Surface Tension and Energy Conservation in a Moving Fluid

TL;DR

This paper analyzes energy transport in a moving fluid with a free surface, incorporating surface tension effects, and derives a conservation law for surface energy considering the Laplace pressure and tangential forces.

Contribution

It introduces a general conservation equation for surface energy in moving fluids, explicitly including surface tension effects and their impact on energy transfer.

Findings

Surface tension contributes to energy conservation at the free surface.

A simple surface area conservation equation is derived.

Surface tension effects vanish in static control volumes, with pressure adjustments.

Abstract

The transport of energy in a moving fluid with a simply connected free surface is analyzed, taking into account the contribution of surface tension. This is done by following a "control volume" with arbitrary, specified velocity, independent of the flow velocity, and determining the rates of energy passing through the boundaries, as well as the energy dissipation in the bulk. In particular, a simple conservation equation for the surface area is written down, which clearly shows the contribution of the Laplace pressure at the free surface and the tangential surface tension forces at its boundary. It emerges as the mechanical conservation law for the surface energy in its general form. For a static control volume, all contributions from surface tension disappear, except that the pressure has to be modified by the Laplace contribution.