Non-stationary heat exchange in cylindrical channel at laminar flow of fluids

Using double integral Laplace–Carson transformation and orthogonal method of Bubnov–Galyorkin, the analytical solution of the non-stationary problem of heat transfer in a cylindrical channel in the laminar flow of fluids was obtained. It has two components: stationary and non-stationary, each part has application only in a certain range of temporal and spatial coordinates. For the stationary Graetz-Nusselt problem on the basis of introduction of the temperature perturbation front and additional boundary conditions it was managed to find an analytical solution that allows the assessment of liquid thermal state with small values of spatial variable, directed along the stream flow. It is not possible to obtain such results using the well-known exact analytical methods because of the poor convergence of infinite series of received solutions.