Power series solutions for laminar plumes in a natural environment

Power series solutions to the boundary layer equations for laminar point and line thermal plumes in natural convection have been derived in terms of recurrent relations. These together with the initial conditions constitute closed-form solutions for any Prandtl number in the region where the series converge. The starting conditions are related to the maximum values of the temperate and velocity profiles. Their values together with those for the radius of convergence of the series have been obtained numerically for different Prandtl numbers, and best-fitting functions have been proposed for the variation observed. The validity of the approach has been tested against the known closed-form solutions giving identical results in the region of convergence. While the utility of the equations does not extend to infinity, the tests conducted indicate that the range of convergence can be potentially extended by using the Euler transform. This is especially true for results involving point heat sources, where it has been shown that, for all Prandtl numbers, the nearest singularity is found in the complex plane and, hence, has no physical significance.