Natural Convection of MHD over a Vertical Wavy Surface in Presence of Porous Media

Abstract

This paper shows the natural convective heat transfer in porous media over the vertical wavy surface and it assumes that the fluid is viscous and in-compressible. This model shows the effects of the inverse of Darcy number. The dimensional partial differential equations are converted into a dimensionless form. The non-linear system of equations is obtained and these equations are solved numerically by the finite difference method. The results are obtained for inverse Darcy number, magnetic parameter, Prandtl number, amplitude of surface, parameter of heat generation and parameter of thermal conductivity, and their effects on the velocity, temperature of the fluid and Nusselt number.

Keywords

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Adel, R. (2022) Natural Convection of MHD over a Vertical Wavy Surface in Presence of Porous Media. Journal of Applied Mathematics and Physics, 10, 3288-3300. doi: 10.4236/jamp.2022.1011218.

1. Introduction

This paper studies the effects of porous medium of a vertical wavy surface in presence of the magnetic field and heat transfer, and uses a different method of numerical solution to solve the problem (finite difference-fully implicit method). Heat generation effects on MHD natural convection flow along a vertical wavy surface with variable thermal conductivity were investigated by Md. Abdul Alim et al. [1].

The study of this paper has a wide application in the fields of geology and biology. Many authors presented solutions for this type of flow just like that, mass and Heat transfer in porous media was studied by Vadasz, P. (ed) [2]. Wang, C.Y. [3] investigated the free convection over a vertical stretching surface. Yao [4] studied the natural convection over a vertical complex wavy surface. The natural convection flow of Non-Darcy over a vertical wavy plate embedded in a non-Newtonian fluid-saturated porous medium was investigated by Hady, F.M., et al. [5]. Cheng, P. and Minkowycz, W. J. [6] studied the free convection along a vertical flat plate with application to transfer from a dike embedded in a porous medium. The free convection flow of boundary layer along a vertical surface in a porous medium with Newtonian heating is also investigated by Lesnic, D., Ingham, D. B., et al. [7]. The effectiveness of convection heat transfer on a flat plate in porous media for power-law fluids has been investigated by Hady, F. M. and Ibrahim, F. S. [8]. Tashtoush and Al-odat [9] investigated the effect of Magnetic field on fluid flow with a variable heat flux over a wavy surface. The natural convection flow with uniform surface temperature over a vertical curvy surface in presence of heat generation/absorption was investigated by Molla et al. [10].

The system of equations will be converted from partial differential equations into dimensionless form. The nonlinear system which is obtained is solved numerically by finite difference (fully implicit method) [11]. The effectiveness of inverse Darcy number, wavy surface amplitude, parameter of magnetic field, Prandtl number, variation of thermal conductivity and heat generation on velocity, temperature and Nusselt number will be obtained.

Formulation of Mathematic

It is considered the free convection boundary layer flow in two-dimensions is steady over vertical wavy surfaces and porous media. It is a viscous incompressible fluid and electrically conducting. There is magnetic field of strength ${B}_{\circ }$. The uniform temperature at the surface ${T}^{\otimes }{}_{w}$ and ${T}^{\otimes }{}_{\infty }$ is the temperature at the fluid, where ${T}^{\otimes }{}_{w}\succ {T}^{\otimes }{}_{\infty }$.

Assume that the surface geometry function ${\sigma }^{\ast }\left({\stackrel{¯}{x}}_{1}\right)$ is arbitrary. Consider the equation of the wavy surface could be determined as:

${\stackrel{¯}{x}}_{2}={\stackrel{¯}{\sigma }}^{\ast }\left({\stackrel{¯}{x}}_{1}\right)=a\mathrm{sin}\left(\frac{n\pi {\stackrel{¯}{x}}_{1}}{l}\right)$ (1)

whereas ${\stackrel{¯}{x}}_{1}$ is the dimensional axis along the curvy surface, ${\stackrel{¯}{x}}_{2}$ is dimensional axis perpendicular to the curvy surface, $l$ is the distance related to the curvy surface and a is the amplitude of curvy surface. It is shown in a Figure 1 that the wavy

Figure 1. Physical model and coordinate system

surface and the system of Cartesian coordinates. Consider that there is no Joule heat effect and neglect the heat generation, viscous dissipation and body forces of the system.

With introducing Boussinesq approximation, the governing equations are steady, laminar and in two-dimensions. Then the following equations are the continuity equation, momentum equations and energy equation are written as:

$\frac{\partial {\stackrel{¯}{q}}_{1}}{\partial {\stackrel{¯}{x}}_{1}}+\frac{\partial {\stackrel{¯}{q}}_{2}}{\partial {\stackrel{¯}{x}}_{2}}=0$ (2)

