On the conditions for existence of unidirectional motions of binary mixtures in the Oberbeck–Boussinesq model

Compatibility conditions are obtained for the nonstationary Oberbeck–Boussinesq equations describing the unidirectional motions of a liquid binary mixture in a horizontal strip. We examine the case of polynomial dependence of temperature on the longitudinal coordinate is considered, and the influence of the dependence of the kind on the remaining unknown functions from the original system. It is shown that a nonstationary unidirectional motion between two solid walls can be described by the Oberbeck–Boussinesq model only for the quadratic or linear law of temperature distribution along the horizontal coordinate. Some initial-boundary value problems are posed.