On Modeling a Layered Viscous Conductive Fluid Flow in a Region Changing in Time

The flow of a viscous conductive incompressible fluid in a time-varying region is investigated. Based on the model of a layered fluid flow, the class of exact solutions of the equations of magnetohydrodynamics in the region moving in time is considered. We study the change in the structure of the fluid flow as a result of the volume effect by the magnetic field and the movement of the boundary of the flow region. The heat dissipation effect due to internal friction and Joule heating is considered. The presented results are relevant to the research on optimizing problems of controlling the dynamics of an incompressible fluid and creating the “digital field” using the domestic technology.