On Stabilization of Multi-Layer Hele-Shaw and Porous Media Flows in the Presence of Gravity

Stabilization of multi-layer Hele-Shaw flows is studied here by including the influence of Rayleigh-Taylor instability in our earlier work (Daripa, J. Stat. Mech. 12:28, 2008a) on stabilization of multi-layer Saffman-Taylor instability. Furthermore, this article goes beyond our previous work with few extensions, improvements, new interpretations, and clarifications on the use of some terminologies. Results of two complete studies have been presented: the first investigates the effect of individually unstable interfaces on the overall stability of the flow, and the second studies the cumulative effect of unstable interfaces as well as unstable internal viscous layers. In each case, modal and absolute upper bounds on the growth rate are reported. Next, these bounds are used to investigate (i) stabilization of long waves on various interfaces; (ii) stabilization of all waves on all interfaces in comparison to pure Taylor instability; (iii) stabilization of disturbances on interior interfaces instead of exterior interfaces. In the first study, notions of partial and total stabilization with respect to the pure Taylor growth rate are introduced. Then necessary and sufficient conditions for partial and total stabilizations are found. Proof of stabilization of long waves on one of the two external interfaces in multi-layer flows is also proved. In the second study, an absolute upper bound is obtained in the presence of stabilizing density stratification across each internal interface even though all interfaces and layers have unstable viscous profiles. Exact results on the upper bounds, and necessary and sufficient conditions for control of instabilities driven by stable/unstable density stratification, unstable viscous layers and unstable interfaces are new and may be relevant to explain observed phenomena in many complex flows generating these kinds of viscous profiles and density stratification as they evolve. The present work builds upon and goes much further in details and new results than our previous work. The gravity effect included here brings with it restrictions which have not been addressed before in this multi-layer context.