Under the gravitational force, when a denser fluid overlies a lighter fluid, the interfacial instability occurs between those two fluids, and it is called Rayleigh-Taylor instability (RTI). The development of nonlinear structures at the two fluid interface consequent to RTI is of much current interest both from theoretical and experimental points of view. The evolution of the structure is called a bubble if the lighter fluid pushes across the unperturbed interface into the heavier fluid and it is termed as a spike if the opposite event takes place. The importance of such instabilities arises in connection with a wide range of applications ranging from astrophysical phenomena such as Supernova remnant to Inertial Confinement Fusion (ICF) problems. A core collapse Super Nova (SN) is driven by an external powerful shock, and strong shocks are the breeding ground of hydrodynamic instabilities like RTI. The interfacial RT instabilities have been investigated by several authors [1] -[11] . In an ICF situation, ablation front of an imploding capsule is subject to the RT instability because dense core is compressed and accelerated by low density ablating plasma. Recent experiments [12] [13] in LLNL shows that RTI at the ablation front of a composite ablator (CH, Si dopped CH) is induced, which requires mitigation for efficient ICF burn. Ponderomotive force due to spatially inhomogeneous laser beam is an effective mechanism for generation of large amplitude self generated magnetic field [14] - [16] in laser produced plasmas. Srivastava et al. [17] analyzed theoretically the spatially inhomogeneous laser beam could generate an axial magnetic field via ponderomotive force. The analytical results were compared with the experimental observation by [18] . The effect of self-generated magnetic field on RTI has been studied by Evans [19] . When the explosion of a Ia type supernova (SNIa) starts in a white dwarf as a laminar deflagration at the center of the star, RTI begins to act [20] [21] . In a white dwarf, magnetic field with strength up to 10⁸ - 10⁹ G exists at the surface and the field may be ~10 times greater near its center. RTI arising during type Ia Supernova explosion is also associated with strong magnetic field.

The effect of magnetic field on RTI was initially studied in details by [22] . In his linear analysis, he has shown that when the magnetic field is normal to the surface of separation of the two fluids, the RTI is shown to be unaffected by the magnetic field for small wave number “k” of the perturbation. However, as k increases, the RTI growth rate does not increase indefinitely with “k” but tends to be a saturation value as [22] . This situation may happen for sufficiently intense magnetic field.

If the external magnetic field is parallel to the interface, it is found in the linear theory that there exists no effect of the magnetic field on the instability [22] . However, in compressible and magnetized fluids, RTI has been shown to be reduced [23] even in linear theory. The RTI growth rate is found to be lowered for continuously accelerated two fluid interface when has component parallel to the magnetic field [24] . The nature of the depression has close resemblance to that of the effect of surface tension [22] and also has useful application in the astrophysical context [20] [25] .

In all the above studies, the RTI was studied in linear approximation. However, in nonlinear theory, it has been studied separately, the evolution of the interfacial fluid structure due to RTI in compressible fluids [26] and that due to presence of magnetic field [27] . The present paper is addressed to the problem of the evolution of the nonlinear interfacial structure caused by RTI in presence of a magnetic field parallel to the surface of separation of the two compressible fluids. With such a geometry, there is no effect of the magnetic field in the classical linear approximation [22] . However, it is no longer the case for nonlinear theory. This may be understood from the following consideration.

In presence of only magnetic field [27] , there exists a magnetic pressure in addition to the usual hydrodynamic pressure. As a result, the RTI driving pressure difference is changed by the inclusion of the magnetic pressure difference [the suffix h(l) corresponds to the dynamical variable associated with the heavier (lighter) fluid]. In our earlier study [27] it has been shown that, the growth rate of RTI may be enhanced or depressed according to whether the extra contribution is positive or negative. Moreover, stabiliza- tion may occur in the sense that the surface of separation executes periodic undulation resulting from time lag in the temporal variation of and. These are entirely nonlinear effects and disappear in the linear appro- ximation.

In compressible fluids [26] , there have been incorporated the effect of compressibility by introducing an extra work function proportional to surface pressure and density gradient scale length. The degree of stabilization or destabilization (depending on algebric sign of the density gradient at the surface) was shown to be proportional to.

In this study, we have also included the compressibility in addition to the magnetic field on RTI. We have developed a model up to second order of nonlinearity in the transverse coordinate system to investigate the combined effect of compressibility and magnetic field by using Layzer’s approach [28] . We have shown that, the growth rate of bubble and spikes is substantially reduced and shows oscillatory stabilization due to the combined effect which is conducive in ICF burn. We are mainly concentrated on a single mode bubble (spike) and our analysis is restricted in it (near the tip). The growth rate of instability is defined by the growth rate of the tip of the bubble (spike). The maximum spatial range of applicability of our solution has also been determined. We have tested the stability criteria of the solution.

The paper is organized as follows: Sec. 2 has been devoted for a brief discussion of Layzer’s model. In Sec. 3 we have discussed the geometry of the problem and the structures of bubble and spike. Basic mathematical model and assumptions regarding the problem have been formulated in Sec. 4. The stability analysis of the model has been studied in Sec. 5. The numerical results regarding the dynamics of the tip of the bubble and spike have been discussed in Sec. 6. Sec. 7 is kept for conclusion.

