy3+η0d2wdy2+γdwdy+dη0dydwdy−γMw=L2,(32)1+43Rd2θdy2+γPrdθdy+λη0dwdy2+λdwdyd2wdy2=0,where(33)γ=ρk0W∞2η0∗2,M=σB02K0ρη02,R=4σ∗θ∞3KK∗,Pr=η0∗Ck,λ=W∞2η0kθ0−θ∞.Here,γ is dimensionless length, M is MHD term coefficient, R is radiation, Pr is Prandtl number and λ is dimensionless quantity. The dimensionless boundary conditions are(34)w0=0,w⟶1asy⟶∞,dwdy⟶0asy⟶∞,θ0=1,θ⟶1asy⟶∞.


## 3.2. Solution for Insulated Plate

Here, we introduce nondimensional temperature parameter(35)θ∗=θ−θ∞θb−θ∞,where θb is bulk temperature. Eckert number is(36)E∗=W∞2cθb−θ∞.The boundary conditions for dimensionless flow are(37)dθdy|y⟶0=0,θy⟶θ∞asy⟶∞.The skin friction and Nusselt number [7] are expressed as(38)Cf=τw1/2ρW2,Nu=yqwkTw−T∞,where(39)Nu=−θ′0.Cf is skin friction coefficient and Nu is Nusselt number. Also,(40)τw=η0dwdy+k0v0d2wdy2y⟶0,qw=−k∂θ∂yy⟶0,and by using similarity transformation, we get(41)12CfRe=1−Nθw′0+w″0,Nu=−θ′0.Here,Re represents Reynold number.


## 4. Reynold’s Model

The viscosity for this model is expressed as(42)η0=e−Nθ,which can be solved by using Maclaurin’s series as(43)η0=1−Nθ.Using the value ofη0 in (31) and (32), we obtain(44)d3wdy3+d2wdy2−Nθd2wdy2+γdwdy−Ndθdydwdy−γMw=L2,(45)1+43Rd2θdy2+γPrdθdy+λdwdy2−λNθdwdy2+λdwdyd2wdy2−1ρCp∂qr∂y=0.


## 5. Vogel’s Model

In this case,(46)μ0=η0∗expDE+θ−θw,which implies the following.The above equation can be written in the form(47)η0=−GG∗1−DθE2.Using (47) in (31) and (32), we get(48)d3wdy3+GG∗d2wdy2−DGG∗E2d2wdy2+γdwdy−DGG∗E2dθdydwdy−γMw=L2,(49)1+43Rd2θdy2+γPrdθdy+λGG∗dwdy2−λDGG∗E2θdwdy2+λdwdyd2wdy2−1ρCp∂qr∂y=0.


## 6. 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]. The solution for (44) and (45) and (48) and (49) is obtained by using shooting technique with Runge–Kutta method [23, 36–38].


### 6.1. Solution for Reynold’s Model

Equations (44) and (45) are for the desired form(50)w‴=−γw′−1−Nθw″+Nθ′w′−γMw,θ″=−11+4/3RγPrθ′−λw′2−λw′w″+Nθw′2−1ρCp∂qr∂y.Now, we define new variables,(51)w=s1,w′=s2,w″=s3,w‴=s3′,θ=s4,θ′=s5,θ″=s5′.By using new variables, we get(52)s1′=s2,s2′=s3,s4′=s5,s3′=−γw′−1−Nθw″+Nθ′w′−γMw,s5′=−11+4/3RγPrθ′−λw′2−λw′w″+Nθw′2−1ρCp∂qr∂y.Along with boundary conditions,(53)s10=0,s1∞=1,s20=1,s2∞=0,s40=1,s4∞=0.


