Physics Research International

Volume 2016, Article ID 5407916, 10 pages

http://dx.doi.org/10.1155/2016/5407916

## Analytic Comparison of MHD Squeezing Flow in Porous Medium with Slip Condition

Department of Mathematics, National University of Computer and Emerging Sciences, FAST, Peshawar Campus, Pakistan

Received 10 October 2015; Accepted 11 January 2016

Academic Editor: Ashok Chatterjee

Copyright © 2016 Inayat Ullah et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The aim of this paper is to compare the efficiency of various techniques for squeezing flow of an incompressible viscous fluid in a porous medium under the influence of a uniform magnetic field squeezed between two large parallel plates having slip boundary. Fourth-order nonlinear ordinary differential equation is obtained by transforming the Navier-Stokes equations. Resulting boundary value problem is solved using Differential Transform Method (DTM), Daftardar Jafari Method (DJM), Adomian Decomposition Method (ADM), Homotopy Perturbation Method (HPM), and Optimal Homotopy Asymptotic Method (OHAM). The problem is also solved numerically using Mathematica solver NDSolve. The residuals of the problem are used to compare and analyze the efficiency and consistency of the abovementioned schemes.

#### 1. Introduction

The study of squeezing flow started in 19th century and it continues to receive considerable attention due to its practical applications in physical and biophysical areas, namely, food industry, chemical engineering, polymer processing, compression, and injection modeling. Stefan [1] accomplished elementary research in this field. Analysis of Newtonian fluid squeezed between two infinite planar plates is studied by Ran et al. [2]. Thin Newtonian liquid films squeezing between two plates were studied by Grimm [3]. Squeezing flow under the influence of magnetic field is broadly applied to bearing with liquid-metal lubrication [4–7].

The study of magnetic field effects on lubrication fluid films has attracted many scientists for a number of years. The flows of electrically conducting fluid through porous medium have attained incomparable status and have been the limelight of concern of many researchers in the last few decades. The particular applications are investigated in the study of ground water flow, irrigation problems, crude petroleum recovery, heat-storage beds, thermal and insulating engineering, chromatography, chemical catalytic reactors, and many more. Hughes and Elco [8] investigated the dynamics of an electrically conducting fluid in the presence of magnetic field between two parallel disks, one rotating at a constant angular velocity, for two cases, an axial magnetic field with a radial current and a radial magnetic field with an axial current. They discovered that the magnetic field affects the load capacity of the bearing and that the frictional torque on the rotor becomes zero for both the cases by applying electrical energy through the electrodes to the fluid. Ullah et al. studied the squeezing flow, in a porous medium, of a Newtonian fluid under the influence of imposed magnetic field [9]. The velocity profile of the fluid is discussed in the last work by considering various relations between the values of Reynolds and Hartmann number.

High order nonlinear boundary value problems arise in the study of squeezing flow of Newtonian as well as non-Newtonian fluids. The exact solution of these problems is sometimes difficult to find due to the mathematical complexity of Navier-Stokes equations. In order to solve these problems, various seminumerical techniques are widely used. We discuss here one by one these techniques and apply them to obtain the velocity profile of the fluid.

Homotopy Perturbation Method (HPM) was first introduced by He [10, 11]. Marinca et al. [12, 13] introduced OHAM for approximate solution of nonlinear problems of thin film flow of a fourth-grade fluid down a vertical cylinder and for the study of the behavior of nonlinear mechanical vibration of electrical machines. It is scrutinized that HPM and HAM are the special cases of OHAM [14].

Differential Transform Method (DTM) was initially introduced by Zhou in 1986 [15]. Islam et al. [16] successfully applied this technique for squeezing flow of a Newtonian fluid in porous medium channel. Ullah et al. [17] investigated the squeezing fluid flow under the influence of magnetic field with slip boundary condition using DTM. Ayaz [18] studied the applications of two-dimensional DTM in case of partial differential equations. Hassan [19] compared DTM with ADM in solving PDEs.

Adomian [20, 21] , in , introduced Adomian Decomposition Method for solving nonlinear functional equations. The technique is based on the decomposition of solution of nonlinear operator equation in a series of functions. Wazwaz [22] introduced the modified form of ADM and used it in many BVPs successfully. The basic idea of Daftardar Jafari Method (DJM) is introduced by Daftardar-Gejji et al. [23, 24] to solve fractional boundary value problems with Dirichlet boundary conditions. The solution of fifth- and sixth-order boundary value problem using DJM is studied by Ullah et al. and they got excellent results [25].

The goal of this research paper is to solve the model of squeezing flow of a Newtonian fluid in a porous medium with MHD effect by using HPM, OHAM, DTM, ADM, NIM, and the Mathematica solver NDSolve. Furthermore, to check the efficiency of each scheme, the residuals of the problem are used. Preparation of the model and basic ideas of the mentioned techniques along with their applications are discussed in the respective sections.

#### 2. Problem Modeling

The continuity and momentum equation for steady squeezing flow in a porous medium under the influence of magnetic field, as shown in Figure 1, areHere is the velocity vector, is the material time derivative, and is the Cauchy stress tensor given by with . is the total magnetic field given by . and represent the imposed and induced magnetic fields, respectively. is Darcy’s resistance given by [26, 27]The magnetohydrodynamic force can be written as follows: Suppose that the magnetic field is applied along -axis and the plates are nonconducting. For small velocity , the gap distance between the plates changes slowly with time so that it can be taken constant. The flow is axisymmetric with -axis perpendicular to plates and at the plates. The components of for the present case are . If is the generalized pressure and the flow is steady then by comparing components the Navier-Stokes equations (1) can be written asIntroducing stream function [9], eliminating the generalized pressure from (4) and (5), and using the transformation and the boundary conditions at at We havesubject to boundary conditionsIntroducing nondimensional parameters, Omitting , (8) and (9) becomeWe solve (11) and (12) by fixing , , and to find the particular solution in each case for comparison purpose.