${\stackrel{¯}{q}}_{1}\frac{\partial {\stackrel{¯}{q}}_{1}}{\partial {\stackrel{¯}{x}}_{1}}+{\stackrel{¯}{q}}_{2}\frac{\partial {\stackrel{¯}{q}}_{1}}{\partial {\stackrel{¯}{x}}_{2}}=-\frac{1}{{\rho }^{\ast }}\frac{\partial {\stackrel{¯}{p}}^{\ast }}{\partial {\stackrel{¯}{x}}_{1}}+{\upsilon }^{\otimes }{\nabla }^{2}{\stackrel{¯}{q}}_{1}+{g}^{\ast }{\beta }^{\otimes }\left({T}^{\otimes }-{T}^{\otimes }{}_{\infty }\right)-\frac{{\sigma }_{\circ }{\beta }_{\circ }{}^{2}}{{\rho }^{\ast }}{\stackrel{¯}{q}}_{1}-\frac{{\upsilon }^{\otimes }}{{k}^{\ast }}{\stackrel{¯}{q}}_{1}$ (3)

${\stackrel{¯}{q}}_{1}\frac{\partial {\stackrel{¯}{q}}_{2}}{\partial {\stackrel{¯}{x}}_{1}}+{\stackrel{¯}{q}}_{2}\frac{\partial {\stackrel{¯}{q}}_{2}}{\partial {\stackrel{¯}{x}}_{2}}=-\frac{1}{{\rho }^{\ast }}\frac{\partial {\stackrel{¯}{p}}^{\ast }}{\partial {\stackrel{¯}{x}}_{2}}+{\upsilon }^{\otimes }{\nabla }^{2}{\stackrel{¯}{q}}_{2}-\frac{{\upsilon }^{\otimes }}{{k}^{\ast }}{\stackrel{¯}{q}}_{2}$ (4)

${\stackrel{¯}{q}}_{1}\frac{\partial {T}^{\otimes }}{\partial {\stackrel{¯}{x}}_{1}}+{\stackrel{¯}{q}}_{2}\frac{\partial {T}^{\otimes }}{\partial {\stackrel{¯}{x}}_{2}}=-\frac{{K}^{\oplus }}{{\rho }^{\ast }{c}^{\ast }{}_{p}}{\nabla }^{2}{T}^{\otimes }+\frac{{Q}^{\ast }{}_{\circ }\left({T}^{\otimes }-{T}^{\otimes }{}_{\infty }\right)}{{\rho }^{\ast }{c}^{\ast }{}_{p}}$ (5)

The boundary conditions

$\begin{array}{l}\text{at}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\stackrel{¯}{x}}_{2}={\stackrel{¯}{x}}_{2}{}_{\text{\hspace{0.17em}}\text{\hspace{0.17em}}w}={\stackrel{¯}{\sigma }}^{\ast }\left({\stackrel{¯}{x}}_{1}\right):{\stackrel{¯}{q}}_{1}=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\stackrel{¯}{q}}_{2}=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{T}^{\otimes }={T}^{\otimes }{}_{w}\\ \text{at}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\stackrel{¯}{x}}_{2}\to \infty :{\stackrel{¯}{q}}_{1}=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{T}^{\otimes }={T}^{\otimes }{}_{\infty },\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\stackrel{¯}{p}}^{\ast }={p}^{\ast }{}_{\infty }\end{array}$ (6)

where ${\stackrel{¯}{q}}_{1},{\stackrel{¯}{q}}_{2}$ are the velocity components in direction of ${\stackrel{¯}{x}}_{1},{\stackrel{¯}{x}}_{2}$ respectively, the density ${\rho }^{\ast }$, ${\stackrel{¯}{p}}^{\ast }$ is the dimensional pressure of fluid, ${\nabla }^{2}$ the Laplacian operator, ${\beta }^{2}{}_{o}$ the magnetic induction, ${\sigma }_{\circ }$ the electrical conduction, ${\beta }^{\otimes }$ the thermal expansion coefficient, ${\upsilon }^{\otimes }$ the kinematic viscosity, ${g}^{*}$ the acceleration of gravity, ${K}^{\oplus }$ the thermal conductivity of fluid, ${c}^{\ast }{}_{p}$ the specific heat at the constant pressure, ${k}^{*}$ the saturated porous Medium permeability, ${p}^{\ast }{}_{\infty }$ the outside pressure of the boundary layer of the fluid and ${Q}_{o}{}^{*}$ is the constant of heat generation.

Transformed the curvy surface into a flat surface and boundary layer approximation by Using Prandl’s transposition theorem. The dimensionless variables are:

