The Effect of Suction and Injection on MHD Flow Between Two Porous Concentric Cylinders Filled with Porous Medium

This Paper deals wit the effects of suction (injection) on magnetohydrodynamic (MHD) steady flow of a viscous and electrically conducting fluid in an annular porous region between two concentric cylinders. The inner cylinder is rotating with uniform angular velocity and the outer one is fixed. The two cylinders are porous with uniform permeability. It is assumed that the suction rate at the inner cylinder is equal to the injection rate at the outer cylinder. A uniform axial magnetic field was applied perpendicular to the flow direction. The flow resistance presented by the porous medium is governed by the Darcy law. By using similarity transformation, the governing partial differential equations have been transformed to a system of nonlinear ordinary differential equations. The solution of the obtained system in its general form has been obtained. Analytical expression for velocity field is obtained in terms of Bessel function of first and second kind. The effects of various parameters such as susction (injection), magnetic and permeability parameters on the flow are discussed and the obtained results are presented graphically. The obtained figures show that, the velocity distribution increased with the increase of permeability parameter of the the porous medium and with suction process. On the other hand, the velocity distribution decreased with the increase of magnetic parameter and with injection parameter.


Introduction
Flow in porous medium has been extensively investigated due to its numerous applications in geophysics, petroleum industry and chemical engineering etc. Many authors deal with the flow through porous cylinders [1][2][3][4][5][6][7]. It is known that even for the Newtonian fluid, if the cylinder surfaces are porous, a uniform suction applied on it can sensibly change the boundary layer structure, reduce the drag and hinder viscous diffusion of vorticity, [8,9]. The flows of many other fluid models have been studied in this geometry, but we shall not discuss them here, [10,11]. Terrill [12] carried out a detailed study of the laminar fow through a porous annulus by assuming the swirl to be zero and presented a series solution for small suction or injection. In recent years, the requirements of modern technology have stimulated interest in fluid flow studies, which involve the interaction of several phenomena. One such study is related to the flows of fluid through porous medium due to their applications in many branches in science and technology, viz. in the fields of agricultural engineering to study the underground water resources, in petroleum technology to study the movement of natural gas, oil, water through the oil reservoirs and in chemical engineering for filtration and purification processes. Such problems have also important applications in geo-thermals reservoirs and geo-thermal energy extractions. It is obvious that in order to utilize the geo-thermal energy to maximum, one should have a complete and precise knowledge of the amount of perturbations needed to generate flow in geo-thermal fluids. Abu-hijleh [13,14] analyzed convection heat transfer from a cylinder with porous medium. Hamza et al. [15] have concedred Poiseuille flow between two coaxial porous cylinders with slip on inner cylinder. Sharma et al. [16] investigates the unsteady flow of viscous incompressible fluid through porous medium induced by periodically heated half filled concentric cylindrical annulus placed horizontally. The effect of porous inserts on the natural convection heat transfer in a vertical open-ended annulus has been numerically investigated by Kiwan and Al-Zahrani [17]. Recently, the problem of two-phase unsteady MHD flow between two concentric cylinders of infinite length has been analysed by Jha et al. [18] when the outer cylinder is impulsively started. Some recent available literature dealing with the flows in second grade fluid can be found in Tan and Masuoka [19], Fetecau and Fetecau [20], Hayat et al. [21], Erdogan and Imrak [22], Sahoo [23], Ariel [24], Hayat et al. [25,26,27]. The effects of porous medium have been investigated by Hayat et al. [28] on the steady flow of a third grade fluid between two stationary porous plates. The governing nonlinear equations are solved by a homotopy analysis method.
