## Jacob Bear Short Course – Day 3

I had an epiphany today. It has to do with dispersion. If you are at a scale, where you look at little pieces of solids which have either and / or air around them. Let’s call this microscopic scale. Say, you are trying to describe how a solute moves *by advection* on that scale, then you would do this by a term that represents the velocity of the fluid times the concentration of the solute. Now, if you want to go one scale up, to a “macroscopic” scale. Then you have to average. It turns out, that the result of this averaging are *two* terms, both represent a velocity times a concentration, but one is the advective flux from before, and the other one is a dispersive flux.

When I heard this this morning for the first time I though, ok, fair enough, but where’s the dispersion tensor coming in? There are two parts to this answer. (1) dispersion has at the “beginning” been called “*mechanical spreading*” — a phenomenon that is caused by pure fluid mechanics. (2) the dispersion tensor comes only in when people realized, that this second part that arose when averaging solute advection from microscale to macroscale, can be described by a constant times a gradient of the solute’s concentration. Tata, and the constant is the dispersion tensor, the entries of which are the dispersivities times the velocity. The discussion of what dispersivities are and how they relate to each other is a whole other story.

And it turns out that the dispersivities in non-isotropic cases are actively researched, foremost by Jacob Bear himself, “30 to 40 years” after he dealt with dispersion for the first time!

Note: I have no idea, what exactly arxiv.org does, I want to point out what I wrote in the impressum, that is that I am not responsible to content of sites I link to (disclaimer), however both current articles by Jacob Bear are available on arxiv.org, that is (see here and here).