Radial fin? What's that? Picture a flat disk, with a heat source of a certain radius in the center. There is a convection coefficient on the top and bottom surfaces of the disk.

A classic  application of this is the type of baseboard radiator in the home where I grew up -- a copper tube (forced hot water from the furnace) with individual metal fins. Okay, those fins were rectangular, but we can  approximate that away with some mathematical tricksmanship (equivalent radius, anyone?). The whole assembly was hidden behind a cosmetic cover with an adjustable vent. I realize that many readers will not be familiar with this system... and besides, most of us aren't going around analyzing our home radiators. (That might be fun, come to think of it...) Seriously, though, why am I bringing this up?

In an electronics  cooling context, there are several places where the radial fin concept, with its associated mathematical solution, might apply. Besides the straightforward heat sink, with a center stem or heat pipe and radial fins,  there is the heat source mounted on a circuit board. The circuit board always participates in the cooling, even if there is another heat path (heat sink) available.  Of course, if the heat source has a really good  heat sink on it, the circuit board cooling will be a minor effect. But even though some components have big heat sinks, there are often neighboring low-power components that rely on the circuit board to spread and  dissipate the heat. You can think of this as a "thermal territory" (see Ake Malhammar's description), where a certain board "acreage" is needed to support the power dissipation and temperature limit for a component. Large territory? Low temperature. Territory too small (components too close together)? The component is starved for dissipation capacity, and needs to increase its temperature to dissipate the heat imposed by the component.

So much for the concept. How can we actually use it for analysis? Bruce Guenin wrote about the circuit board application in Electronics Cooling. In the article, among other things, he describes the heat transfer between circuit board traces and the  air using the circular fin approximation. Table 3 of the article shows the exact mathematical solution; for a circuit board he approximates the effective board thickness (its conductivity isn't uniform).

The Bessel function solution looks intimidating, but Excel (and I'm sure other spreadsheets too) has them built in.

Just one more caveat: use this or any idealized mathematical description to give you design guidance -- not to try for 100% accuracy. For example, you  might use it to answer the question "Does this component need a heat sink or is board cooling going to be OK, given the proximity of its neighbors?" If you're unsure, design in the heat sink...you might be able to eliminate it later, when you discover that the component power estimates were too high by 75%!