A paradox in Hele-Shaw displacements

Abstract

The Saffman–Taylor instability occurs when a less viscous Stokes fluid is displacing a more viscous one, in a rectangular Hele-Shaw cell. We study the case when all surface tensions on the interfaces are zero. Consider N liquid-layers with constant viscosities, inserted between the two initial fluids. The corresponding growth constants become infinite with increasing wave numbers, even if N is very large. On the contrary, a single intermediate liquid with a suitable variable viscosity gives us an almost stable flow. We compute the total amount of intermediate liquid used in both models.