Energy equation for irrotational theories of gas-liquid flow:: viscous potential flow (VPF), viscous potential flow with pressure correction (VCVPF), dissipation method (DM)

The effects of viscosity on the irrotational motions of spherical cap bubble, Taylor bubbles in round tubes, Rayleigh-Taylor and Kelvin-Helmholtz instability described in previous chapters were obtained 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 viscous potential flow 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. We shall 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 is zero at each point on the free surface is dropped in the irrotational approximations. In general, you get an irrotational shear stress from the irrotational analysis. The discrepancy between this shear stress and the zero shear stress required for exact solutions will generate a vorticity layer. In many cases this layer is thin and its influence on the bulk motion and on the irrotational effects of viscosity is minor. Exact solutions cannot be the same as any irrotational approximation for cases in which the irrotational shear stress does not vanish.