MHD Flow and Heat Transfer of a Jeffrey Fluid over a Porous Stretching/Shrinking Sheet with a Convective Boundary Condition

A steady two-dimensional (x, y) boundary layer flow of an incompressible non-Newtonian Jeffrey liquid over a stretching sheet with thermal radiation and heat source is analysed. In the Cartesian coordinate system, the linear velocity of stretching sheet as $u_{w}(x)=b x$ (where b is a real number) is toward the x-axis and y-axis is opposite to it [see Figure 1]. Also, it is considered that the stretching surface is heated due to convection from a hot liquid at temperature T_f. Additionally, a uniform magnetic field of strength B₀ is forced normal to the plate along the y-axis and the induced magnetic field is neglected because the magnetic Reynolds number is very small. The expressions for the Cauchy and the extra stress tensors τ and S₁ in a Jeffery non-Newtonian liquid are defined as follows [1]:

$\tau=-p I+S_{1}$    (1)

$S_{1}=\frac{\mu}{1+\lambda}\left[A_{1}+\lambda_{1}\left(\frac{\partial}{\partial t}+V . \nabla\right) A_{1}\right]$    (2)

In which p is the pressure. The Rivlin-Ericksen tensor A₁ is defined as $A_{1}=\nabla V+(\nabla V)^{\prime}$, where V is the velocity field vector.

The governing equations of non-Newtonian liquid under the assumption of boundary layer approximations is given by (Ref. [8, 23]).

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Figure 1. Geometry of the problem

$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$    (3)

$u \frac{\partial u}{\partial x}+v \frac{\partial u}{\partial y}$

$=\frac{v}{1+\lambda}\left\{\frac{\partial^{2} u}{\partial y^{2}}+\lambda_{1}\left[\begin{array}{c}\frac{\partial u}{\partial y} \frac{\partial^{2} u}{\partial x \partial y}+u \frac{\partial^{3} u}{\partial x \partial y^{2}} \\ -\frac{\partial u}{\partial x} \frac{\partial^{2} u}{\partial y^{2}}+v \frac{\partial^{3} u}{\partial y^{3}}\end{array}\right]\right\}$

$-\frac{v}{K^{*}} u-\frac{\sigma B_{0}^{2}}{\rho} u$    (4)

$u \frac{\partial T}{\partial x}+v \frac{\partial T}{\partial y}$

$=\alpha^{*} \frac{\partial^{2} T}{\partial y^{2}}-\frac{1}{\rho c_{p}} \frac{\partial q_{r}}{\partial y}$

$+\frac{Q_{0}}{\rho c_{p}}\left(T-T_{\infty}\right)$    (5)

The boundary conditions:

$\left.\begin{array}{r}u=u_{w}(x)=b x, v=v_{w}, \\ -k \frac{\partial T}{\partial y}=h\left(T_{f}-T\right) \text { at } \quad y=0 \\ u=0, u^{\prime} \rightarrow 0, T \rightarrow T_{\infty} \\ \text { as } y \rightarrow \infty\end{array}\right\}$    (6)

Utilizing the Rosseland approximation, the radiative heat flux q_ris defined as [36]:

$q_{r}=-\frac{4 \sigma^{*}}{3 K_{s}} \frac{\partial T^{4}}{\partial y}$    (7)

where, K_s and σ^* is the mean absorption coefficient and Stefan-Boltzmann constant respectively. It is assumed that the variation in temperature within the flow is such that T⁴ can be expanding in Taylor series about T_∞ and neglecting higher-order terms beyond the first degree in (T-T_∞), we get:

$T^{4} \approx 4 T_{\infty}^{3} T-3 T_{\infty}^{4}$    (8)

Now, differentiating Eq. (7) w. r. to y and using Eq. (8), we get:

$\frac{\partial q_{r}}{\partial y}=-\frac{16 T_{\infty}^{3} \sigma^{*}}{3 K_{s}} \frac{\partial^{2} T}{\partial y^{2}}$    (9)

Using Eq. (9) in Eq. (5), we obtain:

$u \frac{\partial T}{\partial x}+v \frac{\partial T}{\partial y}$

$=\sigma \frac{\partial^{2} T}{\partial y^{2}}+\frac{1}{\rho c_{p}} \frac{16 T_{\infty}^{3} \sigma^{*}}{3 K_{s}} \frac{\partial^{2} T}{\partial y^{2}}$

$+\frac{Q_{0}}{\rho c_{p}}\left(T-T_{\infty}\right)$    (10)

The following similarity transformations are introduced.

