# Finite Element Solution of an Unsteady MHD Flow through Porous Medium between Two Parallel Flat Plates.

1. Introduction

Theoretical study of magnetohydrodynamics (MHD) flow problems are frequently encountered in cooling systems of nuclear reactors, MHD generators, blood flow measurements, pumps, and accelerators.

Due to coupling of the equations for electrodynamics and fluid mechanics, exact solution is possible only for some simple situations. By using several numerical techniques, such as finite element method (FEM), finite volume method (FVM), and boundary element method (BEM), approximate solution for the MHD flow problems can be obtained.

Gapta and Singh  obtained the exact solutions for unsteady flow in some special cases. Ram and Mishra  investigated the unsteady flow through magnetohydrodynamic porous media. Singh and Lal  studied the FEM solution of time-dependent MHD flow equations. Ram and Jain  have discussed MHD free convective flow through a porous medium in a rotating fluid. Reddy and Bathaiah  have analyzed the Hall effects on MHD flow through a porous straight channel. Lee and Dulikravich  proposed FDM scheme for the 3-dimensional unsteady MHD flow together with temperature field. Sheu and Lin  presented a convection-diffusion-reaction model for solving the unsteady MHD flow using a FDM scheme. The stabilized FEM for solution of the 3-dimensional time-dependent MHD flow equations was given by Ben Salah and et al. . Chauhan and Rastogi  have studied the Hall effects on MHD slip flow and heat transfer through a porous medium over an accelerated plate in a rotating system. Saha and Chakrabarti  have investigated the impact of magnetic field strength on magnetic fluid flow through a channel. Moniem and Hassanin  have developed a solution of MHD flow past a vertical porous plate through a porous medium under oscillatory suction. Sa'adAldin and Qatanani  have studied the unsteady MHD flow through two parallel porous flat plates. Sivaiah and Srinivasa-Raju  have discussed the finite element solution of heat and mass transfer in MHD flow of a viscous fluid past a vertical plate under oscillatory suction velocity. Yuksel and Ingram  have investigated the numerical analysis of a finite element method, Crank-Nicolson discretization for MHD flows at small magnetic Reynold number. Beg et al.  have developed a finite element and network electrical simulation of rotating magnetofluid flow in nonlinear porous media with inclined magnetic field and Hall currents. Sa'ad Aldin and Qatanani  have studied the analytical and finite difference methods for solving unsteady MHD flow through porous medium between two parallel flat plates.

In this work, the finite element solution for the unsteady magnetohydrodynamics (MHD) flow of an electrically conducting, incompressible viscous fluid past through porous medium between two parallel plates in the presence of a transverse magnetic field and Hall effect is considered. A comparison study has been carried out between the finite difference and the finite element solutions. A case study is analyzed with both the finite element method (FEM) and the finite difference method (FDM), namely, the implicit scheme presented in . It was found that the finite element method (FEM) is more accurate for solving these type of problems.

2. Formulation of the Problem

We consider an unsteady flow of an electrically conducting, incompressible viscous fluid past through porous medium between two parallel plates with Hall effect. Let the x-axis be taken along the plates and y-axis be normal to the plates. The fluid is subjected to a constant transverse magnetic field of strength [B.sub.0] in the y direction, with the flow being considered in the x direction, as illustrated in Figure 1. The governing equations for the unsteady, viscous incompressible flow of an electrically conducting fluid for the Brinkman-extended Darcy model are as follows :

Equation of continuity is

[nabla] x q = 0. (1)

Equation of motion is

[partial derivative]q/[partial derivative]t + (q - [nabla]) q = 1/[rho] [nabla]p + [mu]/[rho] [[nabla].sup.2] q - [mu]/[rho] q/k + 1/[rho] J x B. (2)

General Ohm's law is

J + w[tau]/[B.sub.0] J x B = [sigma] [E + q x B + - 1/[[rho].sub.e][n.sub.e] [nabla][P.sub.e]. (3)

Gauss's law of magnetism is

[nabla] x B = 0, (4)

where q is the velocity vector, [rho] is the fluid density, p is the pressure, J is the current density, B is the magnetic vector, [mu] is the coefficient of viscosity, [sigma] is the electrical conductivity, k is the permeability of the medium, w is the electron frequency, t is the electron collision time, [[rho].sub.e] is the electric charge, [n.sub.e] is the number density of electron, [P.sub.e] is the electron pressure, and E is the electric field.

