The theory of viscous potential flow with a pressure correction is applied to problems such as a rising bubble or drop in a viscous fluid, the decay of free gravity waves, and capillary instability. We first derive a relation between the pressure correction and the irrotational shear stress. The pressure correction is then expanded as a harmonic series and the coefficient of the principal mode can be explicitly computed. Our pressure corrections are confirmed by independent results in the case of a rising spherical gas bubble (Kang and Leal 1988) and in the case of the decay of standing free waves (Prosperetti 1976). In the problem of capillary instability, the growth rates computed using the viscous potential flow with a pressure correction are almost indistinguishable from the exact fully viscous solution.
The formulation of the pressure correction can be extended to problems involving the interface of two viscous fluids in which the flows are assumed to be irrotational. We apply the pressure correction to the analysis of capillary instability of two fluids. Comparison of our growth rates and the exact solution shows that the agreement is satisfactory in the maximum growth region but poor for long waves.