An Instability Phenomenon in Hele-Shaw Displacements

Abstract

We study the linear instability of the displacement of an Oldroyd-B fluid by air in a Hele-Shaw cell, motivated by possible applications in Secondary Oil Recovery (SOR) process. Most numerical methods have usually failed when the Weissenberg numbers W i (appearing in the constitutive relations) are near 1. We get an approximate formula of the growth rate with a blow-up for some particular W i = O(1). Therefore the instability at large Weissenberg numbers is due to the model, and not to the computational methods. Our growth rate is quite similar to Saffman-Taylor's formula (obtained when a Newtonian liquid is displaced by air) if W 1 = W 2 .