A computational analysis is submitted for the two-dimensional transient conjugate heat transfer problem. The periodic problem was solved under the heat flux boundary condition outside. The thick-walled pipe is assumed to have two regions. Both regions are infinite in length. Initially an isothermal condition is defined in the upstream region with hydrodynamically developed flow, and sinusoidal change in a time fluid medium is defined by the external surface of pipes in the downstream region. The convection problem with axial fluid conduction was analyzed simultaneously for fluid and solid media. Significant heat is transferred to the upstream region, which preheats the fluid. Heat transfer characteristics change periodically with outside fluid temperature, but these characteristics are the same for all axial positions at steady periodic state conditions. Net heat transfer value is zero at this period. The angular frequency and five dimensionless parameters characterize the flow regime in terms of axial and radial conduction and convection heat transfer characteristics significantly.