Influence of Radiation on Non-Newtonian Fluid in the Region of Oblique Stagnation Point Flow in a Porous Medium: A Numerical Study

Abstract

This article addresses the nonlinear radiation effect on two-dimensional oblique stagnation point flow in a porous medium. Constitutive equations of viscoelastic fluid are employed in the mathematical development of the relevant problem. The resulting nonlinear analysis is computed using Chebyshev Spectral Newton Iterative Scheme. A comparative study of present results with that of previous studies have been made in a limiting sense and shown through tabular values. Excellent agreement is noted which clearly shows that the used numerical scheme is stable and the results are highly accurate. Impact of sundry variables in the flow quantities of interest are discussed. It is observed that shearing parameter $\gamma$ helps to increase the fluid velocity. Thermal boundary layer thickness can be controlled due to small value of radiation parameter and surface heating parameter. It is also noted that with the increase in value of porosity parameter K, the velocity increases but the momentum boundary layer thickness decreases in the region of stagnation point. Moreover, the streamlines are plotted to predict the flow pattern.

Appendix A

Consider the steady, two-dimensional, incompressible Darcy flow of second grade fluid near the oblique stagnation point over an impermeable surface. The surface is placed at $y=0$, and the porous medium occupies upper half plane $y > 0$. The non-Newtonian fluid is passing through the porous medium in the positive x direction. It is assumed that the fluid is transparent to the radiation so the radiation term will only appear in energy equation of solid phase Vafai (2005). Thus the governing equations for steady, two-dimensional, incompressible Darcy flow of second grade fluids near the oblique stagnation point are

$$\begin{aligned}&{\varvec{\nabla }} \cdot \bar{{V}}=0, \end{aligned}$$

(30)

$$\begin{aligned}&\rho \frac{\hbox {d}\bar{{V}}}{\hbox {d}t}={\varvec{\nabla }} \cdot {\varvec{\tau }} -\frac{\mu }{k_1}\bar{{V}}, \end{aligned}$$

(31)

Energy equation for solid phase:

$$\begin{aligned} ({1-\phi }){\varvec{\nabla }} \cdot \left( {k_s {\varvec{\nabla }} \bar{{T}}_s } \right) -{\varvec{\nabla }} \cdot q_{\mathrm{r}} =0, \end{aligned}$$

(32)

Energy equation for fluid phase:

$$\begin{aligned} \phi {\varvec{\nabla }} \cdot \left( {k_f {\varvec{\nabla }} \bar{{T}}_{\mathrm{f}} } \right) -\rho c_p V \cdot {\varvec{\nabla }} \bar{{T}}_{\mathrm{f}} =0, \end{aligned}$$

(33)

where $\bar{{V}}=({u, v, 0})$ is velocity profile, $\bar{{T}}_{\mathrm{f}}$ is temperature of fluid, $\bar{{T}}_{\mathrm{s}}$ is temperature of solid, $\rho$ is density, $k_{1}$ is Darcy permeability parameter, $\hbox {d}/{\hbox {d}t}$ is material derivative, ${\varvec{\tau }}$ is Cauchy stress tensor, $k_{\mathrm{s}}$ is the thermal conductivity of solid, $k_{\mathrm{f}}$ is the thermal conductivity of fluid, $c_{\mathrm{p}}$ is specific heat at constant pressure, $({1-\phi })$ is the ratio of area covered by solid to the total covered area of the medium and $q_{\mathrm{r}}$ is radiative heat flux. We assumed that the no net heat transfer from solid to fluid or fluid to solid so the heat transfer in parallel from both phases. For the simplification of Eqs. (32) and (33), we assumed that there is local thermal equilibrium i.e. $\bar{{T}}_{\mathrm{f}} =\bar{{T}}_{\mathrm{s}} =\bar{{T}}$, so by adding Eqs. (32) and (33) we get

$$\begin{aligned} -\rho C_p V.{\varvec{\nabla }} \bar{{T}}+k_{\mathrm{eff}} {\varvec{\nabla }}^{2}\bar{{T}}-{\varvec{\nabla }} \cdot q_{\mathrm{r}} =0. \end{aligned}$$

