Calculation of heat transfer upon the laminar flow of liquid in a cylindrical channel in the presence of axial thermal conductivity

An analytical method that takes into account axial thermal conductivity was proposed for the calculation of eigenvalues and eigenfunction in a problem of heat exchange for laminar liquid flow in a cylindrical channel. The method was based on the use of the special hypergeomertic confluent function. With this function, it is possible to find the precise fudical values of eigenvalues and eigenfunctions at specific relations of the Bio and Peclet numbers. Moreover, the recommended approach makes it possible to carry out the necessary mathematical calculation operation with an arbitrary combination of the named similarity numbers with a sufficient degree of accuracy. This creates the corresponding dimensionless complex $\alpha$. Such an approach makes it possible to considerably limit (reduce) the number of weighty members of an infinite series of the applicable hypergeometric function. With the use of the named functions, the derived strict and approximate analytical solutions can be applied in the theoretical analysis of a wide class of thermal physics problems, including nonlinear problems.