Spatiotemporal linear stability of viscoelastic Saffman-Taylor flows

TL;DR

This paper conducts a detailed linear stability analysis of viscoelastic Saffman-Taylor flows in a Hele-Shaw cell, identifying stability regimes and the effects of inertia and elasticity on interface behavior.

Contribution

It provides the first comprehensive spatiotemporal stability analysis of Oldroyd-B fluids in this context, including scaling laws and stability criteria.

Findings

Critical Reynolds number diverges as (E(1-ν))^{-5/3}

Critical wavenumber scales as (E(1-ν))^{-2/3}

Absolutely unstable regions occur at high Reynolds and elasticity numbers

Abstract

A comprehensive, temporal and spatiotemporal linear stability analyses of a (driven) Oldroyd-B fluid with Poiseuille base flow profile in a horizontally aligned, square, Hele-Shaw cell is reported to identify the viable regions of topological transition of the advancing interface. The dimensionless groups governing stability are the Reynolds number, $Re = \frac{b^2 \rho \mathcal{U}_0}{12 \eta_0 L}$, the elasticity number, $E = \frac{12 \lambda (1-\nu)\eta_0}{\rho b^2}$ and the ratio of solvent to polymer solution viscosity, $\nu = \frac{\eta_s}{\eta_0}$; here $b$ is the cell gap, $L$ is the length/width of the cell, $\mathcal{U}_0$ is the maximum velocity of the mean flow, $\rho$ is the density of the driven fluid and $\lambda $ is the relaxation time. Excellent agreement on the size of the relative finger width between our model and the experiments in the Stokes and the inertial, Newtonian regime is found. In the asymptotic limit $E(1-\nu) \ll 1$, the critical Reynolds number, $Re_c$ (defined as the largest Reynolds number beyond which all wavenumbers are temporally unstable) diverges as per the scaling law $Re_c \sim \left[E(1-\nu)\right]^{-5/3}$ and the critical wavenumber increases as $\alpha_c \sim \left[E(1-\nu)\right]^{-2/3}$. The temporal stability analysis stipulate that (a) the destabilizing influence of the inertial, (b) the destabilizing impact of finite boundaries near the wall, and (c) the stabilizing (destabilizing) impact of elasticity combined with low (high) fluid inertia . The Briggs idea of analytic continuation is deployed to classify regions of absolute and convective instabilities, as well as the evanescent modes. The phase diagram reveals the presence of absolutely unstable region at high values of Reynolds and elasticity number, confirming the role of fluid inertia in triggering a pinch-off.