On back flow of boundary layers in two-dimensional unsteady incompressible heat conducting flow

In this paper, we study the back-flow problem of boundary layers in two-dimensional unsteady incompressible heat conducting flow. For a kind of monotonic initial and incoming flow, we prove that the first critical point of the tangential velocity profile with respect to the normal variable, if exists, must appear on the boundary if the pressure gradient and temperature in the data are suitable. This critical point is the back-flow point. Moreover, we give a condition on the growth rate of the initial tangential velocity such that there is a back-flow point in the boundary layer.