From viscous fingering to elastic instabilities

Abstract

An analytical and numerical study of the linear Saffman–Taylor instability for a Maxwell viscoelastic fluid is presented. Results obtained in a rectangular Hele–Shaw cell are complemented by experiments in a circular cell corroborating the universality of our main result: The base flow becomes unstable and the propagating disturbances develop into crack-like features. The full hydrodynamics equations in a regime where viscoelasticity dominates show that perturbations to the pressure remain Laplacian. Darcy’s law is expressed as an infinite series in the cell thickness. An unique dimensionless parameter λ¯, equivalent to a relaxation time, controls the growth rate of the perturbation. λ¯ depends on the applied gradient of pressure, the surface tension, the cell thickness, and the elastic modulus of the fluid. For small values of λ¯, Newtonian behavior dominates whereas for higher values of λ¯ viscoelastic effects appear. For the critical value λ¯=λc¯≃10 a blowup is predicted and fracture-like patterns are observed.