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

Abstract

Capillary instability of a liquid cylinder immersed in another liquid is analyzed using viscous potential flow. An effect of viscosity on the irrotational motion may be introduced by evaluating the viscous normal stress at the liquid–liquid interface on the irrotational motions. In a second approximation, the explicit effects of the discontinuity of the shear stress and tangential component of velocity which cannot be resolved pointwise in irrotational flows, can be removed in the mean from the power of traction integrals in the energy equation by the selection of two viscous corrections of the irrotational pressure. The actual resolution of these discontinuities presumably takes place in a boundary layer which is not computed or needed. We include the 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.