Surface tension and energy conservation in a moving fluid

In what sense do the surface tension forces in a fluid surface contribute to the energy budget, and what are the correct boundary conditions at a free surface of a moving viscous liquid? In two recent papers (Bhagat et al. J. Fluid Mech. (2018), 851, R5 and Bhagat & Linden, J. Fluid Mech. (2020), 896, A25) it has been claimed that surface tension forces, F,  give a power term F·v, where v is the fluid velocity at the surface, and that this energy, for a stationary flow is balanced by viscous dissipation at the surface, thereby changing the standard dynamic free surface boundary condition into a balance between surface tension and viscosity. In this work we show that this is not correct: the velocity v determining the power is not the velocity of the liquid, but the velocity of the control surface. For a static control volume, all surface energy contributions from surface tension thus disappear, except the terms coming from the Laplace pressure. We show this by deriving a simple conservation equation for the surface area of a part of a moving liquid, which clearly reveals the contribution of the Laplace pressure at the free surface and the tangential surface tension forces at its boundary.