The effect of magnetic field on the flow becomes important when the flowing fluid is conducting fluid. An understanding of MHD flows is important to the control of liquid metal pumps, of MHD power generators, and of liquid metal heat exchangers in nuclear fusion reactors. Experiments and numerical simulations have been carried out to reveal the behavior of flow under the influence of a magnetic field. Mahapatra [29] has investigated the problems of unsteady motion of a viscous conducting liquid between two porous nonconducting infinite concentric circular cylinders rotating with various angular velocities for some time in presence of a radial field. Khan et al. [30] found the analytic solution for flow of a MHD Sisko fluid through a porous medium by introducing the Darcy's law using the homotopy analysis method. In the limiting case, the obtained solution reduces to the well known solutions for a Newtonian fluid in non porous and porous media. Hayat et al. [31] investigated the MHD flow of a non-Newtonian fluid filling up the porous space in achannel with compliant walls. They used constitutive equations of a Jeffery fluid. Pantokatoras and Fang [32] investigate the Poiseuille and Couette flow in a fluid saturated Darcy-Brinkman porous medium channel with an electrically conducting fluid under the action of a magnetic and electric field. Exact analytical solutions are derived for fluid velocity. Zhao et al. [33] extend the previous work [32] to the case with a Darcy-Brinkman-Forchheimer porous medium. Srivastava et.al. [34] have concedred Poiseuille and Couette flow of an electrically conducting fluid through a porous medium of variable permeability under the transverse magnetic field. They used the Brinkman equation for flow through the porous medium and obtained a numerical solution for velocity and the volumetric flow rate using the Galerkin method.
Magnetic fluids can be considered as liquid mixtures made of magnetic particles chains and small molecules of solvent. At rest, the chains of suspensions are randomly entangled and they do not set up the suspensions structure. When the fluid is in motion, the chains tend to align themselves parallel to the direction of flow. This tendency increases with increasing shear rate, so that the effective viscosity decreases [35,36]. The orientation of magnetic particles in solvent under the influence of an external magnetic field is of great importance owing to the possibility of changes in structure and the products formation. Under the effect of magnetic field these particles may rearrange themselves taking the same direction as the magnetic field lines (i.e. oriented parallel to the magnetic lines), a circumstance that lead to an increase in suspension viscosity. Sheikhzadeh et al. [37] studied numerically the flow field and the heat transfer of a non Newtonian fluid in an axisymmetric channel with a permeable wall. Santhosh et al. [38,39] investigate the two fluid model for the flow of a Jeffrey fluid in tubes of small diameters in the presence of a magnetic field.
Fakour et al. [40] studied the heat transfer process in nanofluid and MHD flow in a channel with permeable walls. Aberkane et al. [41] study numerically the effect of an axial magnetic field imposed on incompressible flow of electrically conductive fluid between two horizontal coaxial cylinders. The effect of heat generation due to viscous dissipation is also taken into account. A finite difference implicit sheme was used in the numerical solution to solve the governing equations of convection flow and mass transfer. Aminfar et al. [42] studied experimentally the effects of using magnetic nanofluid and also applying an external magnetic field on the critical heat flux of sub-cooled flow boiling in vertical annulus. Seth and Singh [43] studied theoretically the effect of Hall current and a uniform transverse magnetic field on unsteady MHD Couette flow of class-II in a rotating system. Verma and Dixit [44] have concerned the MHD laminar steady flow of a viscous incompressible fluid in an annular porous region between two coaxial cylindrical pipes under the uniform transverse magnetic field. Beg et al. [45] presented a mathematical model for the steady, axisymmetric MHD flow of viscous, Newtonian, incompressible, electrically conducting liquid in a high porous regime intercalated between two concentric rotating cylinders in the presence of a radial magnetic field. The flow field of a thirdgrad non-Newtonian fluid in the annulus of rotating concentric cylinders has been investigated by Dizaji et al. [46] in the presence of magnetic field.
In the present problem we have studied the steady and laminar flow of a viscous, incompressible fluid in an annular porous concentric cylinders filled saturated porous medium under the uniform axial magnetic field. Our interest is in understanding the interaction between the viscous fluid and porous medium and the effect of suction or injection at the boundary. Exact solution is obtained and the results are presented for many cases. The Darcy's law is used for flow through a porous medium. The velocity field is obtained and exhibited graphically. The efect of various parameters has been analysed. It has been shown that, the efects of permeability and magnetic parameters have strong effects on flow characteristics.

