Pressure corrections for potential flow analysis of capillary instability of viscous fluids

Abstract

Funada & Joseph (Intl J. Multiphase Flow, vol. 28, 2002, p. 1459) analysed capillary instability assuming that the flow is irrotational but the fluids are viscous (viscous potential flow, VPF). They compared their results with the exact normal-mode solution of the linearized Navier–Stokes equations (fully viscous flow, FVF) and with the irrotational flow of inviscid fluids (inviscid potential flow, IPF). They showed that the growth rates computed by VPF are close to the exact solution when Reynolds number is larger than $O(10)$ and are always more accurate than those computed using IPF. Recently, Joseph & Wang (J. Fluid Mech., vol. 505, 2004, p. 365) presented a method for computing a viscous correction of the irrotational pressure induced by the discrepancy between non-zero irrotational shear stress and the zero-shear-stress boundary condition at a free surface. The irrotational flow with a corrected pressure is called the viscous correction of VPF (VCVPF). Here we compute the pressure correction for capillary instability in cases in which one fluid is viscous and the other fluid is a gas of negligible density and viscosity. The growth rates computed using VCVPF are in remarkably good agreement with the exact solution FVF.