Kinetic derivation of diffuse-interface fluid models

Abstract

We present a full derivation of capillary fluid equations from the kinetic theory of dense gases. These equations involve van der Waals' gradient energy, Korteweg's tensor, Dunn and Serrin's heat flux as well as viscous and heat dissipative fluxes. Starting from macroscopic equations obtained from the kinetic theory of dense gases, we use a new second order expansion of the pair distribution function in order to derive the diffuse interface model. The capillary extra terms and the capillarity coefficient are then associated with intermolecular forces and the pair interaction potential.

Kinetic derivation of Cahn-Hilliard fluid models

“…For the sake of simplicity, the kinetic derivation is performed in the situation where the capillary coefficients are independent of temperature. As a result, we obtain the Euler/van der Waals equations for fluid mixtures, extending previous results for single species fluids [15]. The internal energy includes density gradient terms, the pressure tensor involves a generalized Korteweg type tensor and the heat flux includes a Dunn and Serrin type contribution.…”

“…However, using ∇•v = ∇v:I, these gradients' product terms (2.27) may also be considered as velocity derivative terms. This then leads to an alternative form of entropy production as well as to unphysical transport fluxes as already established in the situation of a single species fluid [15]. Similarly, various terms may be regrouped differently in the expression of the entropy production rate, as notably…”

“…The proper heat flux has been obtained by Dunn and Serrin in the framework of rational thermodynamics [13]. These equations have alternatively been obtained from Hamiltonian considerations by Gavrilyuk and Shugrin [14] and deduced from the kinetic theory of dense gases by Giovangigli [15].…”

“…We establish, however, that the resulting model essentially agrees with those obtained in [2,19,20,23,24], while some other models are found to be unphysical. Thermodynamic methods using entropy production rates are incidentally found to be ambiguous for diffuse interface models since there are various terms in the entropy production rate that involve products of several gradients as for single species fluids [15]. Cahn-Hilliard fluid models also yield discontinuous interface models when the capillarity coefficients go to zero [2].…”

“…These results have then been generalized to the situation of nonisothermal fluids [46]. Still considering single species fluids, a full derivation of nonisothermal capillary fluid equations from the kinetic theory of dense gases has been obtained by Giovangigli [15]. However, to the best of the author's knowledge, a complete derivation of Cahn-Hilliard fluid mixture models from the kinetic theory of dense gas mixtures is still missing is the object of this work.…”