### 6.2. Solution for Vogel’s Model

In this solution, (48) and (49) are(54)w‴=−γw′−1−DE2GG∗w″+DGG∗E2θ′w′−γMw,θ″=−11+4/3R−γPrθ′−1+DθE2GG∗λw′2−λw′w″−1ρCp∂qr∂y.As previous case,(55)s3′=−γw′−1−DE2GG∗w″+DGG∗E2θ′w″−γMw,s5′=−11+4/3R−γPrθ′−1+DθE2GG∗λw′2−λw′w″−1ρCp∂qr∂y.with the same boundary conditions as in (53).


## 6.1. Solution for Reynold’s Model

Equations (44) and (45) are for the desired form(50)w‴=−γw′−1−Nθw″+Nθ′w′−γMw,θ″=−11+4/3RγPrθ′−λw′2−λw′w″+Nθw′2−1ρCp∂qr∂y.Now, we define new variables,(51)w=s1,w′=s2,w″=s3,w‴=s3′,θ=s4,θ′=s5,θ″=s5′.By using new variables, we get(52)s1′=s2,s2′=s3,s4′=s5,s3′=−γw′−1−Nθw″+Nθ′w′−γMw,s5′=−11+4/3RγPrθ′−λw′2−λw′w″+Nθw′2−1ρCp∂qr∂y.Along with boundary conditions,(53)s10=0,s1∞=1,s20=1,s2∞=0,s40=1,s4∞=0.


## 6.2. Solution for Vogel’s Model

In this solution, (48) and (49) are(54)w‴=−γw′−1−DE2GG∗w″+DGG∗E2θ′w′−γMw,θ″=−11+4/3R−γPrθ′−1+DθE2GG∗λw′2−λw′w″−1ρCp∂qr∂y.As previous case,(55)s3′=−γw′−1−DE2GG∗w″+DGG∗E2θ′w″−γMw,s5′=−11+4/3R−γPrθ′−1+DθE2GG∗λw′2−λw′w″−1ρCp∂qr∂y.with the same boundary conditions as in (53).


