Linear instability of viscoelastic interfacial Hele-Shaw flows: A Newtonian fluid displacing an UCM fluid

We theoretically study the linear stability of the Saffman-Taylor problem where a viscous Newtonian fluid (with viscosity eta(l)) displaces an Upper Convected Maxwell (UCM) fluid (with viscosity eta(r)) in a rectilinear HeleShaw cell. The dispersion relation is given by the roots of a cubic polynomial with coefficients depending on wavenumber along with several dimensionless groups as parameters. Using Routh-Hurwitz stability criterion, we show that the viscosity ratio eta(r)/eta(l) still plays a decisive role in determining stability (stable if eta(r)/eta(l) <= 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, capillary number 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, (ii) stress becomes singular when the wavenumber exceeds a certain value; and (iii) a resonance phenomenon occurs when eta(r)/eta(l) is large, where the growth rate increases very rapidly near certain wavenumber and eventually becomes singular when the fluid is inviscid.

Automated summary

We theoretically study the linear stability of the Saffman-Taylor problem and find that the viscosity ratio eta(r)/eta(l) plays a decisive role in determining stability.