On the Conditions for Existence of Unidirectional Motions of Binary Mixtures in the Oberbeck—Boussinesq Model

Abstract

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.

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

“…3. While constructing exact solutions, the temperature and concentration are often taken in the form of polynomials of the second [2,7,9,11,12] or first [2-4, 13-17] degree in the horizontal coordinate x. In our case, due to the fact that θ 0 (x, y) must satisfy (12), the representation of θ 0 (x, y) in polynomial form can be a polynomial in x of the degree at most 3.…”

“…The study in this article is an extension and generalization of the results of [8][9][10]. In particular, in [8] the compatibility analysis and construction of the new exact solutions were carried out for the equations of unidirectional steady flow of a binary mixture with nonlinear buoyancy force and the density being an arbitrary function of temperature and concentration.…”

“…In particular, in [8] the compatibility analysis and construction of the new exact solutions were carried out for the equations of unidirectional steady flow of a binary mixture with nonlinear buoyancy force and the density being an arbitrary function of temperature and concentration. The article [9] contains the most detailed investigation of these flows with a linear dependence of density on the state parameters. In [10], it is shown that the unidirectional motion of a homogeneous liquid in the classical Oberbeck-Boussinesq model exists only under a linear dependence of temperature on the longitudinal coordinate along the liquid layer.…”