Linear instability of viscoelastic interfacial Hele-Shaw flows: Upper convected Maxwell (UCM) fluid displacing another UCM fluid

In continuation of the talk by Dr. Daripa, the displacing fluid is replaced with another UCM fluid. $\mathcal{F}$ now becomes a quartic polynomial rather than cubic with one additional parameter involved, namely the relaxation time contrast $\lambda^r/\lambda^l$. Qualitatively, many features and conclusions found in the previous Newtonian-UCM setup are still valid in this case. The long wave stability is still determined by the viscosity contrast $\eta^r/\eta^l$, however always unstable to short waves. The critical wavenumber beyond which instability occurs depends only on the relaxation time of the displacing fluid $\lambda^l$ and flow speed. The resonance can still happen but only over a relatively narrow region in the parameter domain and may be avoided in multiple ways.