(34)

where $k_{\mathrm{eff}} =\left( {\phi k_{\mathrm{f}} +\left( {1-\phi } \right) k_{\mathrm{s}}} \right)$ is the effective thermal conductivity for both fluid and porous medium. Upon using the Rosseland approximation for radiation, one can obtain Hossain and Takhar (1996)

$$\begin{aligned} q_\mathrm{r} =-\frac{4\sigma ^{*}}{3(\alpha _r +\sigma _s )}{\varvec{\nabla }} \bar{{T}}^{4}, \end{aligned}$$

(35)

where $\alpha _{r}$, $\sigma ^{*}$ and $\alpha _{s}$ are the Rosseland mean absorption coefficient, Stefan–Boltzmann constant and the scattering coefficient, respectively. The rheological equation of second grade fluid can be expressed as

$$\begin{aligned} {\varvec{\tau }} =-pI+\mu \mathbf{A}_{1} +\alpha _{1} \mathbf{A}_2 +\alpha _2 \left( {\mathbf{A}_1 } \right) ^{2}, \end{aligned}$$

(36)

$\bar{{p}}$ is the pressure, $\mu$ is the dynamic viscosity of the fluid, $\alpha _{1}$ and $\alpha _{2}$ are normal stress moduli and the tensors $\mathbf{A}_{1}$ and $\mathbf{A}_{2}$ are the first and second Rivlin-Eriksen tensors which can be calculated as

$$\begin{aligned} \mathbf{A}_1= & {} \left( {{\varvec{\nabla }} \mathbf{V}} \right) +\left( {{\varvec{\nabla }} \mathbf{V}} \right) ^{T},\end{aligned}$$

(37)

$$\begin{aligned} \mathbf{A}_2= & {} \frac{\hbox {d}{} \mathbf{A}_1 }{\hbox {d}t}+\mathbf{A}_1 \left( {{\varvec{\nabla }} \mathbf{V}} \right) +\left( {{\varvec{\nabla }} \mathbf{V}} \right) ^{T}{} \mathbf{A}_{1} , \end{aligned}$$

(38)

where $\hbox {d}/{\hbox {d}t}$ is defined as

$$\begin{aligned} \frac{\hbox {d}\left( {\cdot } \right) }{\hbox {d}t}=\frac{\partial \left( {\cdot } \right) }{\partial t}+\mathbf{V}.\left[ {{\varvec{\nabla }} \left( {\cdot } \right) } \right] . \end{aligned}$$

(39)

The thermodynamic constraints for the fluid model (Eq. 36) are compatible with Clausius-Duhem inequality and it is assumed that the free energy density of the fluid be locally at rest are (see Ref. Garg and Rajagopal 1990)

$$\begin{aligned} \mu \ge 0, \alpha _1 \ge 0, \alpha _1 +\alpha _2 =0. \end{aligned}$$

(40)

If $\alpha _{1} =\alpha _{2} =0$, then Eq. (36) reduces to Cauchy stress tensor for Newtonian fluid. After using Eqs. (35–40), Eqs. (30), (31) and (34) will take the following component form

$$\begin{aligned} \frac{\partial \bar{{u}}}{\partial \bar{{x}}}+\dfrac{\partial \bar{{v}}}{\partial \bar{{y}}}= & {} 0,\end{aligned}$$

(41)