We assume E to be negligible and the magnetic Reynold's number is small so that magnetic induction effect is ignored. Moreover, in the absence of pressure gradient, the ion-slip effects and electron pressure gradient, we have

[mathematical expression not reproducible], (5)

[j.sub.x] = m[j.sub.z], (6)

[j.sub.y] = 0, (7)

[j.sub.z] = [sigma][B.sub.0]u - m[j.sub.x]. (8)

Solving (6) and (8), we have

[mathematical expression not reproducible]. (9)

As the plates are infinite, there is no x dependence. Consequently, (2) and (3) take the following form:

[mathematical expression not reproducible], (10)

where u is the axial velocity, v is the kinematic viscosity, and m = w[tau] is the Hall parameter. The initial and boundary conditions are given by

u = 0, t [less than or equal to] 0, (11)

u = 0, y = [+ or -] h, t > 0.

Upon introducing the nondimensional quantities,

[mathematical expression not reproducible], (12)

where M is the Hartman number, K is the Darcy parameter, and V is the mean velocity of the fluid. Then, the partial differential equations (10) together with the initial and boundary conditions (11) become

[mathematical expression not reproducible], (13)

0 = [partial derivative]P/[partial derivative]Y, (14)

subject to the initial and boundary conditions:

U = 0, T [less than or equal to] 0, (15)

U = 0, Y = [+ or -] 1, T > 0.

In virtue of (14), the pressure is independent of Y; then it is a function of T only. In this case, we can take the pressure gradient as a constant quantity; that is,

[partial derivative]P/[partial derivative]X = -[P.sub.0], (16)

where [P.sub.0] > 0; thus (13) becomes

[partial derivative]U/[partial derivative]T = [P.sub.0] + [[partial derivative].sup.2]U/[partial derivative][Y.sup.2] - aU (17)

subject to the initial and boundary conditions:

U = 0, T [less than or equal to] 0, (18)

U = 0, Y = [+ or -] 1, T > 0,

where a = ([M.sup.2]/(l + [m.sup.2]) + 1/K).

3. Finite Element Method

3.1. Variational Formulations and Galerkin Approximation. The dimensionless partial differential equation (17) subject to the initial and boundary conditions (18) is solved by weighted residual Galerkin finite element method. The standard approach to deriving a Galerkin scheme is to multiply both sides of (17) by a test function [psi] [member of] [H.sup.1.sub.0] [-1, 1] and integrate over the domain

[mathematical expression not reproducible], (19)

where

[H.sup.1.sub.0] := {[psi] [member of] [L.sup.2] [-1, 1], [partial derivative][psi]/[partial derivative]Y [member of] [L.sup.2] [-1,1], [psi](-1) (20)

= [psi](1) = 0}.

Integrating by parts, we obtain

<[partial derivative]U/[partial derivative]T, [psi]) = ([P.sub.0], [psi]) - ([partial derivative]U/[partial derivative]Y, [partial derivative][psi]/[partial derivative]Y) - (aU, [psi]), (21)

where (*, *) denotes the [L.sup.2]-inner product and {U, [psi]) = 0 for T [less than or equal to] 0.

We shall approximate the solution of (21) by assuming that U and [psi] lie in finite dimensional subspace of [H.sup.1.sub.0] [-1,1] for each T. Let [[phi].sub.i] [member of] [H.sup.1.sub.0] [-1,1] for i = 1,...,W and assume that the set [[phi].sub.i],..., [[phi].sub.W] is linearly independent. Further, let [S.sub.h] be a partition of the interval [-1,1] into subintervals -1 = [Y.sub.0] < [Y.sub.1] < *** < [Y.sub.w] < [Y.sub.w+1] = 1. Now we define the finite dimensional space [V.sub.h] spanned by [[phi].sub.i],..., [[phi].sub.w] as

[V.sub.h] := {v [member of] [H.sup.1.sub.0] [-1,1], (22)

v is piecewise linear function on [S.sub.h]}.

