A New Class of Exact Solutions to Magnetohydrodynamics Equations for Describing Convective Flows of Binary Fluids

A new family of exact solutions to the equations of magnetohydrodynamics of an incompressible fluid is presented. The convective fluid flows in a rectangular Cartesian coordinate system are considered. Convection in a conducting fluid is induced by thermal factors and the solute. Thus, the announced exact solution describes thermal diffusion in magnetic fluids. Exact solutions are constructed taking into account the Soret and Dufour crossed dissipative effects. In the article, the Lin–Sidorov–Aristov class is used to construct the exact solution. The velocity field and the magnetic field are described by linear forms with respect to two spatial coordinates. Linear form coefficients depend on the third coordinate and time. The pressure, temperature and solute concentration are described by quadratic forms. A system of equations for finding unknown functions for hydrodynamic fields is given. This system is overdetermined. The article presents an exact solution for determining unknown functions describing steady Stokes flows of convection of a binary conducting fluid. When constructing the exact solution to a nonlinear system of magnetohydrodynamics, all terms for the convective derivative are assumed to be equal to zero (convective and diffusive mixing of a continuous medium is not taken into account).