$\begin{array}{l}{x}_{1}=\frac{{\stackrel{¯}{x}}_{1}}{l},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{x}_{2}=\frac{{\stackrel{¯}{x}}_{2}-{\sigma }^{\ast }}{l}G{r}^{\otimes }{}^{\frac{1}{4}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{p}^{\ast }=\frac{{l}^{2}}{{\rho }^{\ast }{\upsilon }^{\otimes }{}^{2}}G{r}^{\otimes }{}^{-1}{\stackrel{¯}{p}}^{\ast }\\ {q}_{1}=\frac{{\rho }^{\ast }l}{{\mu }^{\otimes }}G{r}^{\otimes }{}^{-\frac{1}{2}}{\stackrel{¯}{q}}_{1},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{q}_{2}=\frac{{\rho }^{\ast }l}{{\mu }^{\otimes }}G{r}^{\otimes }{}^{-\frac{1}{4}}\left({\stackrel{¯}{q}}_{2}-{\sigma }^{\ast }{}_{{x}_{1}}{\stackrel{¯}{q}}_{1}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\sigma }^{\ast }{}_{{x}_{1}}=\frac{\text{d}{\sigma }^{\ast }}{\text{d}{x}_{1}}\\ G{r}^{\otimes }=\frac{{g}^{\ast }{\beta }^{\otimes }\left({T}^{\otimes }{}_{w}-{T}^{\otimes }{}_{\infty }\right)}{{\upsilon }^{\otimes }{}^{2}}{l}^{3},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\theta }^{\otimes }=\frac{{T}^{\otimes }-{T}^{\otimes }{}_{\infty }}{{T}^{\otimes }{}_{w}-{T}^{\otimes }{}_{\infty }}\end{array}$ (7)

whereas ${x}_{1},{x}_{2}$ are the dimensionless coordinates and ${q}_{1},{q}_{2}$ are the dimensionless velocity components in ${x}_{1},{x}_{2}$ directions, ${\theta }^{\otimes }$ the function of dimensionless temperature, ${\mu }^{\otimes }$ the dynamic viscosity, $G{r}^{\otimes }$ the Grashof number and ${p}^{\ast }$ the dimensionless pressure by substituting the previous dimensionless variables into the equations (2)-(5). After ignoring terms of smaller orders of magnitude of $G{r}^{\otimes }$ (the Grashof number), the system of equations are obtained, then the governing equations become in form:

$\frac{\partial {q}_{1}}{\partial {x}_{1}}+\frac{\partial {q}_{2}}{\partial {x}_{2}}=0$ (8)

$\begin{array}{l}{q}_{1}\frac{\partial {q}_{1}}{\partial {x}_{1}}+{q}_{2}\frac{\partial {q}_{1}}{\partial {x}_{2}}\\ =-\frac{\partial {p}^{\ast }}{\partial {x}_{1}}+G{r}^{\otimes }{}^{\frac{1}{4}}{\sigma }^{\ast }{}_{{x}_{1}}\frac{\partial {p}^{\ast }}{\partial {x}_{2}}+\left({\sigma }^{\ast }{{}_{{x}_{1}}}^{2}+1\right)\frac{{\partial }^{2}{q}_{1}}{\partial {x}_{2}{}^{2}}+{\theta }^{\otimes }-{M}^{\ast }{q}_{1}-D{a}^{\oplus }{}^{-1}{q}_{1}\end{array}$ (9)

$\begin{array}{l}{\sigma }^{\ast }{}_{{x}_{1}}\left({q}_{1}\frac{\partial {q}_{1}}{\partial {x}_{1}}+{q}_{2}\frac{\partial {q}_{1}}{\partial {x}_{2}}\right)\\ =-G{r}^{\otimes }{}^{1/4}\frac{\partial {p}^{\ast }}{\partial {x}_{2}}+{\sigma }^{\ast }{}_{{x}_{1}}\left({\sigma }^{\ast }{{}_{{x}_{1}}}^{2}+1\right)\frac{{\partial }^{2}{q}_{1}}{\partial {x}_{2}{}^{2}}-{\sigma }^{\ast }{{}_{{x}_{1}}}_{{x}_{1}}{q}_{1}{}^{2}-D{a}^{\oplus }{}^{-1}{\sigma }^{\ast }{}_{{x}_{1}}{q}_{1}\end{array}$ (10)

$\begin{array}{l}{q}_{1}\frac{\partial {\theta }^{\otimes }}{\partial {x}_{1}}+{q}_{2}\frac{\partial {\theta }^{\otimes }}{\partial {x}_{2}}\\ =\frac{1}{P{r}^{\oplus }}\left({\sigma }^{\ast }{{}_{{x}_{1}}}^{2}+1\right)\left({\gamma }^{\ast }{\theta }^{\otimes }+1\right)\frac{{\partial }^{2}{\theta }^{\otimes }}{\partial {x}_{2}{}^{2}}+{q}^{\ast }{\theta }^{\otimes }+\frac{1}{P{r}^{\oplus }}\left({\sigma }^{\ast }{{}_{{x}_{1}}}^{2}+1\right){\gamma }^{\ast }{\left(\frac{\partial {\theta }^{\otimes }}{\partial {x}_{2}}\right)}^{2}\end{array}$ (11)

whereas ${\gamma }^{*}$ the parameter of thermal conductivity, ${q}^{*}$ the parameter of heat generation, ${M}^{\otimes }$ the magnetic field, Prandtl number is $P{r}^{\oplus }$ the and $D{a}^{\oplus }$ is Darcy number.