$$\begin{aligned} \bar{{u}}\dfrac{\partial \bar{{u}}}{\partial \bar{{x}}}+\bar{{v}}\dfrac{\partial \bar{{u}}}{\partial \bar{{y}}}= & {} -\dfrac{1}{\rho }\dfrac{\partial \bar{{p}}}{\partial \bar{{x}}}+\nu \left( {\dfrac{\partial ^{2}\bar{{u}}}{\partial \bar{{x}}^{2}}+\dfrac{\partial ^{2}\bar{{u}}}{\partial \bar{{y}}^{2}}} \right) +\dfrac{\alpha _1 }{\rho }\left\{ {\dfrac{\partial }{\partial \bar{{x}}}\left[ {\begin{array}{l} 2\bar{{u}}\dfrac{\partial ^{2}\bar{{u}}}{\partial \bar{{x}}^{2}}+2\bar{{v}}\dfrac{\partial ^{2}\bar{{u}}}{\partial \bar{{x}}\partial \bar{{y}}} \\ \quad +4\left( {\dfrac{\partial \bar{{u}}}{\partial \bar{{x}}}} \right) ^{2}+2\dfrac{\partial \bar{{v}}}{\partial \bar{{x}}}\left( {\dfrac{\partial \bar{{v}}}{\partial \bar{{x}}}+\dfrac{\partial \bar{{u}}}{\partial \bar{{y}}}} \right) \\ \end{array}} \right] } \right. \nonumber \\&+\,\left. {\dfrac{\partial }{\partial \bar{{y}}}\left[ {\begin{array}{l} \left( {\bar{{u}}\dfrac{\partial }{\partial \bar{{x}}}+\bar{{v}}\dfrac{\partial }{\partial \bar{{y}}}} \right) \left( {\dfrac{\partial \bar{{v}}}{\partial \bar{{x}}}+\dfrac{\partial \bar{{u}}}{\partial \bar{{y}}}} \right) \\ \quad +2\dfrac{\partial \bar{{u}}}{\partial \bar{{x}}}\dfrac{\partial \bar{{u}}}{\partial \bar{{y}}}+2\dfrac{\partial \bar{{v}}}{\partial \bar{{x}}}\dfrac{\partial \bar{{v}}}{\partial \bar{{y}}} \\ \end{array}} \right] } \right\} \nonumber \\&+\dfrac{\alpha _2 }{\rho }\dfrac{\partial }{\partial \bar{{x}}}\left[ {4\left( {\dfrac{\partial \bar{{u}}}{\partial \bar{{x}}}} \right) ^{2}+\left( {\dfrac{\partial \bar{{v}}}{\partial \bar{{x}}}+\dfrac{\partial \bar{{u}}}{\partial \bar{{y}}}} \right) ^{2}} \right] -\dfrac{\nu }{k_1 }\bar{{u}}, \end{aligned}$$

(42)

$$\begin{aligned} \bar{{u}}\dfrac{\partial \bar{{v}}}{\partial \bar{{x}}}+\bar{{v}}\dfrac{\partial \bar{{v}}}{\partial \bar{{y}}}= & {} -\dfrac{1}{\rho }\dfrac{\partial \bar{{p}}}{\partial \bar{{y}}}+\nu \left( {\dfrac{\partial ^{2}\bar{{v}}}{\partial \bar{{x}}^{2}}+\dfrac{\partial ^{2}\bar{{v}}}{\partial \bar{{y}}^{2}}} \right) +\dfrac{\alpha _1 }{\rho }\left\{ {\dfrac{\partial }{\partial \bar{{x}}}\left[ {\begin{array}{l} 2\dfrac{\partial \bar{{u}}}{\partial \bar{{x}}}\dfrac{\partial \bar{{u}}}{\partial \bar{{y}}}+2\dfrac{\partial \bar{{v}}}{\partial \bar{{x}}}\dfrac{\partial \bar{{v}}}{\partial \bar{{y}}} \\ \quad +\left( {\bar{{u}}\dfrac{\partial }{\partial \bar{{x}}}+\bar{{v}}\dfrac{\partial }{\partial \bar{{y}}}} \right) \left( {\dfrac{\partial \bar{{v}}}{\partial \bar{{x}}}+\dfrac{\partial \bar{{u}}}{\partial \bar{{y}}}} \right) \\ \end{array}} \right] } \right. \nonumber \\&+\,\left. {\dfrac{\partial }{\partial \bar{{y}}}\left[ {\begin{array}{l} 2\dfrac{\partial \bar{{u}}}{\partial \bar{{y}}}\left( {\dfrac{\partial \bar{{v}}}{\partial \bar{{x}}}+\dfrac{\partial \bar{{u}}}{\partial \bar{{y}}}} \right) +4\left( {\dfrac{\partial \bar{{v}}}{\partial \bar{{y}}}} \right) ^{2} \\ 2\bar{{v}}\dfrac{\partial ^{2}\bar{{v}}}{\partial \bar{{y}}^{2}}+2\bar{{u}}\dfrac{\partial ^{2}\bar{{v}}}{\partial \bar{{x}}\partial \bar{{y}}} \\ \end{array}} \right] } \right\} \nonumber \\&+\dfrac{\alpha _2 }{\rho }\dfrac{\partial }{\partial \bar{{y}}}\left[ {4\left( {\dfrac{\partial \bar{{v}}}{\partial \bar{{y}}}} \right) ^{2}+\left( {\dfrac{\partial \bar{{v}}}{\partial \bar{{x}}}+\dfrac{\partial \bar{{u}}}{\partial \bar{{y}}}} \right) ^{2}} \right] -\dfrac{\nu }{k_1 }\bar{{v}}, \end{aligned}$$

