Hydromagnetic flow between two parallel walls

An exact solution to the two-dimensional problem of a viscous, incompressible, and electrically-conducting fluid flow between two parallel walls in the presence of a constant magnetic field normal to the walls is obtained when the velocity normal to the walls is zero and the magnetic Reynolds number is very small so that the induced magnetic field can be neglected. One of the walls is fixed and the other moves in its own plane with a velocity which varies periodically with time about a constant mean. The moving wall temperature is kept constant. Two cases are considered: (1) the fixed wall has a constant temperature, and (2) the fixed wall is insulated against heat (adiabatic wall). Expressions for the velocity field, the temperature field, skin friction, and heat transfer are derived.