By multiplying Equation (10) by ${\sigma }^{\ast }{}_{{x}_{1}}$. The equation becomes:

$\begin{array}{l}{\sigma }^{\ast }{{}_{{x}_{1}}}^{2}\left({q}_{1}\frac{\partial {q}_{1}}{\partial {x}_{1}}+{q}_{2}\frac{\partial {q}_{1}}{\partial {x}_{2}}\right)\\ =-{\sigma }^{\ast }{}_{{x}_{1}}G{r}^{\otimes }{}^{1/4}\frac{\partial {p}^{\ast }}{\partial {x}_{2}}+{\sigma }^{\ast }{{}_{{x}_{1}}}^{2}\left({\sigma }^{\ast }{{}_{{x}_{1}}}^{2}+1\right)\frac{{\partial }^{2}{q}_{1}}{\partial {x}_{2}{}^{2}}-{\sigma }^{\ast }{}_{{x}_{1}}{\sigma }^{\ast }{{}_{{x}_{1}}}_{{x}_{1}}{q}_{1}{}^{2}-D{a}^{\oplus }{}^{-1}{\sigma }^{\ast }{{}_{{x}_{1}}}^{2}{q}_{1}\end{array}$ (12)

In this problem, the inviscide flow field is at rest, therefore $\frac{\partial {p}^{\ast }}{\partial {x}_{1}}=0$. By adding the Equations (9), (10), thus can be eliminate $\frac{\partial {p}^{\ast }}{\partial {x}_{2}}$ from these equations. The equation is:

$\begin{array}{l}{q}_{1}\frac{\partial {q}_{1}}{\partial {x}_{1}}+{q}_{2}\frac{\partial {q}_{1}}{\partial {x}_{2}}\\ =\left({\sigma }^{\ast }{{}_{{x}_{1}}}^{2}+1\right)\frac{{\partial }^{2}{q}_{1}}{\partial {x}_{2}{}^{2}}-\frac{{\sigma }^{\ast }{}_{{x}_{1}}{\sigma }^{\ast }{}_{{x}_{1}{x}_{1}}}{{\sigma }^{\ast }{{}_{{x}_{1}}}^{2}+1}{q}_{1}{}^{2}-D{a}^{\oplus }{}^{-1}{q}_{1}-\frac{{M}^{\ast }}{{\sigma }^{\ast }{{}_{{x}_{1}}}^{2}+1}{q}_{1}+\frac{1}{{\sigma }^{\ast }{{}_{{x}_{1}}}^{2}+1}{\theta }^{\otimes }\end{array}$ (13)

The governing equations could be in form:

$\frac{\partial {q}_{1}}{\partial {x}_{1}}+\frac{\partial {q}_{2}}{\partial {x}_{2}}=0$ (14)

$\begin{array}{l}{q}_{1}\frac{\partial {q}_{1}}{\partial {x}_{1}}+{q}_{2}\frac{\partial {q}_{1}}{\partial {x}_{2}}\\ =\left({\sigma }^{\ast }{{}_{{x}_{1}}}^{2}+1\right)\frac{{\partial }^{2}{q}_{1}}{\partial {x}_{2}{}^{2}}-\frac{{\sigma }^{\ast }{}_{{x}_{1}}{\sigma }^{\ast }{}_{{x}_{1}{x}_{1}}}{{\sigma }^{\ast }{{}_{{x}_{1}}}^{2}+1}{q}_{1}{}^{2}-D{a}^{\oplus }{}^{-1}{q}_{1}-\frac{{M}^{\otimes }}{{\sigma }^{\ast }{{}_{{x}_{1}}}^{2}+1}{q}_{1}+\frac{1}{{\sigma }^{\ast }{{}_{{x}_{1}}}^{2}+1}{\theta }^{\otimes }\end{array}$ (15)

$\begin{array}{l}{q}_{1}\frac{\partial {\theta }^{\otimes }}{\partial {x}_{1}}+{q}_{2}\frac{\partial {\theta }^{\otimes }}{\partial {x}_{2}}\\ =\frac{1}{P{r}^{\oplus }}\left({\sigma }^{\ast }{{}_{{x}_{1}}}^{2}+1\right)\left({\gamma }^{\ast }{\theta }^{\otimes }+1\right)\frac{{\partial }^{2}{\theta }^{\otimes }}{\partial {x}_{2}{}^{2}}+{q}^{\ast }{\theta }^{\otimes }+\frac{1}{P{r}^{\oplus }}\left({\sigma }^{\ast }{{}_{{x}_{1}}}^{2}+1\right){\gamma }^{\ast }{\left(\frac{\partial {\theta }^{\otimes }}{\partial {x}_{2}}\right)}^{2}\end{array}$ (16)

by using the dimensionless formulation (7) in The boundary conditions (6), it becomes in form:

$\begin{array}{l}\text{at}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{x}_{2}=0:{q}_{1}=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{q}_{2}=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\theta }^{\otimes }=1\\ \text{at}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{x}_{2}\to \infty :{q}_{1}=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\theta }^{\otimes }=0\end{array}$ (17)

the physical quantities such as ${\tau }_{w}$ is the shearing stress concerning the coefficient of skin friction ${C}^{\ast }{}_{f}$ and the rate of heat transfer with reference to Nusselt number $N{u}^{\ast }$, their equations:

${C}^{\ast }{}_{f}=\frac{2{\tau }_{w}}{{\rho }^{\ast }{U}^{2}}$ and $N{u}^{\ast }=\frac{{q}_{w}{x}_{1}}{{k}_{\infty }\left({T}^{\otimes }{}_{w}-{T}^{\otimes }{}_{\infty }\right)}$ (18)

By using the dimensionless formulation (7), the coefficient of local skin friction ${C}^{\ast }{}_{f}$ and the heat transfer rate with reference to Nusselt number Nu will be like that:

${C}^{\ast }{}_{f}{\left(\frac{G{r}^{\otimes }}{{x}_{1}}\right)}^{1/4}/2=\sqrt{1+{\sigma }^{\ast }{{}_{{x}_{1}}}^{2}}\frac{{\partial }^{2}{q}_{1}}{\partial {x}_{2}{}^{2}}$ (19)

$N{u}^{\ast }{\left(G{r}^{\otimes }/{x}_{1}\right)}^{-1/4}=-\left(1+{\gamma }^{\ast }\right)\sqrt{1+{\sigma }^{\ast }{{}_{{x}_{1}}}^{2}}\frac{\partial {\theta }^{\otimes }}{\partial {x}_{2}}$ (20)

whereas ${\tau }_{w}={\left({\mu }^{\ast }\stackrel{¯}{n}•\nabla \stackrel{¯}{{q}_{1}}\right)}_{{x}_{2}=0}$ and ${q}_{w}=-k{\left(\stackrel{¯}{n}•\nabla {T}^{\otimes }\right)}_{{x}_{2}=0}$.

2. Method of Solution

The Equations (14)-(16) are nonlinear partial differential and boundary conditions (17) can be solved numerically by finite difference fully (implicit method). By using the implicit method The derivatives w.r.t x1 and x2 are approximated by using central difference.

For example the momentum equation of q1:

$\begin{array}{l}{q}_{1}\frac{\partial {q}_{1}}{\partial {x}_{1}}+{q}_{2}\frac{\partial {q}_{1}}{\partial {x}_{2}}\\ =\left({\sigma }^{\ast }{{}_{{x}_{1}}}^{2}+1\right)\frac{{\partial }^{2}{q}_{1}}{\partial {x}_{2}{}^{2}}-\frac{{\sigma }^{\ast }{}_{{x}_{1}}{\sigma }^{\ast }{}_{{x}_{1}{x}_{1}}}{{\sigma }^{\ast }{{}_{{x}_{1}}}^{2}+1}{q}_{1}{}^{2}-D{a}^{\oplus }{}^{-1}{q}_{1}-\frac{{M}^{\otimes }}{{\sigma }^{\ast }{{}_{{x}_{1}}}^{2}+1}{q}_{1}+\frac{1}{{\sigma }^{\ast }{{}_{{x}_{1}}}^{2}+1}{\theta }^{\otimes }\end{array}$ (15)

$\begin{array}{l}\frac{\theta {q}_{1}{{}^{n+1}}_{i,j}\left({q}_{1}{{}^{n+1}}_{i+1,j}-{q}_{1}{{}^{n+1}}_{i-1,j}\right)+\left(1-\theta \right){q}_{1}{{}^{n}}_{i,j}\left({q}_{1}{{}^{n}}_{i+1,j}-{q}_{1}{{}^{n}}_{i-1,j}\right)}{2\Delta {x}_{1}}\\ +\frac{\theta {q}_{2}{{}^{n}}_{i,j}\left({q}_{1}{{}^{n+1}}_{i,j+1}-{q}_{1}{{}^{n+1}}_{i,j-1}\right)+\left(1-\theta \right){q}_{2}{{}^{n}}_{i,j}\left({q}_{1}{{}^{n}}_{i,j+1}-{q}_{1}{{}^{n}}_{i,j-1}\right)}{2\Delta {x}_{2}}\\ =\frac{1}{\Delta {x}_{2}{}^{2}}\left[\theta \left[{\left({\sigma }^{\ast }{{}_{{x}_{1}}}^{2}+1\right)}^{n}{}_{i,j+\frac{1}{2}}\left({q}_{1}{{}^{n+1}}_{i,j+1}-{q}_{1}{{}^{n+1}}_{i,j}\right)-{\left({\sigma }^{\ast }{{}_{{x}_{1}}}^{2}+1\right)}^{n}{}_{i,j-\frac{1}{2}}\left({q}_{1}{{}^{n+1}}_{i,j}-{q}_{1}{{}^{n+1}}_{i,j-1}\right)\right]\end{array}$