$u=\operatorname{axf}^{\prime}(\eta)$,

$v=-\sqrt{a v} f(\eta), \theta(\eta)$

$=\frac{T-T_{\infty}}{T_{f}-T_{\infty}}, \eta=y \sqrt{\frac{a}{v}}$,

$\psi=x \sqrt{a v} f(\eta)$    (11)

where, the stream function ψ defined as $u=\frac{\partial \psi}{\partial y}, v=-\frac{\partial \psi}{\partial x}$.

Utilizing Eq. (11), the Eqns. (4)-(5) are reduced into the following 3^rd and 2^nd order ordinary differential equations.

$f^{\prime \prime \prime}+\beta\left(f^{\prime \prime 2}-f f^{\prime \prime \prime}\right)+(1+\lambda)\left(f f^{\prime \prime}-f^{\prime 2}\right)$

$-(1+\lambda)\left(M+\frac{1}{K}\right) f^{\prime}=0$    (12)

$\left(1+\frac{4}{3} R\right) \theta^{\prime \prime}+\operatorname{Pr} f \theta^{\prime}+Q \theta=0$    (13)

and Eq. (6) becomes,

$f=S, f^{\prime}=b / a=\alpha$,

$\theta^{\prime}=-\gamma[1-\theta(0)]$ at $\eta=0$    (14)

$f^{\prime}=0, \quad f^{\prime \prime}(\eta)=0$,

$\theta=0$ as $\eta \rightarrow \infty$    (15)

where, $\beta=\lambda_{1} a$, $K=\frac{v}{a K^{*}}$, $M=\frac{\sigma B_{0}^{2}}{\rho a}$, $P r=\frac{v}{\sigma}$, $R=\frac{4 \sigma^{*} T_{\infty}^{3}}{K_{1} k}$ $Q=\frac{v Q_{0}}{\rho c_{p}}$, $S=-\frac{v_{w}}{\sqrt{a v}}$ (S>0 for suction and S<0 for injection), $\alpha=\frac{b}{a}$ and $\gamma=\frac{h}{k} \sqrt{\frac{v}{a}}$.

The physical quantities of interest are which is defined as $R e_{x}^{\frac{1}{2}} C_{f}=\frac{1}{1+\lambda}\left(f^{\prime \prime}(0)+\beta f^{\prime \prime}(0)\right)$.

The surface drag coefficient (C_f) and the rate of thermal energy (Nu) is defined as:

$C_{f}=\frac{\tau_{w}}{\frac{1}{2} \rho u_{w}^{2}}, \quad N u_{x}=\frac{x\left(q_{w}+q_{r}\right)}{k\left(T_{f}-T_{\infty}\right)}$    (16)

where, $\tau_{w}=\frac{\mu}{1+\lambda}\left\{\left(\frac{\partial u}{\partial y}\right)+\lambda_{1}\left[u \frac{\partial^{2} u}{\partial x \partial y}+u \frac{\partial^{2} v}{\partial x^{2}}+v \frac{\partial^{2} u}{\partial y^{2}}\right]\right\}$ and $q_{w}=-k\left(\frac{\partial T}{\partial y}\right)_{y=0}$, q_w is heat transfer from the sheet.

Substituting the value of $\tau_{w}$ and q_w into Eq. (16) we get the skin friction and local Nusselt number in dimensionless form as follows:

$c_{f}=2\left(\operatorname{Re}_{x}\right)^{-\frac{1}{2}} \alpha^{-\frac{3}{2}}$

$\cdot\left[\frac{1}{1+\lambda}\left\{\begin{array}{l}f^{\prime \prime}(0)+\beta\left(f^{\prime}(0) f^{\prime \prime}(0)\right. \\ \left.-f(0) f^{\prime \prime}(0)\right)\end{array}\right\}\right]$    (17)

$N u_{x} \operatorname{Re}_{x}^{-1 / 2}=-\left(1+\frac{4}{3} R\right) \theta^{\prime}(0)$    (18)

where, $R e_{x}=\frac{u_{w}(x)}{v}$ is the local Reynolds number.

