Exact solution of the Couette–Poiseuille type for steady concentration flows

This article presents a new exact solution for predicting the properties of the velocity field, pressure, and impurity distribution in steady shear flows of viscous incompressible fluids in an extended horizontal layer. The solutal convection was described using a mathematical model based on the Oberbeck–Boussinesq equations with a linear dependence of density on concentration. It was assumed that the layer at one of its boundaries (the lower one) is impermeable to the substance (impurity) dissolved in the fluid so that the fluid-sticking effect applies to it. It was demonstrated that the flow is induced by an inhomogeneous distribution of impurities and pressure at the upper boundary of the layer. A uniform distribution of velocities is set at the upper boundary. The obtained solution belongs to the Ostroumov–Birikh and Lin–Sidorov–Aristov classes. The velocity field was described by the two-dimensional Couette profile, i.e., both velocity components depend on the vertical transverse coordinate. The concentration and pressure were described by linear forms relative to horizontal (longitudinal) coordinates, with coefficients depending on the third coordinate. The structure of the exact solution is such that the incompressibility equation is identically satisfied. Thus, an overdetermined, quadratically nonlinear partial differential system was resolved. After the substitution in the stationary system of the Oberbeck–Boussinesq equations supplemented by the equations of diffusion and incompressibility, the unknown functions that determine the hydrodynamic fields were found by integrating the system of ordinary differential equations. This system is of the 13th order and admits an exact polynomial solution that can be used to describe the occurrence of several counterflow zones and the nonmonotonic nature of the specific kinetic energy with up to two zeros. The obtained exact solutions illustrate the multiple stratifications of the shear stress, pressure, and concentration fields. Therefore, hydrodynamic fields have a complex topology defined by the dependence of velocities, pressure, and concentration on the transverse coordinate.