Exact solution of the problem of convective heat exchange for a circular cylinder at small Prandtl numbers (Q1892753)

The problem of the heat exchange of a circular cylinder in an incompressible flow at small Prandtl numbers \(\text{Pr}\ll 1\) is solved. The solution is expressed in terms of Mathieu functions. The isotherm patterns, the distribution of the local Nusselt number Nu along the cylinder surface for different Péclet numbers \(\text{Pe}\leq 10\), and the dependence of the average Nusselt number \(\langle\text{Nu}\rangle\) on Pe are given. A comparison is made between the Nusselt numbers Nu and the temperature profiles in the neighborhood of the windward stagnation point, obtained by solving the energy equation and the equation of the thermal boundary layer. The thermal boundary layer equation is shown to be appropriate for calculating the heat transfer at the windward stagnation point for \(\text{Pr}\ll 1\) and \(\text{Pe}\gtrsim O(1)\).