The dissipation approximation and viscous potential flow

Dissipation approximations have been used to calculate the drag on bubbles and drops and the decay rate of free gravity waves on water. In these approximations, viscous effects are calculated by evaluating the viscous stresses on irrotational flows. The pressure is not involved in the dissipation integral, but it enters into the power of traction integral, which equals the dissipation. A viscous correction of the irrotational pressure is needed to resolve the discrepancy between the zero-shear-stress boundary condition at a free surface and the non-zero irrotational shear stress. Here we show that the power of the pressure correction is equal to the power of the irrotational shear stress. The viscous pressure correction on the interface can be expressed by a harmonic series. The principal mode of this series is matched to the velocity potential and its coefficient is explicitly determined. The other modes do not enter into the expression for the drag on bubbles and drops. They vanish in the case of free gravity waves.