Viscous potential flow analysis of capillary instability

Abstract

Capillary instability of a viscous fluid cylinder of diameter D surrounded by another fluid is determined by a Reynolds number J= VDρ l/ μ l, a viscosity ratio m= μ a/ μ l and a density ratio ℓ= ρ a/ ρ l. Here V= γ/ μ l is the capillary collapse velocity based on the more viscous liquid which may be inside or outside the fluid cylinder. Results of linearized analysis based on potential flow of a viscous and inviscid fluid are compared with the unapproximated normal mode analysis of the linearized Navier–Stokes equations. The growth rates for the inviscid fluid are largest, the growth rates of the fully viscous problem are smallest and those of viscous potential flow are between. We find that the results from all three theories converge when J is large with reasonable agreement between viscous potential and fully viscous flow with J>O(10). The convergence results apply to two liquids as well as to liquid and gas.