Linear Instability of Interfacial Hele-Shaw Flows of Viscoelastic Fluids
29 Pages Posted: 8 Apr 2022
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 ηr/ηl (displaced/displacing fluid), relaxation time contrast λr/λl 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 ηr/ηl > 1). The elastic effect of the displacing layer always destabilizes short waves (unstable if wavelength is shorter than λlU 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 λlU or λrU); (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 ηr/ηl and λr/λl 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
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