A long-wave estimation for the damping coefficient at a flat water–water vapour interface with a phase transition

Gravitational–capillary waves, arising in a system of thick layers of two aggregate states of a substance, are studied using the example of water and its vapour. Their initially flat interface is in hydrostatic and thermodynamic equilibrium at atmospheric pressure and the corresponding saturation temperature $100\,^{\circ }\text{C}$, provided that a balance is maintained for heat fluxes transported through immovable phases in the process of heat conduction. From the problem of linear stability for small perturbations, estimation relations for their damping coefficient and eigenfrequency shift are obtained, including the factors of viscosities of the media and phase transition. The first two contributions, proportional to the kinematic viscosity of the liquid and the square root of the kinematic viscosity of its vapour, respectively, are consistent, accurate to additionally accounted for capillary forces, with the result of Dore’s analysis, which is a development of the weakly viscous Lamb theory for ‘deep water’. Numerical calculations have shown that, with increasing wavelength, the accuracy of the proposed approximation increases, especially for the wave damping coefficient, and the contribution of the viscosity of the light phase increases in percentage terms. As for the phase transition effect, which was previously overestimated using the quasi-equilibrium approximation, it remains insignificant at the level of heat fluxes acceptable in the model of thick layers of homogeneous media. The factor of thermocapillarity is found to be very weak; it becomes qualitatively noticeable only on the background of disappearing heating. Here, together with the factor of dependence of the saturation temperature on the phase pressure, it determines a non-zero correction to the wave damping coefficient.