[Chavaraddi, 4(1): January, 2015] ISSN: 2277-9655 Scientific Journal Impact Factor: 3.449 (ISRA), Impact Factor: 2.114 http: // www.ijesrt.com © International Journal of Engineering Sciences & Research Technology [525] IJESRT INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY EFFECT OF SURFACE ROUGHNESS ON KELVIN-HELMHOLTZ INSTABILITY IN PRESENCE OF MAGNETIC FIELD Krishna B. Chavaraddi*, Vishwanath B. Awati, Priya M. Gouder *Department of Mathematics, Shri Siddeshwar Government First Grade College, Naragund- 582 207, India Department of Mathematics, Rani Channamma University, Belgaum-590 020, India Department of Mathematics, K.L.E. Society’s Dr. M. S. Sheshagiri College of Engineering & Technology, Belgaum - 590008, India ABSTRACT We study the effect of surface roughness on Kelvin-Helmholtz instability (KHI) in a fluid layer above by a porous layer and below by a rigid surface in presence of transverse magnetic field. A simple theory based on fully developed flow approximations is used to derive the dispersion relation with surface roughness for the growth rate of KHI. We replace the effect of boundary layer with Beavers and Joseph slip condition as well as roughness condition at the rigid surface. The dispersion relation is derived using suitable boundary and surface conditions and results are discussed graphically. The magnetic field is found to be stabilizing and the influence of the various parameters involved in the problem on the interface stability is thoroughly analyzed. KEYWORDS: KHI, magnetic field, BJ-slip condition, porous layer, dispersion relation, surface roughness. INTRODUCTION Kelvin-Helmholtz instability is one of the basic instabilities of two-fluid systems, which affects an interface. In Engineering, Kelvin -Helmholtz instability plays an essential role in transition from stratified to slug flow in horizontal pipes explored by Simmons [2]. Lord Kelvin first examined Kelvin- Helmholtz instability in 1910. An inviscid linear analysis of the phenomenon, which is applicable in case of two liquids with similar densities, can be found in various textbooks, for example in Chandrasekhar [3], and Drazin & Reid [4]. The problem becomes much more complicated for large density differences, which appears in case of liquid and gas. For example the instability of sea surface appears at wind speeds significantly lower than the critical wind speed given by linear inviscid analysis Gondret & Rabaud [5]. This phenomenon called “subcritical” Kelvin-Helmholtz instability (with high density difference) was analyzed by Meignin [6] and was found to be result of nonlinear analysis. Kelvin-Helmholtz instability appears in stratified two-fluid flows, in the presence of a small disturbance and relative velocity that is larger than critical. The disturbance causes change of the velocity field. Because of the continuity equation, the velocity of one fluid increases and of the other one decreases. The change in velocity field changes pressure (Bernoulli’s equation). Pressure force is increasing the disturbance; surface tension force and gravity force are decreasing the disturbance. If the pressure force is larger than the sum of surface tension and gravity forces, the Kelvin-Helmholtz instability occurs. A linear theory of the KHI for parallel flow in porous media was introduced by Bau[7] for the Darcian and non-Darcian flows. In both cases, Bau found that the velocities should exceed some critical value for the instability to manifest itself. The instability of plane interface between two uniform superposed fluids through a porous medium was investigated by Kumar [8]. They used linear stability analysis to obtain a characteristic equation for the growth of the disturbance. The nonlinear Kelvin -Helmholtz instability of a horizontal interface between a magnetic inviscid incompressible liquid and an inviscid laminar subsonic magnetic gas is investigated in the presence of a normal magnetic field by Zakaria[9]. El-Sayed [10] investigated the RTI problem of rotating stratified conducting fluid layer through porous medium in the presence of an inhomogenous magnetic field. This problem corresponds physically (in astrophysics) to the RTI of an equatorial section of a planetary magnetosphere or of stellar atmosphere when rotation and magnetic field are perpendicular to