New bounds for stabilizing Hele-Shaw flows

We consider the problem of displacement processes in a three-layer fluid in a Hele-Shaw cell modeling enhanced processes of oil recovery by polymer flooding. The middle layer sandwiched between water and oil contains polymer-thickened water. We provide lower bounds on the length of the intermediate layer and on the amount of polymer in the middle layer for stabilizing the leading front to a specified level. We also provide an upper bound on the growth rate of instabilities for a given viscous profile of the middle layer.