We study the motion of miscible liquids in porous media with the speed determined by Darcy's law. The two basic examples are the displacement of viscous liquids and the motion induced by gravity. Such motion often is unstable and creates patterns called viscous fingers (picture attached).

We concentrate on important for applications property of viscous fingers - speed of their propagation. The work is inspired by the results of F. Otto and G. Menon for a simplified model, called transverse flow equilibrium (TFE). In this work a rigorous upper bound was proved using the comparison principle. At the same time numerical experiments suggest that the actual speeds are better than Otto-Menon estimates.

We consider a two-tubes model -- the simplest model we were able to construct which includes transverse liquid flow. For this model for the gravitational fingers we were able to find families of travelling waves and found the relation between original model and TFE simplification. The main tool in the proof in normal hyperbolicity. For viscous liquid it seems that the phenomenon is the same but up to now it is work in progress.

This is a joint work with Yulia Petrova and Yalchin Efendiev.