$\begin{array}{l}+\left(1-\theta \right)\left[{\left({\sigma }^{\ast }{{}_{{x}_{1}}}^{2}+1\right)}^{n}{}_{i,j+\frac{1}{2}}\left({q}_{1}{{}^{n}}_{i,j+1}-{q}_{1}{{}^{n}}_{i,j}\right)-{\left({\sigma }^{\ast }{{}_{{x}_{1}}}^{2}+1\right)}^{n}{}_{i,j-\frac{1}{2}}\left({q}_{1}{{}^{n}}_{i,j}-{q}_{1}{{}^{n}}_{i,j-1}\right)\right]\right]\\ -\frac{{\left({\sigma }^{\ast }{}_{{x}_{1}}\right)}_{i,j}{\left({\sigma }^{\ast }{}_{{x}_{1}{x}_{1}}\right)}_{i,j}}{1+{\left({\sigma }^{\ast }{{}_{{x}_{1}}}^{2}\right)}_{i,j}}{\left({q}_{1}{{}^{n}}_{i,j}\right)}^{2}-D{a}^{\oplus }{}^{-1}\left({q}_{1}{{}^{n}}_{i,j}\right)-\frac{{M}^{\otimes }}{1+{\left({\sigma }^{\ast }{{}_{{x}_{1}}}^{2}\right)}_{i,j}}\left({q}_{1}{{}^{n}}_{i,j}\right)+\frac{1}{{\left({\sigma }^{\ast }{{}_{{x}_{1}}}^{2}\right)}_{i,j}+1}{\theta }_{i,j}{{}^{\otimes }}^{n}\end{array}$

In the above, ${\theta }^{\otimes }$ is a weighting factor. Take the value of ${\theta }^{\otimes }=0.9$.

This equation will be written in form:

$\begin{array}{l}{A}_{i,j}{q}_{1}{{}^{n+1}}_{i-1,j}+{B}_{i,j}{q}_{1}{{}^{n+1}}_{i,j-1}+{C}_{i,j}{q}_{1}{{}^{n+1}}_{i,j}+{D}_{i,j}{q}_{1}{{}^{n+1}}_{i+1,j}+{E}_{i,j}{q}_{1}{{}^{n+1}}_{i,j+1}\\ ={F}_{i,j}{q}_{1}{{}^{n}}_{i-1,j}+{G}_{i,j}{q}_{1}{{}^{n}}_{i,j-1}+{K}_{i,j}{q}_{1}{{}^{n}}_{i,j}+{Q}_{i,j}{q}_{1}{{}^{n}}_{i+1,j}+{R}_{i,j}{q}_{1}{{}^{n}}_{i,j+1}+{O}_{i,j}\end{array}$

where

$\begin{array}{l}{A}_{i,j}=\frac{-\theta {q}_{1}{{}^{n+1}}_{i,j}}{2\Delta {x}_{1}}\\ {B}_{i,j}=\frac{-\theta {q}_{2}{{}^{n}}_{i,j}}{2\Delta {x}_{2}}-\frac{\theta {\left({\sigma }^{\ast }{{}_{{x}_{1}}}^{2}+1\right)}^{n}{}_{i,j+\frac{1}{2}}}{\Delta {x}^{2}{}_{2}}\\ {C}_{i,j}=\frac{\theta {\left({\sigma }^{\ast }{{}_{{x}_{1}}}^{2}+1\right)}^{n}{}_{i,j+\frac{1}{2}}}{\Delta {x}^{2}{}_{2}}+\frac{\theta {\left({\sigma }^{\ast }{{}_{{x}_{1}}}^{2}+1\right)}^{n}{}_{i,j-\text{\hspace{0.17em}}\frac{1}{2}}}{\Delta {x}^{2}{}_{2}}\\ {D}_{i,j}=\frac{\theta {q}_{1}{{}^{n+1}}_{i,j}}{2\Delta {x}_{1}}\\ {E}_{i,j}=\frac{\theta {q}_{2}{{}^{n}}_{i,j}}{2\Delta {x}_{2}}-\frac{\theta {\left({\sigma }^{\ast }{{}_{{x}_{1}}}^{2}+1\right)}^{n}{}_{i,j+\frac{1}{2}}}{\Delta {x}^{2}{}_{2}}\end{array}$

$\begin{array}{l}{F}_{i,j}=\frac{\left(1-\theta \right){q}_{1}{{}^{n}}_{i,j}}{2\Delta {x}_{1}}\\ {G}_{i,j}=\frac{\left(1-\theta \right){q}_{2}{{}^{n}}_{i,j}}{2\Delta {x}_{2}}+\frac{\left(1-\theta \right){\left({\sigma }^{\ast }{{}_{{x}_{1}}}^{2}+1\right)}^{n}{}_{i,j-\text{\hspace{0.17em}}\frac{1}{2}}}{\Delta {x}_{2}{}^{2}}\\ {K}_{i,j}=\frac{\left(\theta -1\right){\left({\sigma }^{\ast }{{}_{{x}_{1}}}^{2}+1\right)}^{n}{}_{i,j+\frac{1}{2}}}{\Delta {x}_{2}{}^{2}}+\frac{\left(\theta -1\right){\left({\sigma }^{\ast }{{}_{{x}_{1}}}^{2}+1\right)}^{n}{}_{i,j-\frac{1}{2}}}{\Delta {x}_{2}{}^{2}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{{\left({\sigma }^{\ast }{}_{{x}_{1}}\right)}_{i,j}{\left({\sigma }^{\ast }{}_{{x}_{1}{x}_{1}}\right)}_{i,j}}{1+{\left({\sigma }^{\ast }{{}_{{x}_{1}}}^{2}\right)}_{i,j}}{q}_{1}{{}^{n}}_{i,j}-D{a}^{\oplus }{}^{-1}-\frac{{M}^{\otimes }}{1+{\left({\sigma }^{\ast }{{}_{{x}_{1}}}^{2}\right)}_{i,j}}\end{array}$

