Viscous contributions to the pressure for the potential flow analysis of two viscous fluids

Capillary instability of a liquid cylinder immersed in another liquid is analyzed based on potential flow solutions. The growth rate of the instability is obtained by considering the normal stress balance at the interface. We derive a viscous correction of the irrotational pressure which presumably arises from a boundary layer induced by the discontinuity of the tangential velocity and shear stress at the interface evaluated using the potential flow solution. We include the viscous irrotational stress and pressure correction in the normal stress balance and compare the computed growth rates to the growth rates of the exact viscous flow solution. The agreement is excellent when one of the liquids is a gas; for two viscous liquids, the agreement is good to reasonable for the maximum growth rates but poor for long waves. Calculations show that good agreement is obtained when the vorticity is relatively small or the irrotational part is dominant in the exact viscous solution. We show that the irrotational viscous flow with pressure corrections gives rise to exactly the same dispersion relation as the dissipation method in which no pressure at all is required and the viscous effect is accounted for by evaluating the viscous dissipation using the irrotational flow.