Точные решения уравнений Обербека – Буссинеска для сдвиговых течений вязкой бинарной жидкости с учетом эффекта Соре

Abstract

The paper considers an exact solution to the equations of thermal diffusion of a viscous incompressible fluid in the Boussinesq approximation with neglect of the Dufour effect for a steady shear flow. It is shown that the reduced system of constitutive relations is nonlinear and overdetermined. A nontrivial exact solution of this system is sought in the Lin–Sidorov–Aristov class. The resulting family of exact solutions allows one to describe steady-state inhomogeneous shear flows. This class generalizes the classical Couette, Poiseuille, and Ostroumov–Birikh solutions. It is demonstrated that the system of ordinary differential equations reduced within this class retains the properties of nonlinearity and overdetermination. A theorem on solvability conditions for the overdetermined system is proved; it is reported that, when these conditions are met, the solution is unique. The overdetermined system is solvable owing to the algebraic identity relating the horizontal velocity gradients, which are linear functions of the vertical coordinate. The constructive proof of the computation of hydrodynamic fields consists in the successive integration of the polynomials, the polynomial degree being dependent on the values of the boundary parameters.