Q1: there is no physical model that gives this equation for arbitrary $m$; the values $m=1,2,3$ appear in viscous flow (see the references by Zurab Silagadze).

Q2: this [tutorial](http://www.cs.vu.nl/~jhulshof/PAPERS/48.pdf) by J. Hulshof seems a good entry point to the mathematical literature (which considers arbitrary $m$).