On estimates for short wave stability and long wave instability in 3-layer Hele-Shaw flows

We consider linear stability of three-layer Hele-Shaw flows with each layer having constant viscosity and viscosity increasing in the direction of a basic uniform flow. While the upper bound results on the growth rate of long waves are well known from our earlier works, lower bound results on the growth rate of short stable waves are not known to-date. In this paper we obtain such a lower bound. In particular, we show in this paper following results: (i) the lower bound for stable short waves is also a lower bound for all stable waves and the exact dispersion curve for most stable eigenvalue intersects the dispersion curve based on the lower bound at a wavenumber where most stable eigenvalue is zero; (ii) the upper bound for unstable long waves is also an upper bound for all unstable waves and the exact dispersion curve for the most unstable eigenvalue intersects the dispersion curve based on upper bound at a wavenumber where the most unstable eigenvalue is zero. Numerical results are provided which support these findings.