To this end, the approximate solution [u.sub.h] is

[u.sub.h] = [W.summation over (i=1)] [u.sub.i] (T) [[phi].sub.i] (Y). (23)

Inserting (23) into (21) and selecting as trial function [psi] the basis function of [u.sub.h], we obtain a system of ODEs:

B[??] = P - Au - Bau, (24)

where [??] = [([partial derivative][u.sub.1]/[partial derivative]T, ..., [partial derivative][u.sub.w]/[partial derivative]T).sup.T] and u = [([u.sub.1],..., [u.sub.w]).sup.T]. Here A is the Stiffness matrix and B is the Mas matrix defined, respectively, as

[mathematical expression not reproducible], (25)

with [[phi].sub.i] ([Y.sub.i]) = [[delta].sub.ij] being the usual finite element basis corresponding to the partition [S.sub.h]. Thus, to compute the entries of the Stiffness matrix A, next, we need to determine [[phi]'sub.i], Y).

Here

[mathematical expression not reproducible],(26)

and then

[mathematical expression not reproducible], (27)

where [h.sub.i] = [Y.sub.i] - [Y.sub.i-1].

Using (26) and (27), with a uniform mesh [h.sub.i] = h, we get

[mathematical expression not reproducible], (28)

[mathematical expression not reproducible], (29)

[mathematical expression not reproducible]. (30)

Scheme (24) is called semidiscretization, since [u.sub.h] is still a continuous function of T .

3.2. Time Stepping. In this section, we consider the semidiscretization in time. We first discretize the time interval (0, T) into a uniform grid with size k = T/N. Approximating the derivative in (24) at time level [T.sup.n] by the Crank-Nicolson scheme with [u.sup.0] = 0, we have

B([u.sup.n] - [u.sup.n-1]/k) = P - A ([u.sup.n] - [u.sup.n-1]/2) (31)

- Ba ([u.sup.n] - [u.sup.n-1]/2).

Then, we rewrite the Crank-Nicolson method as

(B + k/2(A + Ba))[u.sup.n] (32)

= (B - k/2 (A + Ba)) [u.sup.n-1] + kP.

Thus, we have the full discretization which is simply a combination of discretization in space and time:

U = [u.sup.n.sub.h]. (33)

4. Numerical Results and Discussion

To show the efficiency of the FEM described in the previous parts and to draw a comparison with the FDM, we present some examples. These tests are chosen such that there exist analytical solutions for them to give an obvious overview of the methods presented in this work.

Numerical Example 1. As an application to the FEM and FDM, we consider the following test case with the values M = 1, m = 1, K = 0.1, [P.sub.0] = 1, T = 0.25 fixed, and Y [member of] [-1,1]. Next the computed matrices A, B, and P given in (28), (29), and (30), respectively, are used in (32) to obtain the velocity U(Y,T).

Table 1 compares the exact values for the velocity U(Y, T) with both the FEM and the FDM values (for more details on the exact and FDM solutions, see ). A further comparison between the exact, FEM, and FDM values for the velocity can be observed in Figure 2. A plot of the absolute error that resulted from the FEM can be seen in Figure 3. Figure 4 compares the absolute errors obtained from the FEM and the FDM solutions.

Numerical Example 2. As for another test case, we take M = 2, m = 1, K = 0.3, [P.sub.0] = 1, Y = 0.5 fixed, and T [member of] [0,1]. Figure 5 compares the exact, FEM, and FDM values for the velocity. Figure 6 presents a plot of the absolute error that resulted from the FEM. A comparison between the absolute errors obtained from the FEM and the FDM solutions can be seen in Figure 7.

