Energy equation for irrotational theories of gas–liquid flow: viscous potential flow, viscous potential flow with pressure correction, and dissipation method

Abstract

Viscous potential flow We obtained the effects of viscosity on irrotational motions of spherical cap bubbles, Taylor bubbles in round tubes, and RT and KH instabilities described in previous chapters by evaluating the viscous normal stress on potential flow. In gas–liquid flows, the viscous normal stress does not vanish and it can be evaluated on the potential. It can be said that, in the case of gas–liquid flow, the appropriate formulation of the irrotational problem is the same as the conventional one for inviscid fluids with the caveat that the viscous normal stress is included in the normal stress balance. This formulation of VPF is not at all subtle; it is the natural and obvious way to express the equations of balance when the flow is irrotational and the fluid viscous. In this chapter we use the acronym VPF, viscous potential flow, to stand for the irrotational theory in which the viscous normal stresses are evaluated on the potential. In gas–liquid flows we may assume that the shear stress in the gas is negligible so that no condition need be enforced on the tangential velocity at the free surface, but the shear stress must be zero. The condition that the shear stress be zero at each point on the free surface is dropped in irrotational approximations.