Q1: there is no physical model that gives this equation for arbitrary $m$; only $m=1,2,3$ appear in viscous flow (see the references by Zurab Silagadze); an instructive derivation for $m=3$ based on a variational principle is given in A variational approach to thin film hydrodynamics of binary mixtures (2015).
Q2: this tutorial by J. Hulshof seems a good entry point to the literature.