Properties of thermocapillary fluids and symmetrization of motion equations

•Internal energy depends on mass density, volume energy and their spatial derivatives.•A Legendre transformation induces a quasi-linear system of conservation laws.•Governing equations are stable at equilibrium positions.

The equations of fluid motions are considered in the case of internal energy depending on mass density, volume entropy and their spatial derivatives. The model corresponds to domains with large density gradients in which the temperature is not necessarily uniform. The new general representation is written in symmetric form with respect to the mass and entropy densities. For conservative motions of perfect thermocapillary fluids, Kelvin's circulation theorems are always valid. Dissipative cases are also considered; we obtain the balance of energy and we prove that equations are compatible with the second law of thermodynamics. The internal energy form allows to obtain a Legendre transformation inducing a quasi-linear system of conservation laws which can be written in a divergence form and the stability near equilibrium positions can be deduced. The result extends classical hyperbolicity theory for governing-equations' systems in hydrodynamics, but symmetric matrices are replaced by Hermitian matrices.