## 7. Graphical Results and Discussion

In graphical portray, Figure1 explains the physical geometry of the problem. Figure 2 gives portray of N=0.1,2.2,4.3,6.4 for Reynold’s model on temperature field. Figure 3 shows the behavior of R=0.1,0.3,0.6,1.0 for Reynold’s model on temperature portray. Figure 4 renders effects of γ=0.1,2.5,3.0,3.4 for Reynold’s model on velocity field. Figure 5 describes effects of W=0.2,3.2,6.2,9.4 on Reynold’s model for velocity 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 γ=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 γ=0.1,0.2,0.3,0.4 for temperature field of Reynold’s model. Figure 15 depicts the influence of γ 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 γ for skin friction. Figure 17 shows effects for Vogel’s model on Nusselt number for γ and D=0.1,0.5,1.0. Figures 18–20 give the stream lines of γ=0.1,0.5,0.9 for Reynold’s model. Figures 21–23 tell 3−D structures of γ=0.1,0.5,0.9 for Reynold’s model. Tables 1–6 give the change in temperature for Reynold’s model on N,W,R,γ,λandPr at the wall. Tables 7–15 depict the change in temperature for Vogel’s model on G,D,E,G∗,R,N,γ,λandPr at the wall. Table 16 elucidates the ináuence of N,W,R,γ,λandPr on Nusselt number. Table 17 explicates the consequences of G,G∗,D,E,R,N,γ,λandPr for Vogel’s model. 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,γ,λandPrfor skin friction coefficient of Vogel’s model.Figure 1
Physical geometry of the problem.Figure 2
Effects ofN on temperature portray for Reynold’s model.Figure 3
Influences ofR on temperature outline for Reynold’s model.Figure 4
Impact of on velocity field for Reynold’s model.Figure 5
Effects ofW on velocity profile for Reynold’s model.Figure 6
Impact ofD on temperature outline for Vogel’s model.Figure 7
Effect ofG∗ on temperature field for Vogel’s model.Figure 8
Impact ofG on temperature profile for Vogel’s model.Figure 9
Influence of on velocity profile for Vogel’s model.Figure 10
Effect ofγ on temperature outline for Vogel’s model.Figure 11
Influences ofN on velocity field for Vogel’s model.Figure 12
Impact ofR on temperature outline for Vogel’s model.Figure 13
Effects ofN on velocity profile for Reynold’s model.Figure 14
Impact ofγon temperature field for Reynold’s model.Figure 15
Effects ofγand N on Nusselt number for Reynold’s model.Figure 16
Effects ofγand D on skin friction for Vogel’s model.Figure 17
Effects ofγand D on Nusselt number for Vogel’s model.Figure 18
Stream lines forγ=0.1 for Reynold’s model.Figure 19
Stream lines forγ=0.5 for Reynold’s model.Figure 20
Stream lines forγ=0.9 for Reynold’s model.Figure 21
The portray of3−D graph of Reynold’s model for γ=0.1Figure 22
The portray of3−D graph of Reynold’s model for γ=0.5Figure 23
The portray of3−D graph of Reynold’s model for γ=0.9Table 1
The values of change inγ for temperature of Reynold’s model at the wall.
γ0.10.20.30.4N=0.1R=0.6Pr=7.7−0.8769141−1.1638242−1:4937560−1.8544489λ=1.5W=0.2Table 2
The values of change inW for temperature of Reynold’s model at the wall.
W0.11.12.13.1N=0.1R=0.6Pr=7.7−0.8718237−0.9246222−0.9818317−1.04378λ=1.5γ=0.1Table 3
The values of change inR for temperature of Reynold’s model at the wall.
R0.61.01.41.8N=0.1W=0.2Pr=7.7−0.8769141−0.7838524−0.7274551−0.6896815λ=1.5γ=0.1Table 4
The values of change inλ for temperature of Reynold’s model at the wall.
λ1.52.02.53.0N=0.1W=0.2Pr=7.7−0.8769141−0.9210312−0.9650590−1.0089976R=1.5γ=0.1Table 5
The values of change inPr for temperature of Reynold’s model at the wall.
Pr7.78.89.910.10N=0.1W=0.2λ=2.5−0.9650590−1.0069910−1.0499959−1.0579279R=1.5γ=0.1Table 6
The values of change inN for temperature of Reynold’s model at the wall.
N0.10.40.71.0Pr=8.8W=0.2λ=2.5−1.0069910−1.0285733−1.0497506−1.0702968R=1.5γ=0.1Table 7
