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

Abstract

We theoretically study the linear stability of the Saffman–Taylor problem where a viscous Newtonian fluid (with viscosity η l ) displaces an Upper Convected Maxwell (UCM) fluid (with viscosity η r ) in a rectilinear Hele-Shaw 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 η r / η l still plays a decisive role in determining stability (stable if η r / η l ≤ 1 ). If η r / η 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 η r / η l is large, where the growth rate increases very rapidly near certain wavenumber and eventually becomes singular when the displacing fluid is inviscid. • Explicit formulas for the dispersion relation. • Singular behaviors in the velocity, stress, and growth rate. • Viscosity contrast determines long wave stability. • Destabilizing effect if increasing flow speed, Deborah number or Capillary number. • More unstable than Newtonian case with shorter most unstable wave.