Initial boundary value problem on the motion of a viscous heat-conducting liquid in a vertical pipe

The initial-boundary value problem arising in a modeling an unsteady unidirectional convective flow in vertical heat exchangers with an arbitrary cross section is researched. An a priori estimate in $L_2$ is obtained and uniqueness of the problem solution is proved. For a rectangular and circular sections solution was found in the form of double Fourier series. Sufficient conditions for stabilization of solution to rest with increasing time are given.