Pressure corrections for the potential flow analysis of Kelvin–Helmholtz instability with heat and mass transfer Mukesh Kumar Awasthi ⇑ , Rishi Asthana, G.S. Agrawal Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, India article info Article history: Received 25 June 2011 Received in revised form 13 December 2011 Accepted 21 December 2011 Available online 8 February 2012 Keywords: Pressure correction Potential flow Kelvin–Helmholtz stability Incompressible fluid Viscous pressure abstract Pressure corrections for the viscous potential flow analysis of Kelvin–Helmholtz instability at the inter- face of two viscous fluids have been carried out when there is heat and mass transfer across the interface. Both fluids are taken as incompressible and viscous with different kinematic viscosities. In viscous poten- tial flow theory, viscosity enters through normal stress balance and effect of shearing stresses is com- pletely neglected. We include the viscous pressure in the normal stress balance along with irrotational pressure and it is assumed that this viscous pressure will resolve the discontinuity of the tangential stres- ses at the interface for two fluids. It has been observed that heat and mass transfer has destabilizing effect on the stability of the system. A comparison between viscous potential flow (VPF) solution and viscous contribution to the pressure for potential flow (VCVPF) solution has been made and it is found that the effect of irrotational shearing stresses stabilizes the system. Ó 2012 Elsevier Ltd. All rights reserved. 1. Introduction When two superposed fluid layers of different physical param- eters move parallel to each other with a relative horizontal veloc- ity, the instability of the plane interface between the two fluids is called Kelvin–Helmholtz instability [1,2]. The Kelvin–Helmholtz instability occurs in various situations such as wind blowing over the ocean, meteor entering the earth atmosphere and in oil explo- ration industry etc. When two fluids are divided by an interface, the interfacial instability is usually discussed without considering heat and mass transfer across the interface. On the other hand, the transfer of mass and heat across the interface is very important in many situ- ations such as boiling heat transfer in chemical engineering and in geophysical problems. Hsieh [3,4] formulated the problem of Rayleigh–Taylor instability and Kelvin–Helmholtz instability with heat and mass transfer across the liquid vapour interface. Hsieh [4] found that when the vapor layer is hotter than the liquid layer, the effect of heat and mass transfer tends to inhibit the growth of instability. Nayak and Chakrborty [5] established the formulation of Kelvin–Helmholtz instability of the cylindrical interface be- tween the liquid and vapor phases with heat and mass transfer. Lee [6] has studied the Kelvin–Helmholtz instability of inviscid flu- ids taking heat and mass transfer into the account and observed that the heat and mass transfer has no effect on the linear inviscid analysis while it plays an important role in the nonlinear analysis. Viscous potential flow theory has played an important role in studying various stability problems. Tangential stresses are not considered in viscous potential theory and viscosity enters through normal stress balance [7]. The no slip condition at the boundary is not enforced in viscous potential theory. The viscous potential flow analysis of Kelvin–Helmholtz instability has been studied by Funa- da and Joseph [8]. They have observed that the stability criterion for viscous potential flow is given by the critical value of relative velocity. Funada and Joseph [9] studied the viscous potential flow analysis of capillary instability and observed that viscous potential flow is better approximation of the exact solution than the inviscid model. Funada and Joseph [10] extended their work of capillary instability for viscoelastic fluids of Maxwell type and observed that the growth rates are larger for viscoelastic fluids than for the equivalent Newtonian fluids. In the viscous potential flow theory tangential stresses are not considered and viscosity enters through normal stress balance. Wang, Joseph and Funada [11] presented the idea that there exist viscous pressure along with irrotational pressure in the normal stress balance and it is assumed that this viscous pressure will re- solve the discontinuity of tangential stresses for two fluids at the interface. Wang, Joseph and Funada [12] carried out the viscous contributions to the irrotational pressure for potential flow analy- sis of capillary instability taking a viscous fluid and another fluid of negligible viscosity to resolve the discontinuity of the tangential velocity and shear stress at the interface. The effect of irrotational 0017-9310/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2012.01.032 ⇑ Corresponding author. Tel.: +91 01332 285157. E-mail address: mukeshiitr.kumar@gmail.com (M.K. Awasthi). International Journal of Heat and Mass Transfer 55 (2012) 2345–2352 Contents lists available at SciVerse ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt