Linear instability of viscoelastic interfacial Hele-Shaw flows: Newtonian fluid displacing an upper-convected Maxwell fluid

We theoretically study linear stability of two-phase interfacial flow problem where a viscous Newtonian fluid displaces an Upper convected Maxwell (UCM) fluid in a rectilinear Hele-Shaw cell. The dispersion relation is found to be a root of a cubic polynomial $\mathcal{F}$ with coefficients depending on wavenumber along with several other dimensionless groups as parameters. Through Routh–Hurwitz stability criterion, we found the viscosity contrast $\eta^r/\eta^l$ still plays a decisive role in determining the stability (stable if $\eta^r/\eta^l \leq 1$). If $\eta^r/\eta^l>1$, the flow is more unstable than an identical Newtonian-Newtonian setup and the most unstable wavenumber is larger. Increasing Deborah number $De$, capillary number $Ca$ or flow speed worsens the instability. Elasticity has a variety of effects and can give rise up to three types of singular behaviors: (i) there exists infinitely many distinct wavenumbers at which the velocity becomes singular, and (ii) stress becomes singular when the wavenumber exceeds a certain value; and (iii) a resonance phenomenon is discovered when $\eta^r/\eta^l$ is large, where the growth rate increases very rapidly near a certain wavenumber and eventually becomes singular if displacing fluid becomes inviscid.