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The stability of stratified of incompressible, viscoelastic plasma through a porous medium in the presence of the quantum mechanism is considered. The dispersion relation is obtained using the normal mode technique. The behavior of growth rate with respect to the quantum effect, strain retardation time and stress relaxation time are examined in the presence of porosity of the porous medium, the medium permeability, kinematic viscosity. It is shown that, the presence of quantum term stabilizes a certain wave number band, whereas the system is unstable for all wave numbers in the absence of quantum term. The considered parameters beside the quantum term will bring about more stability on the considered system.

The Rayleigh-Taylor instability (RTI) is an important hydrodynamic effect that arises when a heavy fluid is accelerated into a lighter one. Similar to pouring water into oil, the heavier fluid, once perturbed, streams to the bottom, pushing the light fluid aside. This notion for a fluid in a gravitational field was first discovered by Rayleigh [

In plasmas, RTI can occur when dense plasma is supported against gravity by the pressure of the magnetic field. The investigation of Rayleigh-Taylor instabilities in magnetized plasma is a problem of considerable interest in space (ionospheric spread-F), fusion (curvature induced instabilities like interchange, ballooning, etc.) and the astrophysical plasmas. For a system of two incompressible plasma superimposed one over the other have been studied by Kalra and Talwar [

The linear growth rate of a finite layer plasma in which the density is continuously stratified exponentially along the vertical is studied by Goldston and Rutherford [

The hydrodynamic instabilities in quantum plasmas have been an important subject of research in the last a few years, where the quantum effects plays an important role in the behavior of the charged plasma particles when the de Broglie wavelength of the charge carriers becomes equal to or greater than the dimension of the quantum plasma system [

The purpose of this work is to examine theoretically the effect of the quantum mechanism on RTI for a finite thickness layer of incompressible viscoelastic plasma through porous media. This layer is confined between two rigid boundaries (). Using the normal mode approach, the dispersion relation is obtained analytically, and numerically analyzed.

Our starting point is the set of equations of the quantum hydrodynamic (QHD) for an plasma model through a porous media (4-10 and 27-34).

Here is the velocity of the fluid, is the density, thermal pressure, is the gravitational acceleration, is permeability of the medium and is the porosity (). Note that when and the system corresponds to a nonporous medium. is the viscosity of plasma. Here is the stress relaxation time, is the strain retardation time.

is represented by Bohm Potential term (also, is called a quantum pressure and for more details on the derivation, see refs. [27,35]), where is the Plank constant, is the electron mass and the ion mass.

To investigate the stability of hydrodynamic motion, we ask how the motion responds to a small fluctuation in the value of any of the flow variables appearing in the Euler equations. If the fluctuation grows in amplitude so that the flow never returns to its initial state, we say that the flow is unstable with respect to fluctuations of that type. Accordingly, we replace the variables in Equations (1) and (2) as in the form and. The quantities with subscripts “0” represent the unperturbed, or “zeroth-order” motion of the fluid, while the quantities with subscripts “1” represent a small perturbation about the zeroth-order quantities (first-order or linearized quantities); that is, and Substituting these expressions into Equations (1)-(4). In particular example of RTI we consider the fluid initially at rest. This means that, so the linearized equations can be easily derived from Equations (1) and (2) as

(5)

We now appeal to the fact that, for many situations of interest in ICF (inertial-confinement fusion), unstable flow occurs at velocities much smaller than the local sound speed. This has the effect that accelerations in the flow are not strong enough to change the density of a fluid element significantly, so the fluid moves without compressing or expanding. In such a situation we call the flow incompressible. Provided that we are well away from shock waves or centers of convergence, the assumption of incompressible flow is often valid. To say that fluid elements move without changing density is to say that the Lagrangian total derivative of density is zero, that

We also linearize this equation, where the first-order quantities, as before become

Comparing this equation to Equation(4), which can be rewritten in expanded form as

we see that subtracting Equation (7) from Equation (8) yields

This is a consequence of the assumption of incompressible flow. So, we can use either Equation (7) or Equation (9) to replace the linearized continuity Equation (4) under this assumption. One can seen that the set of Equations (3), (7) and (9) is complete for describing the quantum effects on the Rayleigh-Taylor instability of incompressible plasma. Now, where

,

The fluid is arranged in horizontal strata, and then is a function of the vertical coordinate and.Then, the system Equations (3), (7) and (9) become as in the appendix (Equations (30)-(37)).

