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 Delta-bar, equivalent to a relaxation time, controls the growth rate of the perturbation. Delat-bar depends on the applied gradient of pressure, the surface tension, the cell thickness, and the elastic modulus of the fluid. For small values of Delta-bar, Newtonian behavior dominates whereas for higher values of Delta-bar viscoelastic effects appear. For the critical value Dalta-bar ~= 10 a blowup is predicted and fracture-like patterns are observed.