Analytical solution of non-linear boundary value problem for MHD flow

Introduction

In the recent years, the investigation of flow over a stretching surface has attracted the attention of research community due to its significant applications in different industries such as extrusion paper production, extrusion of polymer sheets, metal and plastic industries [2] . Fang et al. [3] determined exact solution of the Navier Stokes equations analytically to study the MHD viscous flow under slip conditions over a permeable stretching surface. Magneto hydrodynamic (MHD) flow has important applications in MHD pumps, aerodynamics heating, MHD power generators, accelerators, purification of crude oil, petroleum industries and polymer technology [1] . Xinhua et al. [4] studied the asymmetric flow and heat transfer of viscous fluid between contracting/expanding rotating disks by using Homotopy analysis method. Siddique et al. [5] presented a new exact solution for MHD transient rotation flow of a second grade fluid in a porous space. Adabala Ramachandra Rao et al. [7] determined symmetric and asymmetric solutions of oscillatory MHD flow due to electrically rotating disks. Khan et al. [6] discussed MHD flows of a second grade fluid between two side walls perpendicular to a plate through a porous medium. Aziz et al. [9] discussed the thermal analysis of a longitudinal trapezoidal fin with temperature-dependent thermal conductivity and heat transfer coefficient. Abbasbandy [8] gave the application of Homotopy analysis method to nonlinear equation arising in heat transfer. Some exact solutions for the helical flow of a generalized Oldroyd-B fluid in a circular cylinder were presented by Fetecau et al. [10] . The purpose of this study is obtaining an analytical solution for radial velocity and axial velocity profiles on steady incompressible viscous MHD asymmetric flow of an electrically conducting fluid between two infinite parallel stationary coaxial porous disks of different permeability in the presence of a transverse magnetic field.

Mathematical formulation of the problem

Let us take the cylindrical polar coordinate system ) , , ( z r  for the physical problem. Here, we consider the case of an asymmetric steady laminar incompressible viscous flow of an electrically conducting fluid confined between two large stationary porous disks of infinite radii coinciding with the planes a z   with constant injection velocities 1

V at the upper disk in the presence of a uniform transverse magnetic field of intensity 0 B as shown in Fig. 1.

Fig 1: Geometry of the disks

To observe the effects of different permeability of the disks, we define the permeability parameter as follows:

With the presence of a uniform magnetic field, the governing equations for steady viscous flow of an electrically conducting fluid can be written as:

Where  denotes the density,  the gradient, V the fluid velocity vector, v kinematic viscosity, p the pressure, J the current density, B the total magnetic field so that The uniform stationary magnetic field B is in the transverse direction and the magnetic Reynolds number is taken small. As a consequence the induced magnetic field b is neglected. We further assume that there is no electric field (i.e. 0  E ) as there is no applied polarization voltage. This means that no energy is extracted or added to the fluid system. By employing the above flow assumptions, the electromagnetic body force occurring in eqn. (2.3) takes the following linearized form:

The components of velocity ) , , ( w v u along radial, transverse and axial directions for the present problem can be written as:

In view of eqns. (2.4)-(2.8), the governing eqns. (2.2)-(2.3) for an electrically conducting incompressible fluid in the presence of a uniform magnetic field are given in dimensionless form as:

The boundary condition at the two porous disks for the velocity field is specified as follows:

V are uniform injection velocities at the lower and upper disks respectively.

A similarity transformation can be used to reduce partial differential equations (2.10)-(2.12) into ordinary differential equations as follows:

The velocity components given in eqns. (2.15)

Where k is a constant of integration. The boundary conditions (2.13) in dimensionless form can be written as:

The shear stress on the disks is defined as:

We have to solve eqn. (2.18) subject to the boundary conditions (2.19).

Solution of the problem using the New Homotopy analysis method

New HAM is a non-perturbative analytical method for obtaining series solutions to nonlinear equations and has been successfully applied to numerous problems in science and engineering [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29] . In comparison with other perturbative and non-perturbative analytical methods, New HAM offers the ability to adjust and control the convergence of a solution via the so-called convergencecontrol parameter. Because of this, New HAM has proved to be the most effective method for obtaining analytical solutions to highly non-linear differential equations. Previous applications of New HAM have mainly focused on non-linear differential equations in which the non-linearity is a polynomial in terms of the unknown function and its derivatives. As seen above, the nonlinearity present in electro hydrodynamic flow takes the form of a rational function, and thus, poses a greater challenge with respect to finding approximate solutions analytically. Our results show that even in this case, New HAM yields excellent results. Liao [14][15][16][17][18][19][20][21][22] proposed a powerful analytical method for non-linear problems, namely the New Homotopy analysis method. This method provides an analytical solution in terms of an infinite power series. However, there is a practical need to evaluate this solution and to obtain numerical values from the infinite power series. In order to investigate the accuracy of the New Homotopy analysis method (New HAM) solution with a finite number of terms, the system of differential equations were solved. The New Homotopy analysis method is a good technique comparing to another perturbation method. The New Homotopy analysis method contains the auxiliary parameter h , which provides us with a simple way to adjust and control the convergence region of solution series. The approximate analytical solution of the eqns. ( 14)-( 18) using the Homotopy analysis method [35] is

The corresponding dimensionless radial velocity using the eqn. (2.21) is given by

And the shear stress using the eqn. (2.21) is given by

Results and Discussion

In this section, we present our findings in tabular and graphical forms together with the discussion and their interpretations. The parameters of the study are Permeability parameter A , the Reynolds number R and the Hartmann number M . Fig. 2 depicts the velocity profile along axial direction. It is noted that increasing the permeability parameter A , lowers the axial velocity profile in some fixed values of the other parameter R and M . Fig. 3 shows that increasing the permeability parameter A , rises the radial velocity profile in some fixed values of the other parameter R and M .Fig. 4 shows that the axial velocity profile increases with increase in the magnitude of Reynolds number R and some fixed values of other parameter A and M . Fig. 5 depicts that increasing the magnitude of Reynolds number R , increases the radial velocity near the lower disk and decrease near the upper disk. From Fig. 6, it is observed that the axial velocity profile increases near the lower disk and fall near the upper disk by increasing the values of the Hartmann number M and some fixed values of the other parameter A and R . From Fig. 7, it is noted that the radial velocity profile move towards the lower disk and it decreases by increasing the Hartmann number M and some fixed values of the other parameter A and R .

Conclusion

In this study, an analytical solution of the problem of MHD asymmetric flow of an electrically conducting fluid between two infinite parallel porous disks is investigated using a suitable similarity transformation. The analytical expressions of the dimensionless axial velocity and radial velocity have been derived by the Homotopy analysis method. In HAM, we can choose h in an appropriate way which controls the convergence of the series. This method can be easily extended to solve the non-linear initial and boundary value problems in physical sciences.

Acknowledgement

Researchers express their gratitude to the Secretary Shri S. Natanagopal, The Madura College Board, Madurai, Dr. K. M. Rajasekaran, The Principal and Dr. S. Muthukumar, Head of the Department of Mathematics, The Madura College (Autonomous), Madurai, Tamil Nadu, India for their constant support and encouragement.

The approximate solution of the eqns. (B.2) as follows:

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