Cascade of traveling waves in miscible displacement in porous media

We study the motion of miscible liquids in porous media with the speed determined by Darcy's law. It has a long list of applications in the petroleum industry. 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.
We concentrate on the 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 which contains fingering instability. For the case of gravitational fingers we were able to find families of traveling waves and their speeds – it is significantly different from the Otto-Menon estimates. The proof strongly relies on techniques from dynamical systems: heteroclinic orbits, singularly-perturbed systems, normal hyperbolicity, transversality.
This is a joint work with Yulia Petrova and Yalchin Efendiev.