This paper concerns the linear stability of three viscous fluid layers in porous media. The system is composed of a middle fluid embedded between two semi-infinite fluids, in which the effect of the normal magnetic field is to introduce. The principle aim of this work is to investigate the influence of fluid viscosity and the porosity effect on the growth rate in the presence of normal magnetic field. The parameters governing the layers flow system, the magnetic properties and porosity effects strongly influence the wave forms and their amplitudes and hence the stability of the fluid. The stability criteria are discussed theoretically and numerically and stability diagrams are obtained, where regions of stability and instability are identified. It is found that the stabilizing role for the magnetic field is retarded when the flow is in porous media. Moreover, the increase in the values of permeability parameters plays a dual role, in stability behavior. It has been found that the phenomenon of the dual (to be either stabilizing or destabilizing) role is found for increasing the permeability parameter. It is established that both the viscosity coefficient and the magnetic permeability damps the growth rate, introducing stabilizing influence. The role of the magnetic field and Reynolds number is to increase the amplitude of the disturbance leading to the destabilization state of the flow system, promote the oscillatory behavior. Influence of the various parameters of the problem on the interface stability is thoroughly discussed.
There has been a great deal of interest in magnetic fluids of the stability of hydrodynamic stability. A magnetic fluid, also known in the literature as a ferromagnetic fluid or simply a ferrofluid. In many previous researches has shown the importance of studying the hydrodynamic stability problems, for example capillary-gravity waves of permanent form at the interface between two unbounded magnetic fluids in porous media are investigated in paper [
The authors in paper [
Based on a modified Darcy’s law for a viscoelastic fluid, Sirwah [
In the present work we wish to consider an investigation of stability for flow in a porous medium under the effect of a magnetic field that is normal to the interface between the fluids. The considered system is composed of a viscous fluid layer of finite thickness embedded between two semi-infinite fluids. We have concentrated in this work to investigate the mechanisms of stability of three porous layers of fluids in the presence of normal magnetic field. This paper is organized as follows: This section has presented the motivation for the investigation in addition to relevant background information. In next section, we will give a formulation of the problem statement, including the basic equations of the fluid mechanics and Maxwell’s equations governing the motion of our model. In the third section and its subsections, are concerned with the derivation of the characteristic equation and numerical estimations for stability configuration. The salient results of our analysis are discussed and some important conclusions are drawn in final section of this paper.
The system under consideration is composed of an infinite horizontal viscous fluid sheet of vertical height
acting in the positive y-direction. The two interfaces are parallel and the flow in each phase is every where parallel to each other. The surface deflections are expressed by
First, we will use the dimensionless variables to provide improved insight into the physics and in order to understand hydrodynamic stability better. So we define the corresponding dimensionless variables using the half thickness of the middle fluid sheet L as a length scale. Thus the stream velocity and the time are made dimensionless using
sionless by
sure
The dynamics of the problem are described by the simultaneous solution of three field equations: Maxwell’s equations, Navier-Stokes equation, and the continuity equation. Assuming a quiescent initial state, therefore the base state velocity in the fluid layers is zero in which the flow is steady and fully developed. Fluid flow through a porous medium is often given by the phenomenological Darcy’s equation. Thus, the equations governing two- dimensional motion of a viscous incompressible fluid through porous medium are [
and the equation of continuity will be
where
meability parameter and
zontal gradient operator.
