On Marangoni shear convective flows of inhomogeneous viscous incompressible fluids in view of the Soret effect

Abstract

A new exact solution is obtained for the Oberbeck-Boussinesq equations describing the steady-state layered (shear) Marangoni convection of a binary viscous incompressible fluid with the Soret effect. When layered (shear) flows are considered, the Oberbeck-Boussinesq system is overdetermined. For it to be solvable, a class of exact solutions is constructed, which allows one to satisfy identically the “superfluous” equation (the incompressibility equation). The found exact solution allows the Oberbeck-Boussinesq system of equations to be reduced to a system of ordinary differential equations by the generalized method of separation of variables. The resulting system of ordinary differential equations has an analytical solution, which is polynomial. The polynomial velocity field describes counterflows in the case of a convective fluid flow. It is demonstrated that the components of the velocity vector can have one stagnant (zero) point inside the region under study. In this case, the corresponding component of the velocity vector can be stratified into two zones, in which the fluid flows in opposite directions. The exact solution describing the velocity field for the Marangoni convection of a binary fluid has non-zero helicity, the flow itself being almost everywhere vortex.