c  Copyright, Darbose International Journal of Applied Mathematics and Computation Volume 3(2),pp 131–138, 2011 http://ijamc.psit.in Viscous potential flow analysis of Kelvin-Helmholtz instability of cylindrical interface Mukesh Kumar Awasthi *1 , G. S. Agrawal *2 ∗ Department of Mathematics, Indian Institute of TechnologyRoorkee, Roorkee-247667 India Email: mukeshiitr.kumar@gmail.com Abstract: A linear analysis of Kelvin-Helmholtz instability of cylindrical interface is carried out using viscous potential flow theory. In the inviscid potential flow theory, the viscous term in Navier- Stokes equation vanishes as viscosity is zero. In viscous potential flow, the viscous term in Navier-Stokes equation vanishes as vorticity is zero but viscosity is not zero. Viscosity enters through normal stress balance in viscous potential flow theory and tangential stresses are not considered. Both asymmetric and axisymmetric disturbances are considered. A dispersion relation has been obtained and stability criterion is given in the terms of the critical value of relative velocity. A comparison between inviscid potential flow and viscous potential flow has been made. It has been observed that Reynolds number and inner fluid fraction both have destabilizing effect on the stability of the system. Keywords : Fluid-fluid interfaces; hydrodynamic stability; viscous potential flow; interfacial flows; incompressible fluids. 1 Introduction When two fluids of different physical properties are superposed one over other and are moving with a relative horizontal velocity, the instability occurs at the plane interface. It is called Kelvin- Helmholtz instability [1, 2]. Kelvin-Helmholtz instability occurs in various situations like mixing of clouds, meteor is entering on the earth’s atmosphere etc. Cylindrical geometry is very important while studying stability problems related to liquid jets and cooling of fuel rods by liquid coolants in the nuclear reactor. Nayak and Chakrborty [3] considered the Kelvin-Helmholtz instability of the cylindrical interface of inviscid fluids with heat and mass transfer and showed that plane geometry configuration is more stable than cylindrical one. The Kelvin-Helmholtz instability of a cylindrical flow with a shear layer has been considered by Wu and Wang [4]. Viscous potential theory has played an important role in studying various stability problems. Joseph and Liao [5] have shown that irrotational flow of a viscous fluid satisfies Navier-Stokes equations. Tangential stresses are not considered in viscous potential theory and viscosity enters through normal stress balance. In this theory no-slip condition at the boundary is not enforced so that two dimensional solutions satisfy three dimensional solutions. Various vortocity and cir- culation theorems of inviscid potential flow also hold well in viscous potential flow. Joseph et al. [6] studied viscous potential flow of Rayleigh-Taylor instability. Funada and Joseph [7] have done the viscous potential flow analysis of Kelvin-Helmholtz instability in a channel and found that the stability criterion for viscous potential flow is given by the critical value of the relative velocity. Corresponding author: Mukesh Kumar Awasthi