Effect of Surface Tension on the Kelvin-Helmholtz Instability of Superposed Viscous Fluids in Hydromagnetics Saturating Porous Medium

Veena Sharma¹^*, Radhe Shyam¹, Sudrshna Sharma², Abhishek Sharma³

¹Department of Mathematics and Statistics, Himachal Pradesh University Summer Hill Shimla -171 005 (India)

²Department of Mathematics Government Collage Ghumarwin District Bilaspur (India)

³Student of second year Civil Engineering Bahra University Waknaghat Solan (India)

*Corresponding Author E-mail: veena_math_hpu@yahoo.com

ABSTRACT:

This paper deals with the instability of viscous superposed, fluids saturating porous medium in the presence of horizontal magnetic field and to include the effect of surface tension. Using linear theory and normal mode technique the dispersion relation so obtained is analyzed mathematically for the stable configuration. The effects of medium porosity, surface tension and magnetic field, on the growth rate (imaginary) of the most unstable mode have been investigated numerically. The square of the Alfvn velocity accounting for magnetic field and surface tension have stabilizing effect on the system and medium porosity has destabilizing effect on the system. All these numerical results have been depicted graphically. The results show that the magnetic field and surface tension bring about more stability for a certain wavenumber band on the growth rate of unstable configuration.

KEYWORDS: Kelvin-Helmholtz Instability, horizontal magnetic field, surface tension, porous medium.

INTRODUCTION:

Kelvin-Helmholtz instability occurs when we consider the character of the equilibrium of a stratified heterogeneous fluid in which different layers are in relative motion. The most important case is when two superposed fluids flow one over the other with a relative horizontal velocity, the instability of the plane interface between the two fluids when it occurs in this instance, is known as Kelvin-Helmholtz instability. The experimental demonstration of the Kelvin-Helmholtz instability has been given by Francis (1954). The effect of rotation and a general oblique magnetic field on the Kelvin-Helmholtz instability has been studied by Sharma and Srivastava (1968). Michael (1955) has discussed the stability of a combined current and vortex sheet in a perfectly conducting fluid, while the effect on the Kelvin-Helmholtz instability of a magnetic field transverse to the direction of streaming has been considered by Northrop (1956). There are diverse applications of the Kelvin-Helmholtz instability like to examine the horizontal and temporal variability of the out-of-cloud vertical velocity, the stratospheric gravity wave response to the convection to determine the vertical and spatial extent of turbulence due to gravity wave breaking, to provide a more realistic evolving background flow and convective initiation. A regional scale forecast model is used to force the cloud model the time evolution of the bulent region, effects of model resolution, wave instability and trapping. It is also used in understanding of CIT-generating mechanisms which is extremely important for commercial and other high-attitude aircraft flying above developing convection. The instability of the plane interface separating two uniform superposed streaming fluids under varying assumptions of hydrodynamics and hydromagnetics has been discussed in a treatise by Chandrasekhar (1961). Alterman (1961) has studied the effect of surface tension to the Kelvin-Helmholtz instability of two rotating fluids. Reid (1961) studied the effect of surface tension and viscosity on the stability of two superposed fluids. Bellman and Pennington (1954) further investigated in detail illustrating the combined effects of viscosity and surface tension. The uniform magnetic field along the direction of shear flow parallel to interface have been given by Ofman and Thompson (2011). Recently, Cavus and Kazkapan (2013) have studied magnetic Kelvin-Helmholtz instability in the solar atmosphere and have found that the growth rate of instability increases with velocity shear, it needs higher values of magnetic field in order to stabilize as said in Lapenta and Knoll (2003). The medium has been assumed to be non-porous in these studies.

The flow through porous medium has been of considerable importance in recent years particularly among geophysical fluid dynamics, recovery of crude oil from the pores of reservoir rocks, chemical engineering (absorption, filtration), petroleum engineering, hydrology, soil physics and biophysics etc. The physical properties of comets, meteorites and interplanetary dust strongly suggest the significance of the effect of porosity in astrophysical context given by McDonnel (1978) and Rudaraiah and Srimani (1976). The gross effect, as the fluid slowly percolates through the pores of the rock, is represented by Darcy’s law which states that the usual viscous term in the equations of motion is replaced by the resistance term , where is the viscosity of the fluid, the permeability of the medium (which has the dimension of length squared), and the filter (seepage) velocity of the fluid. Sharma and Kumari (1991) have studied hydromagnetic instability of streaming fluids in porous medium theoretically including surface tension and have found that the magnetic field and surface tension, therefore have stabilizing effect and complete suppress the Kelvin-Helmholtz instability for small wavelengths. The medium porosity reduces the stability range given in terms of a difference in streaming velocities and the Alfvn velocity. Sharma et al. (1980) have studied the Kelvin-Helmholtz instability through porous medium of two superposed plasmas.