In writing Maxwell’s equations for the problem, we supposed that the electro-quasi-static approximation is valid for the problem, and hence the magnetic field equations read
Here,
and thus we have the Laplace equation in the form
where
Solution of the equations of motion cited before is accomplished by utilizing the convenient boundary conditions. The flow field solutions of the above governing equations have to satisfy the kinematic and dynamic boundary conditions at the two interfaces, which can be taken as
where
faces are moving with the fluids lead to
In addition the jump in the shearing stresses is zero across the interfaces, this gives
where,
Furthermore, the Maxwell’s conditions on the magnetic field where no free surface charges are present on the interfaces. The continuity of the normal component of the magnetic displacement at the interfaces reads:
The tangential component of the magnetic field is zero across the interfaces, this requires that from this equation, we have
where, we use the zero order from continuity of the normal component of magnetic field to express both
The completion of the mathematical description of the problem requires an additional interfacial condition determine the shape of the interface between the fluids, which is the dynamical equilibrium boundary condition in which the surface traction suffers a discontinuity due to the surface tension:
These boundary conditions represented here are prescribed at the interface
The analysis of linear theory, as presented in Chandrasekhar book [
To solve the equations for the fluid phases under consideration, the two-dimensional finite disturbances are introduced into the equation of motion and continuity equation as well as the boundary conditions. As a customary in hydrodynamic stability analysis [
where
Eliminating the pressure term from Equations (1) and (2) and using (15) and (17), we obtain the following equation
and
Using the normal mode solution (17) we can obtain the pressure from Equation (1):
Substituting, the solution of the analytical solution of Equation (18) into Equation (15) we get
Also, the solution of the magnetic potential, in view of Equation (5) may be taken the form
Since the disturbances vanish as
In this section, we will derive the dispersion relation controlling the stability behavior of the system. When the obtained solutions of the stream function, magnetic potential and surface tension are inserted into Equations (6)- (12), we have a linear homogeneous system of algebraic equations of the fourteen unknown coefficients
where 0 is a null vector, Z is a vector of unknown coefficients defined as
where the superscript T indicates the matrix transpose. A non-trivial solutions of the unknown coefficients
which represents the linear dispersion equation for surface waves propagating through a viscous layer embedded between two other fluids with the influence of constant horizontal magnetic field. This dispersion relation controls the stability in the present problem. That is, each negative of the real part of w corresponds to a stable mode of the interfacial disturbance. On the other hand, if the real part of w is positive, the disturbance will grow in time and the flow becomes unstable.
It is clear that the eigenvalue relation (24) is somewhat more general and quite complex, since
we get
braic equation for the frequency w which coincides with that obtained by Kwak and Pozrikidis [
In the following, numerical applications are carried out to demonstrate the effects of various physical parameters on the stability criteria of the system. In the present work, we will numerically solve the implicit dispersion relation by means of the Chebyshev spectral tau method [
In this section, the goal is to determine the numerical assess for the stability pictures for surface waves propagating through porous media. In order to present this examination, Equation (24) is used to control the stability behavior, which requires specification of the parameters: the magnetic field, the magnetic permeability, the porosity effect, the density, the viscosity. In the calculations given below all the physical parameters are sought in the dimensionless form as defined above. The stability of fluid sheets corresponds to negative values of the disturbance growth rate (i.e.
To show the effect of changes of the magnetic permeability ratio
correspondingly at higher values of the wave number, further the plane
The examination of the influence of the magnetic field
The examination of change of the lower to the middle fluid viscosity ratio
The influence of magnetic field
A conclusion that may be made from the comparison among the parts (a-c) of
This work is concerned with the influence of the normal magnetic field on the gravitational stability of a viscous fluid sheet of finite thickness. The sheet is embedded between two semi-infinite fluids layers moving in porous media, under the influence of magnetic field. The solutions of the linearized equations of motion under the boundary conditions lead to an implicit dispersion relation between the growth rate and wave number. The parameters governing the layers flow system, the magnetic properties and porosity effects strongly influence the wave forms and their amplitudes and hence the stability of the fluid. The stability criteria have been performed theoretically and numerically in which the physical parameters are put in the dimensionless form. Some stability diagrams have been plotted and discussed, in which the influence of the various parameters of the problem on the interface stability is thoroughly analyzed.
It has been found that the phenomenon of the dual (to be either stabilizing or destabilizing) role is found for increasing the permeability parameter. It is established that both the viscosity coefficient and the magnetic permeability damps the growth rate, introducing stabilizing influence, where a part of its kinetic energy may be absorbed. However, it is expected to be a more careful search would clarify that the motion of the interfacial waves will be more stable with the increase of the values of the viscosity as well as the magnetic permeability. In addition an increase of the lower to the middle fluid viscosity ratio decrease both the growth rate and the stability range of fluid sheet, which give a stabilizing influence on the stability behavior of the waves. This result confirmed the fact that when the lower fluid is more viscous than the upper, thus the system is stable. The role of the magnetic field and Reynolds number is to increase the amplitude of the disturbance leading to the destabilization state of the flow system, promote the oscillatory behavior.