(43)

$$\begin{aligned} \bar{{u}}\dfrac{\partial \bar{{T}}}{\partial \bar{{x}}}+\bar{{v}}\dfrac{\partial \bar{{T}}}{\partial \bar{{y}}}= & {} \dfrac{1}{\rho c_p }\bar{{{\varvec{\nabla }} }} \cdot \left( {\left[ {k_{\mathrm{eff}} +\dfrac{16\sigma ^{*}\bar{{T}}^{3}}{3(\alpha _r +\sigma _s )}} \right] \bar{{{\varvec{\nabla }} }}\bar{{T}}} \right) , \end{aligned}$$

(44)

where $\bar{{u}}$ and $\bar{{v}}$ are the velocity components in $\bar{{x}}$ and $\bar{{y}}$ direction, respectively, $\bar{{p}}(\bar{{x}},\bar{{y}})$ is the pressure function of the fluid and $\nu$ is the kinematic viscosity. Here, the fluid flow is considered as potential flow, in which the flow is impinging obliquely to flat plate and far away from the plate the fluid is moving with velocity $U_{e}(\bar{{x}},\bar{{y}})=a\bar{{x}}+b\bar{{y}}$. The boundary conditions for the present fluid flow are given by

$$\begin{aligned} \bar{{y}}=0 : \quad \bar{{u}}=0, \bar{{v}}=0, \bar{{T}}=T_w , \nonumber \\ \bar{{y}}\rightarrow \infty :\quad \bar{{u}}=a\bar{{x}}+b\bar{{y}}, \bar{{T}}=T_\infty . \end{aligned}$$

(45)

where a, b are the constant having positive values with dimension inverse of time. $T_{\infty }$ is the ambient temperature of the fluid away from the surface and $T_{w}$ is the surface temperature. After using following non-dimensional variables

$$\begin{aligned} x=\bar{{x}}\sqrt{\frac{a}{\nu }}, y=\bar{{y}}\sqrt{\frac{a}{\nu }}, u=\frac{1}{\sqrt{\nu a}}\bar{{u}}, v=\frac{1}{\sqrt{\nu a}}\bar{{v}}, p=\frac{1}{\rho \nu a}\bar{{p}}, T=\frac{\bar{{T}}-T_\infty }{T_w -T_\infty }, \end{aligned}$$

(46)

the Eqs. (41–45) take the following dimensionless form

$$\begin{aligned} \dfrac{\partial u}{\partial x}+\dfrac{\partial v}{\partial y}= & {} 0,\end{aligned}$$