The problem of MHD two-dimensional heat transfer flow of a Jeffrey liquid over a shrinking/stretching sheet is studied numerically. The Jeffrey fluid parameters λ and β are chosen in the range of 0≤λ≤2.0, 0≤β≤0.5. The present non-Newtonian model reduced to a viscous model in the absence of Jeffrey fluid parameters λ and β. A parametric study is conducted on Newtonian and Jeffrey fluids, in order to get the pertinent parameters on various flow fields. The velocity, skin-friction coefficient, temperature, and Nusslet number results are analysed through the plotted in Figures 2-13. Table 1, gives the comparison values of Nusslet number with those of Chen [40] and Grubka and Bobba [41] in the absence of M, Q, R, λ, β, K and γ. It is noticed that the comparison shows a concurrence with the existing outcomes and thus confirms the exactness of numerical code applied in the current work.

Table 1. Comparison results of $N u_{x} R e_{x}^{-1 / 2}$ for different values of Pr (in the absence of K, γ, β, M, R, Q and λ)

Table 2, depicts the effect of suction/injection parameter (S), magnetic field (M) and Deborah number (β) on the skin-friction coefficient and Nusselt number along with a variation of Pr. It is seen that the skin friction coefficient decreased with increasing suction parameter (S>0) and β, while the inverse pattern is seen with S<0 and M. Further, an increase in β and M increases the Nusselt Number.

Figure 2 demonstrates the influence of magnetic field parameter M on the velocity profiles against η for both Newtonian and non-Newtonian fluid cases. It is observed that the increase in M reduces the fluid velocity and boundary layer thickness. Physically, an applied magnetic field generates a reverse force (Lorentz force) in the flow field and causes to oppose the movement of liquid. Also, it is noticed that the dimensionless fluid velocity at the sheet has a higher value for a Newtonian fluid (λ=β=0) than to the non-Newtonian fluid (λ, β>0).

Table 2. Numerical Values of $R e_{x}^{\frac{1}{2}} C_{f}$ and $N u_{x} R e_{x}^{-1 / 2}$ for different values of S, M, β and Pr (other parameters are fixed) (i.e. K=2.0; Q=1.0; R=0.4; α=0.3; λ=0.2; γ=1.0)

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Figure 2. Effect of M on $f^{\prime}(\eta)$

Figures 3(a) and 3(b) illustrate the effect of velocity ratio parameter (α=b/a) on the velocity and temperature distributions. It is noticed that α>0 relates to the sheet stretching while α<0 demonstrate the shrinking sheet respectively. When α = 0 then there is no flow. It is seen that the velocity increases with the increase of α for both the stretching and shrinking cases. Also, it is observed that the rate transport of velocity curve decreases with the increasing distance η normal to the sheet and the velocity vanishes at η=2 from the sheet. Whereas α has a reverse trend in case of the temperature profile.

Figures 4(a) and 4(b) demonstrate the impact of Deborah number β on the velocity and temperature functions for both stretching and shrinking cases. The velocity becomes constant when α=0, decreases for shrinking case and increases for stretching case. It is obvious from the figure that the momentum boundary layer converges quickly for smaller β values. Physically, small β resembles a situation where the material has time to relax (Newtonian nature) and high β relates to non-Newtonian nature. Also, the temperature profile θ(η) reduced with increasing values of β for stretching case.

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Figure 3. Effect of α on (a) f'(η) (b) θ(η)

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Figure 4. Effect of β on (a) f'(η) (b) θ(η)

Figure 5(a) shows the influence of the ratio of relaxation and retardation times parameter λ on the velocity profiles for stretching and shrinking sheets. It is observed that the velocity decreases with the increase of λ in stretching case and reverse in shrinking case. Physically, an increase in λ means a reduction in fluid retardation time which in turn to stop the hastening of fluid motion. On the other hand, the temperature profile increases with the increase of λ which is obvious from Figure 5(b). Physically, an increase in λ leads to a rise in relaxation time and diminish in retardation time, due to this higher temperature and thicker thermal boundary layer.