Some solar activities in the solar atmosphere are created by a Kelvin-Helmholtz instability in the presence of magnetic field and subsequent reconnection processes and Kelvin-Helmholtz instability plays an important role in energy transfer mechanism in the solar atmosphere. The effect of the Kelvin-Helmholtz instability is shown to convert shear flow in compression flow that derives reconnection. Khalil Elcoot (2010) has studied the new analytical approximation forms for non-linear instability of electric porous media. Asthana et al. (2012) have been studied Kelvin-Helmholtz instability of two viscous fluids in porous medium for two dimensional flow. Rudraiah et al. (2011) have studied the study of surface instability of Kelvin-Helmholtz type in a fluid layer bounded above by a porous layer and below by a rigid surface. The effect of porosity in astrophysical context and the plasma outflow occur in regions which are created by the Kelvin-Helmholtz vortices. We believe that the mechanism presented here opens promising possibilities of further investigation. However a clear understanding of the role of the Kelvin-Helmholtz instability in reconnection requires a fully three-dimensional flows.

Keeping in view, the diverse applications stated earlier, a study has been therefore, investigated to examine the effect of the magnetic field, surface tension, kinematic viscosity, medium porosity and the surface tension on the instability of electrically conducting, streaming viscous three dimensional fluids saturating porous medium numerically using the software Mathematica version-5.2.

FORMULATION OF THE PROBLEM AND PERTURBATION EQUATIONS

The initial state whose stability we wish to examine is that of an incompressible, electrically infinitely conducting viscous fluid in which there is a horizontal streaming in the - direction with a velocity through a homogeneous and isotropic porous medium of medium porosity and medium permeability . A uniform horizontal magnetic field and acceleration due to gravity pervade the system. Suppose that at some prescribed level , the density may change discontinuously and bring into play effect due to effective interfacial tension and denote normal to the interface.

(42)

In particular, if is real, expression (17) simply represents the oscillatory waves so that the system is stable. However if has imaginary part, it represents a perturbation which grows exponentially with time that the system is unstable.

In the remaining part of this paper, the analysis of Kelvin-Helmholtz instability mechanism is done by obtained from equation (41) for astrophysical situations in porous medium.

CONCLUSIONS:

A study has been made to investigate numerically the effects of square of the Alfvn velocity, surface tension, and the medium porosity on the instability of superposed viscous fluids in hydromagnetics saturating porous medium. The principal conclusions drawn are as follows:

1). The square of Alfvn velocity in imaginary growth rates of the perturbations the Kelvin-Helmholtz instability has stabilizing effect on the system.

2). Surface tension has large enough stabilizing effect on the system. The effect of surface tension dissipates the energy of any disturbance more than that carried out by the magnetic field. In other words, the role of the square of Alfvn velocity can help the surface tension to find more stability on the Kelvin-Helmholtz instability problem, while the surface tension plays the fundamental role to generate the complete stability.

3). The imaginary growth rate of the perturbations decreases slightly with the increase in medium porosity has very slight stabilizing effect on the system.

REFERENCES:

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Bellman R. and Pennigton R. H., 1954: “Effact of surface tension and viscosity on Taylor instability”.Quart. Appl. Math. 12, 151-162.

Cavus, H. and Kazkapan, D., 2013: “Magnetic Kelvin-Helmholtz instability in the solar atmosphere”. New Astronomy. 25, 89-94.

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Francis, J.R.D., 1954: “Wave motions and the aerodynamic drag on a free oilsurface”. Phil. Mag. Ser.7,4, 695.

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Received on 14.01.2014 Accepted on 01.01.2014

Int. J. Tech. 4(1): Jan.-June. 2014; Page 234-239