(47)

$$\begin{aligned} u\dfrac{\partial u}{\partial x}+v\dfrac{\partial u}{\partial y}= & {} -\dfrac{\partial p}{\partial x}+\left( {\dfrac{\partial ^{2}u}{\partial x^{2}}+\dfrac{\partial ^{2}u}{\partial y^{2}}} \right) +\hbox {We}\left\{ {\dfrac{\partial }{\partial x}\left[ {\begin{array}{l} 2u\dfrac{\partial ^{2}u}{\partial x^{2}}+2v\dfrac{\partial ^{2}u}{\partial x\partial y} \\ \quad +4\left( {\dfrac{\partial u}{\partial x}} \right) ^{2}+2\dfrac{\partial v}{\partial x}\left( {\dfrac{\partial v}{\partial x}+\dfrac{\partial u}{\partial y}} \right) \\ \end{array}} \right] } \right. \nonumber \\&+\left. {\dfrac{\partial }{\partial y}\left[ {\begin{array}{l} \left( {u\dfrac{\partial }{\partial x}+v\dfrac{\partial }{\partial y}} \right) \left( {\dfrac{\partial v}{\partial x}+\dfrac{\partial u}{\partial y}} \right) \\ \quad +2\dfrac{\partial u}{\partial x}\dfrac{\partial u}{\partial y}+2\dfrac{\partial v}{\partial x}\dfrac{\partial v}{\partial y} \\ \end{array}} \right] } \right\} \nonumber \\&+\,\lambda \dfrac{\partial }{\partial x}\left[ {4\left( {\dfrac{\partial u}{\partial x}} \right) ^{2}+\left( {\dfrac{\partial v}{\partial x}+\dfrac{\partial u}{\partial y}} \right) ^{2}} \right] -\dfrac{\nu }{ak_1 }u, \end{aligned}$$

(48)

$$\begin{aligned} u\dfrac{\partial v}{\partial x}+v\dfrac{\partial v}{\partial y}= & {} -\dfrac{\partial p}{\partial y}+\left( {\dfrac{\partial ^{2}v}{\partial x^{2}}+\dfrac{\partial ^{2}v}{\partial y^{2}}} \right) +\hbox {We}\left\{ {\dfrac{\partial }{\partial x}\left[ {\begin{array}{l} 2\dfrac{\partial u}{\partial x}\dfrac{\partial u}{\partial y}+2\dfrac{\partial v}{\partial x}\dfrac{\partial v}{\partial y} \\ \quad +\left( {u\dfrac{\partial }{\partial x}+v\dfrac{\partial }{\partial y}} \right) \left( {\dfrac{\partial v}{\partial x}+\dfrac{\partial u}{\partial y}} \right) \\ \end{array}} \right] } \right. \nonumber \\&\quad +\,\left. {\dfrac{\partial }{\partial y}\left[ {\begin{array}{l} 2\dfrac{\partial u}{\partial y}\left( {\dfrac{\partial v}{\partial x}+\dfrac{\partial u}{\partial y}} \right) +4\left( {\dfrac{\partial v}{\partial y}} \right) ^{2} \\ 2v\dfrac{\partial ^{2}v}{\partial x^{2}}+2u\dfrac{\partial ^{2}v}{\partial x\partial y} \\ \end{array}} \right] } \right\} \nonumber \\&+\,\lambda \dfrac{\partial }{\partial y}\left[ {4\left( {\dfrac{\partial v}{\partial y}} \right) ^{2}+\left( {\dfrac{\partial v}{\partial x}+\dfrac{\partial u}{\partial y}} \right) ^{2}} \right] -\dfrac{\nu }{ak_1 }v, \end{aligned}$$

(49)

$$\begin{aligned} u\dfrac{\partial T}{\partial x}+v\dfrac{\partial T}{\partial y}= & {} \dfrac{1}{\Pr }{\varvec{\nabla }} \cdot \left( {\left[ {1+\dfrac{16\sigma ^{*}T_\infty ^{3}\left( {1+\left( {\dfrac{T_w }{T_\infty }-1} \right) T} \right) ^{3}}{3k_{\mathrm{eff}} (\alpha _r +\sigma _s )}} \right] {\varvec{\nabla }} T} \right) , \end{aligned}$$

(50)

The boundary conditions will become

$$\begin{aligned} y=0 : \quad u=0, v=0, T=1, \nonumber \\ y\rightarrow \infty : \qquad u=x+\frac{b}{a}y, T=0. \end{aligned}$$

(51)