Exact solution for stable convective concentration flows of a Couette type

Abstract

The paper presents an exact solution describing a steady-state diffusion layered Couette-type flow of a viscous incompressible fluid binary mixture, induced by specifying a parabolic wind at one of the boundaries of the flow region. The flow is modeled using the system of equations of concentration convection, which consists of the Navier-Stokes equations (in the Boussinesq approximation), the incompressibility equation, and the equation for changing the concentration of the light phase of a binary mixture. The solution of this nonlinear overdetermined system of equations is sought within the framework of the Lin-Sidorov-Aristov class. The solvability of the reduced system of equations with respect to the components of the velocity field, concentration and pressure fields is shown. A plane horizontal infinite layer of constant thickness is considered as the flow region. A distinctive feature of the presented exact solution is taking into account the impermeability property of a solid hydrophilic surface, which limits the fluid layer under consideration from below. In the course of analyzing the exact solution describing the distribution of the velocity field, it was shown that in some cases the flow can be reduced to a unidirectional one. In the general case, each of the nonzero components of the velocity vector can have at most one zero point inside the layer under consideration, and they will coincide only if the flow is reducible to a unidirectional one. It is shown that the corresponding shear stress field can be stratified into two zones, in each of which the stress retains its sign and changes it upon passing to another zone. The investigation of the concentration field revealed the fundamental possibility of stratification of the background concentration field into two zones compared to the reference value. At the same time, the background pressure field can be divided into three zones in relation to the reference level. Thus, the solution proposed can describe return flows localized (for certain combinations of edge parameters and physical characteristics of the fluid) near the boundaries of the considered fluid layer.