Considering that the perturbation in any physical quantity takes the form

where and are horizontal components of the wave-number vector such that and (may be complex ()) is the frequency of perturbations or the rate at which the system departs from equilibrium thee initial state. Using the expression (10) in the system of equations (30)-(37) in the appendix, they become as in the appendix (see Equations (38)-(45))

Now, eliminating some of the variables from the system of Equations (38)-(42) in the appendix, we have

(11)

and if we consider,

Then Equation (11) becomes

where

For a finite thickness layer of incompressible plasma through porous media bounded on the other interface by a rigid boundary at, and, Equation (13) takes the form

where.

Now, we put the solution of Equation (15) in the form

and substituting into Equation

(15), the dispersion relation may be written as

Now, we insert the following dimensionless quantities in Equation (16)

where is the plasma frequency, and use Equation (12), then the dispersion relation (16) takes the form

From the system of Equations (17)-(21), it is clear that, the effect of strain retardation time is connected by the presence of the resistance term (i.e., the effect of will occur in the presence of), while the effect of stress relaxation time may be occur alone or it is connected by the presence of, and.

Now, where and in the case of and (stable oscillations), then Equation (17) becomes

Here, some special cases are considered from Equation (22) to clarify the different roles of the parameter’s problem.

1) At, we find that, and

So the normalized growth rate from Equation (22) is

This case is considered by Goldston and Rutherford (see ref. [

2) At, we find that, , while is as in Equation (21). So the normalized growth rate from Equation (22) is

This case is studied in Refs. [29-33]. It is clarified that, the quantum term has stabilizing effect on RTI problem. This influence is obvious from Equations (23) and (24), where for all the values of wave number. Also, the system increases as increases through the range, when it starts to decreases asincreases, where

and at the system arrives to complete stability with

3) At we find that, while

and

Then the normalized growth rate from Equation (22) is given by

Also, from Equations (23) and (25) one can see that, the stress relaxation time has a stabilizing role on the given system. This role increases with increasing of the magnitudes of stress relaxation time. While in the presence of quantum term the normalized growth rate in Equation (25) takes the form

from above results it very clear that

4) At, we find that, , while are as in Equations (20) and (21). So the normalized growth rate given by

(27)

corresponds Equation (33) ref. [

(28)

and in the presence of the strain retardation, Equation(28) becomes

(29)

From Equations (28) and (29) one can see that, the stabilizing role that plays the stress strain retardation time on the given system.

In the general case, if we wish to look into the effect of various factors, on the instability of the considered system, Equation (22) is to be numerically solved. Pertaining results are presented in Figures 1-3.

mum instability, for our example in

The role of porosity of the porous medium, the medium permeability, kinematic viscosity with quantum term is explained in ref. [

Figures 2 and 3 are plotted to indicate the influence of the strain retardation time and the stress relaxation time on the problem, respectively, in the presence of quantum effect.

at. The same note for the stress relaxation time is shown in

The effect of quantum term on the Rayleigh-Taylor instability of stratified fluid/plasma through porous media has been studied. The effect of elasticity is revealed through the strain retardation time and the stress relaxation time. It is found that, the critical point for the stability

that occurs in the presence of quantum term remains unchanged by the addition of the other parameters of the problem. Both maximum and critical point for the instability are unchanged by the addition of the strain retardation and the stress relaxation. All growth rates are reduced in the presence of porosity of the medium, the medium permeability, the strain retardation time and the stress relaxation time. These results indicate that quantum effect plays a major role in securing a complete stability for the system at hand, while other parameters are only of secondary significance.

Finally, our select model is more stable than those considered in refs. [29-33]. This discrepancy highlights a stabilizing role due to the presence of both the strain retardation and the stress relaxation on Rayleigh-Taylor instability problem, increasing the dissipation of any disturbance, thus providing an increased stability.

The author would like to thank the referees for their useful suggestions and comments that improved the original manuscript.

Here, if we put

, ,

and

Then, the system Equations (3), (7) and (9) maybe written as

Now, Using the expression (10) in the system of equations (30)-(37) maybe written as

(40)

(43)

(45)