Linear Instability of Interfacial Hele-Shaw Flows of Viscoelastic Fluids

Abstract

We present a study on the role of elasticity in causing fingering or fracturing instability during the immiscible displacement process of a viscoelastic fluid by another viscoelastic fluid in a rectilinear Hele-Shaw cell. Upper convected Maxwell (UCM) models are used for both fluid layers and linear stability analysis is performed in the regime of moderate to large Deborah number. Through a proper scaling scheme and a lubrication approximation, the governing equations can be reduced to a single inhomogenoues partial differential equation with variable coefficients, which is then solved analytically using classical techniques. The solution can be expressed in terms of weighted integrals of Bessel functions of first kind. The solution is then inserted to the interface conditions, which in turn leads to a dispersion relation that is implicitly given by the roots of a quartic polynomial with coefficients depending on the wavenumber, viscosity contrast, relaxation time contrast and a composite parameter that contains capillary number, Deborah number and flow speed. By taking proper limits in the parameter space, the classical Saffman-Taylor result is recovered. We find that the long wave stability is still determined by the viscosity ratio (unstable if displaced fluid is more viscous), however the elastic effect of the displacing fluid always destabilizes short waves (unstable if the perturbation has wavelength shorter than the distance traveled by the fluid bulk within one relaxation time). Elasticity also has a variety of effects but primarily affects the short waves and can give rise up to three types of singular behaviors, thus establishing connection to the fracture fingering pattern observed in experiments. In particular, (i) velocity becomes singular at infinitely many isolated wavenumbers, and (ii) stress becomes singular if wavenumber exceeds certain value (precise values depend on a variety of parameters, but most importantly this can happen even for slow flow), and (iii) temporal growth rate becomes singular at up to two wavenumbers (can always be avoided if flow is slow enough). The following special cases are also considered • Air displaces an UCM fluid • Viscous Newtonian fluid displaces an UCM fluid • UCM fluid displaces air • UCM fluid displaces a viscous Newtonian fluid