Thermal convection in a horizontal channel in the presence of a longitudinal temperature gradient varying with time in accordance with the exponential law is studied. An attempt is made to describe the delay of thermocapillary convection onset owing to the existence of a film on the fluid surface. The difference in the convection flow structures in the cases of fixed and destroyed films is shown. An exact solution for the plane-parallel flow under a surface with the friction proportional to the velocity at a constant longitudinal temperature gradient is derived. The solution of the time-dependent problem for the model with the Bingham properties of the fluid surface is obtained using a difference method.