Temperature and velocity profiles along a singular interface between two immiscible fluids - Vanhaelen, Quentin (Mathematical methods in the applied sciences, 2022)
Bedeaux, Albano, and Mazur (BAM) have developed a convenient method to derive the boundary conditions for non‐equilibrium thermodynamic systems made of two bulk phases separated by a free interface. In their description, the interface is considered as a thermodynamic system characterized by its intensive and extensive variables. The boundary conditions describing the free interface are expressed in terms of fluxes, thermodynamic forces, and interface transport coefficients. Previous works have focused on the derivation of the generalized boundary conditions for different systems using the BAM method, but the corresponding solutions and dynamics have not been widely studied. We consider a theoretical two dimensional system made of two incompressible fluids separated by a deformable interface for which the boundary conditions are obtained using the BAM method. Ignoring the effects of external forces, the dynamics of the system is investigated using the method of multiple scales. The hierarchy of equations is solved up to the second order and we obtain conditions that must be fulfilled by the transport coefficients modeling the cross‐effects of the velocity slips and temperature jumps so that non‐trivial asymptotic solutions for the temperature and velocity profiles can be constructed.