We consider a setup in a Hele-Shaw cell where a fluid of constant viscosity μl occupying a near-half-plane pushes a fluid of constant viscosity μ>μl occupying a layer of length L which in turn pushes another fluid of constant viscosity μr>μ occupying the right half-plane. The fluid upstream has a velocity U. Careful analysis of the dispersion relation arising from linear stability of the three-layer Hele-Shaw flow problem leads to the following specific analytical results all of which are strikingly independent of the length (L) of the middle layer: (i) a necessary and sufficient condition for modal instability; (ii) a critical viscosity of the middle layer that gives the shortest bandwidth of unstable waves; and (iii) a strict upper bound on the growth rate of instabilities, meaning that this upper bound is never reached and hence this upper bound can be improved upon. Results based on exact growth rates are presented which provides some insight into the instability transfer mechanism between interfaces as the parameters of the problem are varied. Numerical evidence that supports the effectiveness of the upper bound is also presented.

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