Potential flow model was first introduced in the nonlinear regime by Layzer in 1955 to study the stability of superposed fluid in a gravitation field in a three dimensional axially symmetric cylindrical or in two dimensional parallel walls from initially slight perturbed position. He obtained an approximate analytical solution consi- dering a perfect incompressible fluid occupying the upper half of a vertical cylinder, being supported by a diaph- ragm against gravity. The lower half of the tube is empty i.e.. The diaphragm is removed initially and applied a small disturbance at the interface then consequent motion has been investigated in 3D tubular and 2D periodic flows. He expanded the flow potentials using the Bessel (3D) and Fourier (2D) series, and retaining the lowest order terms, re-expanded the conservation laws in the vicinity of a regular point of the free surface (bubble tip). In this way, the nonlinear boundary value problem for the Euler equations was reduced to a system of ordinary differential equations.The asymptotic solutions for this dynamical system at short and long times have agreed with the linear theory of Rayleigh and with the results of Taylor, respectively. Though Layzers solution was the local one, as it described the temporal evolution of the flow in the vicinity of the bubble tip only, yet it accounted for the dynamics in the entire domain via the use of the Bessel and Fourier series .

We consider such a system in which heavier fluid lies above the lighter one and their interface is in the X-Z plane (Figure 1). Heavier fluid (density) above this interface is extended to infinity towards positive Y axis and lighter fluid (density) is below the interface and extended to infinity along negative Y axis. An external magnetic field has been applied across the interface towards positive Z axis. The gravity acts along negative Y axis. Since the force due to magnetic field is only in the perpendicular direction of the magnetic field, our study is essentially two dimensional.

We assume the perturbed interface to be of parabolic form which is given by

For a bubble (here the lower fluid is pushing across the interface into the upper fluid with) we have,

and for spike:

here are respectively, the height and curvature of the bubble.

We assume that the fluids are compressible, inviscid and the flow is irrotational. So there exists a velocity po- tential for both heavier and lighter fluids. Both the fluids obey the law of barotropic equation of state i.e., where is speed of sound through the fluid and is adiabatic index of the fluid and is the density. The fluids are infinitely conducting in nature. We consider both high and low , where p is the fluid pressure and B is the magnetic field. We work in a frame in which the interface is at rest,so the zeroth order velocity and we take the first order velocity to be. We assume that the flow is irrotational. Moreover, in this situation the fluctuations are assumed to be incompress- ible, which may not require the fluid itself to be incompressible [29] . Thus we may consider the compressibility of the fluid in this condition. This assumption may be expressed as i.e..

Following [1] velocity potential flow model, we get velocity potentials for heavier fluid as

Velocity potential for lighter fluid is given by

We check the stability of the system given by the set of equations Equation (26), assuming the initial values and to be unity. Let us consider the following coupled equations,

and

There are two sets of fixed points of the system given by the above equations, are

and

where

Now to check the stability of the solutions we calculate the eigen values of the following matrix, called Jacobian (J), at those fixed points

If be the eigen value of the Jacobian, then

The eigen values at the fixed point are

where

So, the system will be stable if

The above mentioned condition also holds for the stability of the fixed point

This has been checked thoroughly by using numerical method. The phase diagram (Figure 2) also ensures the stability of the system under that above mentioned condition.

However, the eigen value of the second set of fixed point is of very lengthy expression and not presented in this paper.

According to the linear analysis,when the externally applied magnetic field is parallel to the interface of separation of the two fluid system (heavier fluid lies over the lighter fluid under the action of gravity), the RTI remain unaffected [22] . However, nonlinear analysis shows a decreasing nature of RTI growth rate for the above configuration [27] . It was shown that, compressibility also decreases the growth rate of RTI [26] [32] . In our

nonlinear study, we consider the compressibility and interface-parallel magnetic field simultaneously, which has not been studied previously.

To verify our model numerically, we have used Runge-Kutta method and set the values of constants in our equations as follows:,

The results of our studies are discussed as follows.

We found in earlier studies that interface parallel magnetic field decreases the growth rate nonlinearly, whereas, compressibility effect also linearly decreases the RTI growth rate for bubble and spikes. When they act together, the resultant decreasing character of growth rate depends on their relative strength. This relative strength depends on the value of. If is small, i.e., the magnetic field dominates the growth rate when compressibility has the dominating effect on the growth rate.

In our calculation, we assume the variation of for both fluids are negligible and their values are almost constant. We also see that, Atwood number has very profound effect on the dynamics of instability. For fixed value of magnetic field, governs the dynamics of the interface. Moreover, in presence of magnetic field, the evolution of the bubble is quite different from that of the spike, near the critical value, for a fixed.

In this paper we have considered single mode bubble (spike) and the whole analysis is valid for the spatial

scale lengths. This particular limit in length scale arises due to our assumption

about the shape of the bubble and spike as a parabola and also the fact that we ignored the higher order terms in the expansion of velocity potential. However, this length scale gives an overview of the effect of magnetic field and compressibility on the growth rate of RT instability. From our general model of RT instability, all previous results concerning effect of a) only compressibility [26] [32] ; b) only magnetic field [27] can be derived, shown in the numerical graphs (Figure 3, Figure 4, Figures 6-8).

Recent experiments [12] [13] on hydrodynamic instability and mix experiments at Nation Ignition Facilities (NIF) reveal that such RTI at the ablation front in composite ablator is primarily responsible for the hot spot mix, which requires mitigation. In our analysis, it is evident that the compressibility in the composite ablator and magnetic field may play a key role in mitigating RTI leading to ignition at the hot spot.

AniruddhaMitra,LabakantaMandal,RajkumarRoychoudhury,ManoranjanKhan, (2015) Combined Effect of Magnetic Field and Compressibility on Rayleigh Taylor Instability. Open Journal of Fluid Dynamics,05,322-338. doi: 10.4236/ojfd.2015.54033