5. Conclusions

MHD flow problems, which have a very important place in physics and engineering, are usually hard to solve analytically. Therefore, it is required to obtain approximate solutions using computational methods. In this work, the problem of unsteady MHD flow through porous medium in the presence of magnetic field between two parallel flat plates has been investigated and solved using the FEM.

A comparison between FFM and FDM has been carried out. The exact results and the numerical results using the FEM have showntobeinclosedagreement. Thiscanclearlybe seen in Figures 2,3,5, and 6. The results of the numerical examples indicate that the FFM is more accurate than the FDM (see Figures 4 and 7). This asserts the ability and reliability of the FEM for solving these types of problems.

https://doi.org/10.1155/2017/6856470

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

 S. G. Gapta and B. Singh, "Unsteady mhd flow in a rectngular chanael under transverse magnatic field," Indian J. Pure Appl. Math, vol. 3, no. 6, pp. 1038-1047, 1972.

 G. Ram and R. S. Mishra, "Unsteady flow through magnetohydrodynamic porous media," Indian Journal of Pure and Applied Mathematics, vol. 8, no. 6, pp. 637-647, 1977.

 B. Singh and J. Lal, "Finite element method for unsteady MHD flow through pipes with arbitrary wall conductivity," International Journal for Numerical Methods in Fluids, vol. 4, no. 3, pp. 291-302, 1984.

 P. C. Ram and R. K. Jain, "MHD free convective flow through a porous medium in a rotating fluid," International Journal of Energy Research, vol. 14, no. 9, pp. 933-939, 1990.

 N. B. Reddy and D. Bathaiah, "Hall effect on MHD flow through a porous straight channel," Defence Science Journal, vol. 32, no. 4, pp. 313-326, 1982.

 S. Lee and G. S. Dulikravich, "Magnetohydrodynamic steady flow computations in three dimensions," International Journal for Numerical Methods in Fluids, vol. 13, no. 7, pp. 917-936, 1991.

 T. W. Sheu and R. K. Lin, "Development of a convection-diffusion-reaction magnetohydrodynamic solver on nonstaggered grids," International Journal for Numerical Methods in Fluids, vol. 45, no. 11, pp. 1209-1233, 2004.

 N. Ben Salah, A. Soulaimani, and W. Habashi, "A finite element method for magnetohydrodynamics," Computer Methods in Applied Mechanics and Engineering, vol. 190, no. 43-44, pp. 5867-5892, 2001.

 D. S. Chauhan and P. Rastogi, "Hall effects on MHD slip flow and heat transfer through a porous medium over an accelerated plate in a rotating system," International Journal of Nonlinear Science, vol. 14, no. 2, pp. 228-236, 2012.

 S. Saha and S. Chakrabarti, "Impact of magnetic field strength on magnetic fluid flow through a channel," International Journal of Engineering Research and Technology, vol. 2, no. 7, pp. 1-8, 2013.

 A. A. Moniem and W. S. Hassanin, "Solution of MHD Flow past a vertical porous plate through a porous medium under oscillatory suction," Applied Mathematics, vol. 4, pp. 694-702, 2013.

 A. Sa'adAldin and N. Qatanani, "Analytical and numerical methods for solving unsteady MHD flow problem," International Journal of Mathematical Sciences and Engineering Applications, vol. 9, no. 2, pp. 307-318, 2015.

 S. Sivaiah and R. Srinivasa-Raju, "Finite element solution of heat and mass transfer flow with Hall current, heat source, and viscous dissipation," Applied Mathematics and Mechanics. English Edition, vol. 34, no. 5, pp. 559-570, 2013.

 G. Yuksel and R. Ingram, "Numerical analysis of a finite element, Crank-Nicolson discretization for MHD flows at small magnetic Reynolds numbers," International Journal of Numerical Analysis and Modeling, vol. 10, no. 1, pp. 74-98, 2013.

