Non-Stationary Flow of a Viscous Incompressible Electrically Conductive Liquid on a Rotating Plate in the Presence of Media Injection (Suction), Considering Induction and Diffusion Effects

The branch of physics known as magnetohydrodynamics (MHD) emerged in the middle of the 20th century. MHD models, being substantially nonlinear, are quite challenging for theoretical study and allow nontrivial consideration only in particular limited cases. Thus, due to the exceptional growth of calculation power, research on MHD is now primarily concentrated on numerical modeling. The achievements are considerable; however, there is a possibility of overlooking some phenomena or missing an optimal approach to modeling and calculating that could be identified with theoretical guidance. The paper presents a theoretical study of a particular class of boundary and initial conditions. The flow of a viscous, electrically conductive fluid on a rotating plate in the presence of a magnetic field is considered. The fluid and the bounding plate rotate together with a constant angular velocity around an axis that is not perpendicular to the plane. The flow is induced by sudden longitudinal vibrations of the plate, injection (suction) of the medium through the plate, and an applied magnetic field directed normal to the plate. The full equation of magnetic induction is used, taking into account both the induction effect and energy dissipation due to the flow of electric currents. An analytical solution of three-dimensional magnetohydrodynamics equations in a half-space bounded by a plate is presented. The solution is given in the form of a superposition of plane waves propagating with certain wave numbers along the y-coordinate axis. For certain regions of system parameters, the vibration of the bounding plate does not cause waves in the media.