On the role of elastic effect on the fingering or fracturing instability in Hele-Shaw flow

This a theoretical study on the role of elastic effect on the fingering or fracturing instability for interfacial Hele-Shaw flow, where a viscous Newtonian fluid (or air) displaces an Oldroyd-B(OB) fluid in a rectilinear Hele-Shaw cell. Linear stability analysis is performed (subject to the lubrication approximation) in the regime of Deborah number being $\boldsymbol{O}(1)$ or larger. We found the viscosity contrast is still the decisive factor in determining long wave stability, similar to the classical Saffman-Taylor prediction. For short wave disturbances, instability occurs, and the flow is almost always more unstable than an identical Newtonian setup and manifest in a finer scale. Short waves disturbances are plagued with singularities (multiple types), thus establishing the connection to the fracturing/cracks observed in experiments. Certain regularizing effect of the Newtonian contribution of the stress from OB is found, (i) without it (becoming upper convected Maxwell (UCM)), at least one additional singular behavior is comes into play, and (ii) more unstable and UCM.