$\begin{array}{l}{Q}_{i,j}=\frac{\left(\theta -1\right){q}_{1}{{}^{n}}_{i,j}}{2\Delta {x}_{1}}\\ {R}_{i,j}=\frac{\left(\theta -1\right){q}_{2}{{}^{n}}_{i,j}}{2\Delta {x}_{2}}+\frac{\left(1-\theta \right){\left({\sigma }^{\ast }{{}_{{x}_{1}}}^{2}+1\right)}^{n}{}_{i,j+\frac{1}{2}}}{\Delta {x}_{2}{}^{2}}\\ {O}_{i,j}=\frac{1}{\left(1+{\sigma }^{\ast }{{}_{{x}_{1}}}^{2}\right)}{\theta }_{i,j}{{}^{\otimes }}^{n}\end{array}$

3. Results

The numerical results are represented by graphs for the velocity q1 and temperature θ and Nusselt number Nu*. Use the different values of parameters to illustrate their effective on the velocity, temperature and Nusselt number. Figure 2 reveals the influence of magnetic parameter MÄ to velocity profiles q1, this figure shows that the increasingof magnetic parameter tends to increase the velocity q1 at the other parameters are constant as prÅ = 0.7, q* = 0.1, a = 0.2, DaÅ−1 = 0.1 and γ* = 2. Figure 3 indicates to the effective of increase the inverse Darcy number tends to increase velocity q1, at prÅ = 0.7, q* = 0.1, a = 0.2, MÄ = 0.2 and γ* = 2. Figure 4 shows that the heat generation q* increase tends to the velocity q1 increase at prÅ = 0.7, DaÅ−1 = 0.1, a = 0.2, MÄ = 0.2 and γ* = 2. Figure 5 illustrated that the thermal conductivity variation increase γ tends to the velocity q1 increase at prÅ = 0.7, DaÅ−1 = 0.1, a = 0.2, MÄ = 0.2 and q* = 0.1. Figure 6 indicates

Figure 2. Velocity distribution q1 for different magnetic parameter MÄ.

Figure 3. Velocity distribution q1 with various inverse Darcy number.

Figure 4. Velocity distribution q1 for different heat generation parameter q*.

Figure 5. Velocity distribution q1 for different thermal conductivity variationγ*.

Figure 6. The temperature θÄ with various magnetic.

to the influence of increase the magnetic field tends to decrease the temperature θÄ, at prÅ = 0.7, q* = 0.1, a = 0.2, DaÅ−1 = 0.1 and γ* = 2. Figure 7 and Figure 8 present the temperature θ* decrease with increase both the inverse of Darcy number and prandtle number prÅ at q* = 0.1, a = 0.2, MÄ = 0.2 and γ* = 2. Figure 9 and Figure 10 indicate to influence of prandtle number and magnetic field on Nusselt number, the increase of prandtle number and magnetic field parameter tends to increase the Nusselt number Nuat q* = 0.1, a = 0.2, DaÅ−1 = 0.1 and γ* = 2. Figure 11 and Figure 12 present increase the Nusselt number cause increase both of inverse Darcy number and heat generation parameter q* at prÅ = 0.7, a = 0.2, MÄ = 0.2 and γ* = 2. Figure 13 shows that the increase in thermal conductivity variation parameter γ tends to increase the Nusselt number Nu* at MÄ = 0.2, DaÅ−1 = 0.1, a = 0.2, q* = 0.1, and prÅ = 0.7.

4. Conclusion

This paper presents the influence of porous medium wavy vertical surface of

Figure 7. The temperature θÄ with various inverse Darcy number DaÅ−1.

Figure 8. The temperature θÄ for different prandtle number PrÅ.

Figure 9. Nusselt number with various prandtle number.

Figure 10. Nusselt number with various magnetic parameter.

Figure 11. Nusselt number with various inverse Darcy number.

Figure 12. Nusselt number with various heat generation.

Figure 13. Nusselt number with various thermal conductivity.

natural convection flow with thermal conductivity variable. The system of dimensional partial differential equations is turned into equations of partial differential whose non-linear, they are solved numerically by using finite difference (fully implicit method). This paper has a new result of influence of inverse Darcy number with the velocity, temperature and Nusselt number. The results have been obtained to velocity, temperature and Nusselt number with various parameters, such as magnetic field, Prandtl number, amplitude of wavelength, heat generation parameter and thermal conductivity variation parameter.

