  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.
