Exact solutions for the description of nonuniform unidirectional flows of magnetic fluids in the Lin–Sidorov–Aristov class

The paper considers the exact integration of magnetic hydrodynamic equations for describing nonuniform unidirectional flows of viscous incompressible fluids. The construction of an exact solution is based on the well-known representation of hydrodynamic fields as the Lin–Sidorov–Aristov class. The 3d magnetic field is described by linear forms with respect to two spatial coordinates (longitudinal, or horizontal). The coefficients of the linear forms depend on the third coordinate and time. In view of the incompressibility condition, the 1D velocity field depends on two coordinates and time. The pressure is shown to be determined by a quadratic form with constant coefficients. These coefficients are determined by pressure distribution on the known (free) boundary. The exact solution is illustrated by the integration of non-1D hydrodynamic fields in the case of the steady motion of a conducting viscous incompressible fluid. This solution is polynomial, and it will be useful for the formulation of new problems of hydrodynamic stability.