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

We theoretically study the linear stability of Saffman–Taylor problem where a viscous Newtonian fluid (with viscosity ηl) displaces an Upper Convected Maxwell (UCM) ηr) in rectilinear Hele-Shaw cell. The dispersion relation is given by roots cubic polynomial with coefficients depending on wavenumber along several dimensionless groups as parameters. Using Routh–Hurwitz criterion, we show that ratio ηr/ηl still plays decisive role determining (stable if ηr/ηl≤1). If ηr/ηl>1, flow more unstable than identical Newtonian–Newtonian setup and most larger. Increasing Deborah number, capillary number or speed worsens instability. Elasticity has variety effects can give rise up to three types singular behaviors: (i) there exists infinitely many distinct wavenumbers at which velocity becomes singular, (ii) stress when exceeds certain value; (iii) resonance phenomenon occurs large, growth rate increases very rapidly near eventually displacing inviscid.