Suppose there is a cylindrical object in a round hole in a block and you want to model diffusion from inside the object to outside the hole. For example, suppose the object is hot (a piston perhaps) and it is losing heat to a block it’s been inserted into, or suppose the object is an electrical conductor and there’s a flow of current across the gap and into the surrounding block. Mechanically, in such a circumstance, the objects aren’t perfectly concentric (the object rests against the side of the hole along a line or at two diagonally opposite points). For that matter, things are never perfectly round. Suppose the hole is what keeps the object in position; the object is resting in the hole in other words.
The finite element method for modeling is pretty typical, and where we can define the gap reasonably well, it would work fine. That is, if you picture a crescent shape in which the points of the crescent touch each other, that would be the shape you’d model diffusion across.
There are two difficulties:
In a mathematical sense there ought to be either a mathematical line of contact (if the hole and object axes are parallel) or two mathematical points of contact (if the axes aren’t parallel). The diffusion rate there would be unlimited by the locally nonexistant gap.
The crescent shaped gap grows thinner and thinner the closer you get to the mathematical contacts, and it takes more and more finite elements to model what happens in that thinner and thinner gap.
I think this problem would have come up any number of times in finite element modeling, and perhaps there’s some well worked out approach to it, some kind of empirical formula or something. Anybody?