The unsteady heat transfer due to a heat source in an MHD stretching sheet flow

The time evolution in the temperature field resulting from the sudden introduction of a heat source into the already fully established steady MHD flow of an electrically conducting fluid past a linearly stretching isothermal surface is considered. The problem is shown to be fully described by two dimensionless parameters, a modified magnetic field strength ?? and a heat source strength Q. Numerical solutions of the initial-value problem show that there is a critical value Q _c of the parameter Q, dependent on ??, such that, for Q<Q _c , the solution approaches a steady state at large times and, for Q>Q _c , the solutions grows exponentially large as time increases. This growth rate is determined through an eigenvalue problem which also determines the critical value Q _c . The limits of Q _c for both small and large values of ?? are discussed.