 A. Beg, S. Rawat, J. Zueco, L. Osmond, and R. Gorla, "Finite element and network electrical simulation of rotating magnetofluid flow in nonlinear porous media with inclined magnetic field and hall currents," Theoretical and Applied Mechanics, vol. 41, no. 1, pp. 1-35, 2014.

 A. Sa'ad Aldin and N. Qatanani, "Analytical and numerical methods for solving unsteady mhd flow through porous medium between two parallel flat plates," An-Najah University Journal for Research, vol. 30, no. 1, pp. 173-186, 2016.

 P. Knabner and L. Angermann, Numerical methods for elliptic and parabolic partial differential equations, vol. 44 of Texts in Applied Mathematics, Springer-Verlag, New York, NY, USA, 2003.

Department of Mathematics, Faculty of Sciences, An-Najah National University, Nablus, State of Palestine

Received 27 January 2017; Accepted 19 April 2017; Published 14 June 2017

Caption: FIGURE 1: Schematic diagram of the system.

Caption: FIGURE 2: The exact, FEM, and FDM values for the velocity.

Caption: FIGURE 3: The absolute error resulted from the FEM approximation.

Caption: FIGURE 4: The absolute error resulted from the FEM and FDM approximations.

Caption: FIGURE 5: The exact, FEM, and FDM values for velocity case 2.

Caption: FIGURE 6: The absolute error resulted from FEM approximation case 2.

Caption: FIGURE 7: The absolute error that resulted from FEM and FDM approximations case 2.
```Table 1: The exact, (FEM) and (FDM) solutions of the velocity U.

Y      Exact solution     FEM solution
[U.sub.E]        [U.sub.FE]

-0.8   0.0439256214739   0.0439263413482
-0.6   0.0662966816784   0.0662977446546
-0.4   0.0773731593814   0.0773743987214
-0.2   0.0824053505275   0.0824066768827
0      0.0838631333124   0.0838644864314
0.2    0.0824792484052   0.0824805760964
0.4    0.0775578709141   0.0775591133358
0.6    0.0666815959272   0.0666826648556
0.8    0.0446904326399   0.0446911643550

Y           [absolute value of           FDM solution
([U.sub.E] - [U.sub.          [U.sub.FD]
FE])] by FEM

-0.8   7.1987427380237 x [10.sup.-7]    0.0439284690437
-0.6   1.0629762063479 x [10.sup.-6]    0.0662354201183
-0.4   1.2393400675991 x [10.sup.-6]    0.0772434172904
-0.2   1.3263552283981 x [10.sup.-6]    0.0822283020686
0      1.3531190140581 x [10.sup.-6]    0.0836692897549
0.2    1.3276912429471 x [10.sup.-6]    0.0823013886500
0.4    1.2424216914302 x [10.sup.-6]    0.0774266269162
0.6    1.0689283097276 x [10.sup.-6]    0.0666185173810
0.8    7.3171502338459 x [10.sup.-7]    0.0446921159260

Y           [absolute value of
([U.sub.E] - [U.sub.
FD])] by FDM

-0.8   0.2847569757748 x [10.sup.-5]
-0.6   0.6126156005674 x [10.sup.-4]
-0.4   0.1297420909940 x [10.sup.-3]
-0.2   0.1770484588176 x [10.sup.-3]
0      0.1938435575730 x [10.sup.-3]
0.2    0.1778597551711 x [10.sup.-3]
0.4    0.1312439979023 x [10.sup.-3]
0.6    0.6307854626647 x [10.sup.-4]
0.8    0.1683286084617 x [10.sup.-5]
```
Title Annotation: Printer friendly Cite/link Email Feedback Research Article; magnetohydrodynamics Sa'adAldin, AbdelLatif; Qatanani, Naji Journal of Applied Mathematics Report Jan 1, 2017 3105 A Mathematical Model of Malaria Transmission with Structured Vector Population and Seasonality. On the Solution of the Eigenvalue Assignment Problem for Discrete-Time Systems. Cooling systems Finite element method Flow (Dynamics) Fluid dynamics Magnetic fields Magnetohydrodynamics Porous materials Viscous flow