The values of change inγ for temperature of Vogel’s model at the wall.
γ0.10.30.50.7N=1.3R=0.1Pr=7.7−2.9582914−4.075563−5.3502953−6.6967343λ=1.7D=1.9E=2.0G∗=1.8G=1.1Table 8
The values of change inλ for temperature of Vogel’s model at the wall.
λ0.73.35.98.5N=1.3R=0.1Pr=7.7−4.2929048−6.9167525−9.1619449−11.076930γ=0.5D=1.9E=2.0G∗=1.8G=1.1Table 9
The values of change inPr for temperature of Vogel’s model at the wall.
Pr7.78.89.910.10N=1.3R=0.1λ=1.7−5.3502950−5.8292081−6.3140881−6.4026828γ=0.5D=1.9E=2.0G∗=1.8G=1.1Table 10
The values of change inN for temperature of Vogel’s model at the wall.
N1.41.51.61.7Pr=9.9R=0.1λ=1.7−6.3140881−6.3323518−6.3506975−6.3691254γ=0.5D=1.9E=2.0G∗=1.8G=1.1Table 11
The values of change inR for temperature of Vogel’s model at the wall.
R0.40.50.60.7Pr=9.9N=1.6λ=1.7−6.3506975−5.6990981−5.1785360−4.7545397γ=0.5D=1.9E=2.0G∗=1.8G=1.1Table 12
The values of change inE for temperature of Vogel’s model at the wall.
E2.02.42.83.2Pr=9.9N=1.6λ=1.7−5.1785360−5.2495374−5.2941906−5.3239363γ=0.5D=1.9R=0.3G∗=1.8G=1.1Table 13
The values of change inD for temperature of Vogel’s model at the wall.
D1.92.42.93.4Pr=9.9N=1.6λ=1.7−5.2941906−5.2615104−5.2295760−5.1983724γ=0.5E=2.8R=0.3G∗=1.8G=1.1Table 14
The values of change inG∗for temperature of Vogel’s model at the wall.
G∗1.82.43.03.6Pr=9.9N=1.6λ=1.7−5.2295760−5.3720192−5.4587700−5.5161120γ=0.5E=2.8R=0.3D=1.9G=1.1Table 15
The values of change inG for temperature of Vogel’s model at the wall.
G1.12.63.13.6Pr=9.9N=1.6λ=1.7−5.4582770−5.3018863−5.1474593−4.9949967γ=0.5E=2.8R=0.3D=1.9G∗=1.1Table 16
Values of Nusselt number for distinct parameters for Reynold’s model.
γPrλNWR−θ′00.17.71.50.10.20.6−0.87691410.12−0.93031800.13−0.98576840.1−0.87691417.8−0.88574477.9−0.89827117.7−0.90197981.6−0.88574471.7−0.89457171.5−0.87691410.2−0.89984460.3−0.90512850.1−0.87691410.3−0.90555610.4−0.91131430.2−0.87691410.7−0.86977100.8−0.84386590.6−0.8769141Table 17
Values of Nusselt number for distinct parameters for Vogel’s model.
γGG∗DENRλPr−θ′00.11.11.81.92.01.30.11.77.7−2.95829140.12−3.06011930.13−3.11195490.1−2.95829141.2−2.90847361.3−2.85911431.1−2.95829141.9−2.98734422.0−3.01362461.8−2.95829142.0−2.94872132.1−2.92935381.9−2.95829142.1−2.97544992.2−2.99058792.0−2.95829141.4−2.96151221.5−2.96473601.3−2.95829140.2−2.75217100.3−2.58316170.1−2.95829141.8−2.83052551.9−2.90834491.7−2.95829147.8−2.67854327.9−2.87623907.7−2.9083449Table 18
Values of skin friction for distinct parameters for Reynold’s model.
NλPrγWR1/2CfRe0.11.77.70.10.20.2−0.16166330.2−0.16832750.3−0.17499330.1−0.16166331.8−0.16159481.9−0.16152631.7−0.16166337.8−0.16163117.9−0.16159917.7−0.16166330.2−0.08677230.3−0.01247270.1−0.16166330.3−0.90555610.4−0.17092300.2−0.18022080.3−0.16205170.4−0.16237340.2−0.1616633Table 19
Values of skin friction for distinct parameters for Vogel’s model.
GG∗DλγEPrNR1/2CfRe2.10.40.30.70.12.07.71.30.2−22.0042822.2−20.5069542.3−26.0495572.1−22.0042820.5−14.8819110.6−11.1341320.4−22.0042820.40.411914050.50.534401940.3−22.0042820.8−22.0049110.9−22.0055360.7−22.0042820.2−21.9685730.3−21.9361790.1−22.0042822.10.247147712.20.214951122.0−22.0042827.8−22.0028757.9−22.0014687.7−22.0042821.4−22.0230731.5−22.0418691.3−22.0042820.2−22.0987660.3−22.5678790.1−22.004282


## 8. 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 inG and the velocity curve decreases with the enlargement of γ, D and N for both models(2)
γandPr are decreases for Reynold’s well as Vogel’s models(3)
The stream lines are sighted to shrink and the3−D graphs bended with the increase in γ of Reynold’s model(4)
Skin friction curve increases with the increase inN, while Nusselt number graph decreases with the enlargement in λ



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*Source: 2927013-2020-11-07.xml*