Nomenclature

a the dimensionless amplitude of the wavy surface

${B}_{ο}$ magnetic induction

${M}^{\ast }$ magnetic field parameter

$l$ characteristic reference wave-length

$N{u}^{\ast }$ local Nusselt number

${\stackrel{¯}{p}}^{\ast }$ the dimensional pressure of fluid

$P{r}^{\oplus }$ Prandtl number

$G{r}^{\otimes }$ Grashof number

${\sigma }_{ο}$ electrical conductivity

${g}^{\ast }$ gravitational acceleration

${T}^{\otimes }$ temperature

${q}_{1},{q}_{2}$ axial and normal dimensionless velocity Components, respectively

${\stackrel{¯}{q}}_{1},{\stackrel{¯}{q}}_{2}$ the velocity components along

( ${\stackrel{¯}{x}}_{1},{\stackrel{¯}{x}}_{2}$ )

${x}_{1},{x}_{2}$ dimensionless coordinates

${\stackrel{¯}{x}}_{1},{\stackrel{¯}{x}}_{2}$ dimensional coordinates

$D{a}^{\oplus }$ Darcy number

${p}^{\ast }$ the dimensionless pressure.

${q}^{\ast }$ heat generation parameter

${Q}_{o}{}^{*}$ heat generation constant

${c}^{\ast }{}_{p}$ specific heat at constant pressure

${k}^{\ast }$ permeability of the saturated permeable Medium

${K}^{\oplus }$ thermal conductivity

Greek Symbol

${\upsilon }^{\otimes }$ kinematics viscosity

${\rho }^{\ast }$ density

${\sigma }^{\ast }\left({\stackrel{¯}{x}}_{1}\right)$ surface geometry function

${\theta }^{\otimes }$ dimensionless temperature

${\mu }^{\otimes }$ the dynamic viscosity

${\beta }^{\otimes }$ coefficient of thermal expansion

${\gamma }^{\ast }$ thermal conductivity variation parameter

${\tau }_{w}$ the shearing stress

Superscripts

$\stackrel{¯}{}$ dimensional quantity

Subscripts

w wall surface

$\infty$ free stream

${x}_{1}$ derivative with respect to x

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

 [1] Alim, Md.A., Karim, Md.R. and Akand, Md.M. (2012) Heat Generation Effects on MHD Natural Convection Flow Along a Vertical Wavy Surface with Variable Thermal Conductivity. America Journal of Computational Mathematics, 2, 42-50.https://doi.org/10.4236/ajcm.2012.21005 [2] Vadasz, P. (2008) Emerging Topics in Heat and Mass Transfer in Porous Media. Springer, New York, 39-62, 85-110. https://doi.org/10.1007/978-1-4020-8178-1 [3] Wang, C.Y. (1989) Free Convection on a Vertical Stretching Surface. Journal of Applied Mathematics and Mechanics, 69, 418-420. https://doi.org/10.1002/zamm.19890691115 [4] Yao, L.S. (2006) Natural Convection along a Vertical Complex Wavy Surface. International Journal of Heat and Mass Transfer, 49, 281-286.https://doi.org/10.1016/j.ijheatmasstransfer.2005.06.026 [5] Hady, F.M., Mohamed, R.A. and Mahdy, A. (2008) Non-Darcy Natural Convection Flow Along a Vertical Wavy Plate Embedded in a Non-Newtonian Fluid Saturated Porous Medium. International Journal of Applied Mechanics and Engineering, 13, 91-100. [6] Cheng, P. and Minkowycz, W.J. (1977) Free Convection about a Vertical Flat Plate Embedded in a Porous Medium with Application to Transfer from a Dike. Journal of Geophysical Research, 82, 2040-2044. https://doi.org/10.1029/JB082i014p02040 [7] Lesnic, D., Ingham, D.B. and Pop, I. (1999) Free Convection Boundary Layer Flow along a Vertical Surface in a Porous Medium with Newtonian Heating. International Journal of Heat and Mass Transfer, 42, 2621-2627.https://doi.org/10.1016/S0017-9310(98)00251-8 [8] Hady, F.M. and Ibrahim, F.S. (1997) Forced Convection Heat Transfer on a Flat Plate Embedded in Porous Media for Power-Law Fluids. Transport in Porous Media, 28, 125-134. https://doi.org/10.1023/A:1006525412728 [9] Tashtoush, B. and Al-Odat, M. (2004) Magnetic Field Effect on Heat and Fluid Flow over a Wavy Surface with a Variable Heat Flux. Journal of Magnetism and Magnetic Materials, 268, 357-363. https://doi.org/10.1016/S0304-8853(03)00547-X [10] Molla, M., Hossain, M.A. and Yao, L.S. (2004) Natural Convection Flow along a Vertical Wavy Surface with Uniform Surface Temperature in Presence of Heat Generation/Absorption. International Journal of Thermal Sciences, 43, 157-163.https://doi.org/10.1016/j.ijthermalsci.2003.04.001 [11] Tannehill, J.C., Anderson, D.A. and Pletcher, R.H. (1997) Computational Fluid Mechanics and Heat Transfer. CRC press, Boca Raton, 247-448.