Formulation of the Problem
We consider here the steady flow of an electrically conducting viscous incompressible fluid contained between two concentric porous cylinders which is filled with a porous medium. The inner cylinder is rotating with uniform angular velocity  around the system axis and the outer one is fixed. The walls of the cylinders being porous with uniform permeability. We use cylindrical coordinate system (r, , z) with the z-axis coincident with the common axis of the cylinders. We assume radius of inner and outer cylinder is 1 R and 2 R , respectively. A uniform axial magnetic field B of strength o B is acting on the axial direction, figure 1. In the analysis, we assume that the induced magnetic field is negligible.
For steady flow the equations governing the flow are the continuity and momentum equations: here V is the velocity, p is the pressure,  is the coefficient of viscosity and f is the body force per unit volume. The presence of a magnetic field and porous medium require that an additional forces be included in the equations of fluid motion aside from the usual pressure and shear forces. The added force takes the form: where J is the current density, B is the magnetic induction vector,  is the Darcy resistance for the fluid in the porous medium,  and k are the porousity and the permeability of the porous medium. The current density may be expressed by the generalized Ohm's law as: in which the terms E , respectively, represent the conduction and induction currents and  is the electrical conductivity. In the present study, we assume that the magnetic field is in z-direction. Therefore, neglecting the electric field E in equation 4 and replacing B by the externally applied field We shall seek an axisymmetric two dimentional solution and thus assume that all variables are independent on the coordinate  due to the symmetry about the z-axis.Therefore, In cylindrical coordinates the velocity field, magnetic force and and the Darcy resistance may be written as: Therefore, the governing equations 1 and 2 in component form are: The solution of the continuity equation, 9, gives r u R u the last equation satisfy the boundary conditions, The following dimensionless quantities are used: after straightforward computations it follows that the dimensionless radial velocity u is given by (after dropping the dimensionless mark "*" for simplicity): and the dimensionless azimuthal velocity  satisfies the boundary value problem: with the boundary conditions and S is the suction (injection) parameter, M is the magnetic parameter and K is permeability parameter. The values of M and K are an index to the relative importance of magnetic forces and permeability of the porous medium respectively. When 0 M  , magnetic forces are absent; when M increases, the magnetic force becomes increasingly important. The value 0 K  is for blocked medium (solid); and when   K (or 0 K / 1  ) the annular region beween the two cylinders becomes a hollow cylinder.

Solution of the Problem
Equation 16 is to be solved subject to the boundary conditions in equation 17. Also equation 16 can be reduced and an exact solution can be found. The classical Couette flow (Newtonian fluid without suction, mamnetic field and porous medium) is obtained as a special case.

Special Cases (a) Couette flow with suction (injection) only
In this case 0   (there is no magnetic field 0 M  and hollow region between the two cylinders   K ).
Therefore, equation 16 takes the form: the solution of the last equation is:

(b) Couette flow with magnetic field and porous medium
For no suction (injection) an analytical solution of the system of equations 16 and 17 can be found. In this case 0 S  and the governing equation is: the solution of the last equation is: where 1 J and 1 Y are the Bessel functions of first and second kind, respectively. Using the boundary conditions 17, 1 A and 2 A are obtained as follows

(c) Couette flow with suction (injection), magnetic field and porous medium
In this case the general solution of the governing equations 16 and 17 is:

Results and Discussion
We noticed a strong correlation among the many parameters upon which the flow depends. Among a variety of numerical experiments we report here only the most significant: we always fixed  . This is because permeability K increases as  decreases.

Conclusions
The main objective of the theoretical solution is to examine in detail the effect of suction and injection on MHD flow of viscous and electrically conducting fluid in an annular porous region between two concentric cylinders. The walls are porous and a suction or injection is applied at one of them. The governing equation are solved in exact manner considering the Lorentz force to account the resistance offered by the magnetic field and Darcy's law to account the flow resistance presented by the porous medium. The obtained velocity profiles possess modified Bessel functions of first and second kind. The results presented show how the solution varies along the flow and exhibits a strong dependence on the suction parameter. The effects of various parameters on the flow characteristics such as suction (injection), magnetic and permeability parameters are studied and obtained results are exhibited graphically. In limiting case when magnetic field and permeability is zero the obtained results reduces to the classical results of Couette flow in an annular cylinder. Therefore, we conclude the following remarks:  An increase in suction (injection) parameter S decrease (increase) the velocity profile monotonically.
 An increase in magnetic parameter M reduces the velocity profile monotonically due to the effect of magnetic force against the flow direction.This is because of the fact that increase in magnetic field increases the Lorentz force in opposite to the direction of flow. So we can use magnetic field to control the fluid velocity.
 An increase in the parameter K yields an effect opposite to that of the magnetic field.