Purely irrotational theories of the effects of viscosity and viscoelasticity on capillary instability of a liquid cylinder

Abstract

Capillary instability of a liquid cylinder can arise when either the interior or exterior fluid is a gas of negligible density and viscosity. The shear stress must vanish at the gas-liquid interface but it does not vanish in irrotational flows. Joseph and Wang [D.D. Joseph, J. Wang, The dissipation approximation and viscous potential flow, J. Fluid Mech. 505 (2004) 365] derived an additional viscous correction to the irrotational pressure. They argued that this pressure arises in a boundary layer induced by the unphysical discontinuity of the shear stress. Wang et al. [J. Wang, D.D. Joseph, T. Funada, Pressure correction for potential flow analysis of capillary instability of viscous fluids, J. Fluid Mech. 522 (2005) 383] showed that the dispersion relation for capillary instability in the Newtonian case is almost indistinguishable from the exact solution when the additional pressure contribution is included in the irrotational theory. Here we extend the formulation for the additional pressure to potential flows of viscoelastic fluids in flows governed by linearized equations, and apply this additional pressure to capillary instability of viscoelastic liquid filaments of Jeffreys type. The shear stress at the gas–liquid interface cannot be made to vanish in an irrotational theory, but the explicit effect of this uncompensated shear stress can be removed from the global equation for the evolution of the energy of disturbances. This line of thought allows us to present the additional pressure theory without appeal to boundary layers. The validity of this purely irrotational theory can be judged by comparison with the exact solutions of Navier–Stokes equations. Here we show that our purely irrotational theory is in remarkably good agreement with the exact solution in linear analysis of the capillary instability of a viscoelastic liquid cylinder.