Thermal boundary layer spherical bubble

In this form the parametric solutions, Eqs. (73) and (74), for the spherically symmetric problem can be applied immediately. Physically this implies that a bubble growing in an initially arbitrary temperature field grows at precisely the same rate as if the initial temperature were averaged in each thin spherical shell surrounding the bubble center. Two special cases are considered in detail (7) the linear thermal boundary layer, of thickness /, next to the heating surface, outside of which the temperature is uniform and (2) the exponential boundary layer, where the temperature is assumed to decay exponentially with distance from the wall. The latter distribution is of the form... 

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