Saffman-Taylor instability of viscoelastic fluids: From viscous fingering to elastic fractures

Abstract

We study the linear stability of an air front pushing on a viscoelastic upper convected Mawxell fluid inside a Hele-Shaw cell. Both theory and experiments involving several viscoelastic fluids prove that a unique dimensionless time parameter $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\lambda}}$ controls all elastic effects. For small values of $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\lambda}}$, Newtonian behavior dominates, while for higher values of $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\lambda}}$ viscoelastic effects appear. We show that the linear growth rate of a small initial perturbation diverges for a critical value $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\lambda}}=\stackrel{\texttildelow{}}{{\ensuremath{\lambda}}_{\mathbf{c}}}\ensuremath{\simeq}\mathbf{1}\mathbf{0}$. Experiments prove that this divergence is associated to a fracturelike pattern instability of the interface. We conclude that the observed fractures come from the Saffman-Taylor instability and that they directly emerge from the linear regime of it.