The effect of permeability parameter K on the velocity profiles is shown in Figure 6. It is observed that in the case of stretching sheet (α>0) the velocity increases with the increase of K and β values. These results are similar to those obtained in Ref. [12]. It is also noticed from the figure that the velocity boundary layer converges quickly for small values of β.

Figures 7(a), 7(b) and 7(c) respectively show the influence of suction/injection parameter S on the velocity and temperature profiles. It is obvious from the figures that f'(η) and θ(η) is found to a decrease with the increase of S>0. On the other hand, inverse behaviour is seen in the injection case (S<0).

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Figure 5. Effect of λ on (a) f'(η) (b) θ(η)

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Figure 6. Effect of K on f'(η)

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Figure 7. Effect of (a) S≥0 on f'(η) (b) S < 0 on f'(η) (c) S>0 on θ(η)

The influence of Biot number γ on θ(η) is plotted in Figure 8. It is evident that the θ(η) increase with the rise in γ for both stretching and shrinking cases. In general, γ=0 represents the no convective heat exchange from the surface of the sheet to the cold fluid which is far away from the sheet. For this situation, it should be understood that the surface is totally insulated. Further, it is observed that the temperature is greater in the case of shrinking when compared to the stretching sheet. These outcomes obviously reinforced from the physical perspective.

The effect of thermal radiation parameter R on the dimensionless temperature profile is offered in Figure 9. It is inspected that the fluid temperature increases by an increase in R values. This is because of the fact that as R increases; the mean absorption coefficient K_s diminishes. Hence, the divergence of radiation heat flux increases. As a result, the rate of radiative heat transfer into the liquid rises.

Figure 10 represents the temperature profile for different values of heat source/sink parameter (Q>0 and Q<0). It is observed that the boundary layer generates the energy, which causes the temperature profile to increase with increasing values of the heat source (Q>0) whereas reverse effects are seen in the heat sink (Q<0) case. This is due to the fact that the rise of Q in the boundary layer makes energy which roots the θ(η) increase. From this, it is concluded that in the engineering and industrial heat transfer applications the heat sink is well appropriate for real cooling of the stretching sheet. It is also important to note that θ(η) decreases with increasing values of the suction parameter (S>0).

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Figure 8. Effect of γ on θ(η)

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Figure 9. Effect of R on θ(η)

Figure 11(a) portrays the attributes of skin friction coefficient concerning λ and β for stretching and shrinking cases. It is found that the skin-friction reduce in case of stretching sheet with rising β and reverse a trend in the case of a shrinking sheet. Figure 11(b) shows the influence of β on Nusselt number for stretching/shrinking sheet. The Nusselt number raised in terms of increase in β for stretching sheet, while it is reversed in case of the shrinking sheet.

Figures 12(a) and 12(b) describe the changes in Nusselt number against Prandtl number Pr for various values of Q and R respectively. It is perceived from the graphs that Nusselt number decline with the augmented values of Q and rises for Pr. The heat transfer coefficient for viscous fluid is more than that of Jeffrey fluid. In addition, the opposite trend is noticed with the rise in R.

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Figure 10. Effect of Q on θ(η)

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Figure 11. Effect of λ against β on (a) $C f_{x} R e_{x}^{1 / 2}$ (b) $N u_{x} R e_{x}^{-1 / 2}$

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Figure 12. Effect of Pr against (a) Q on $N u_{x} R e_{x}^{-1 / 2}$ (b) R on $N u_{x} \operatorname{Re}_{x}^{-1 / 2}$

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Figure 13. Comparison of the velocity distribution for various values of α between present and Hayat et al. [8] work

In order to check the accuracy of the present results, we have analyzed the outcomes of the velocity distribution for different numbers of α in Figure 13 with those reported by Hayat et al. [8] in the absence of M and K. This correlation shows an incredible arrangement.