Linear Instability of Interfacial Hele-Shaw Flows of Viscoelastic Fluids

Abstract

We present a theoretical 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. This is a generalization of Hai & Daripa (2022) where the case of a Newtonian fluid displacing an UCM fluid is considered. The elastic effect of the displacing layer has a significant impact on the overall flow stability. The dispersion relation is implicitly given by the roots of a quartic polynomial with its coefficients depending on a modified wavenumber, viscosity contrast η^r/η^l (displaced/displacing fluid), relaxation time contrast λ^r/λ^l and a composite parameter inversely related to the flow speed U (depth averaged). Viscous effect is still the dominant mechanism in determining long wave stability (unstable if η^r/η^l > 1). The elastic effect of the displacing layer always destabilizes short waves (unstable if wavelength is shorter than λ^lU and shorter the wavelength is, more unstable the flow becomes). In addition, three types of singular behaviors are found all of which are associated with elastic effects: (i) velocity becomes singular at infinitely many isolated wavenumbers (precise values are inversely proportional λ^lU or λ^rU); (ii) stress becomes singular if wavenumber exceeds certain value (can happen even for slow flow); and (iii) growth rate becomes singular at up to two wavenumbers if η^r/η^l and λ^r/λ^l fall within certain range but this can always be avoided if flow is slow enough. The special cases of an UCM fluid displacing air or a viscous Newtonian fluid are also considered.

Keywords: Viscoelastic Hele-Shaw Flows, Linear Stability Theory, Dispersion Relations, Singularity, Fracture in Fluids