Mode-coupling approach to non-Newtonian Hele-Shaw flow.
The Saffman-Taylor viscous fingering problem is investigated for the displacement of a non-Newtonian fluid by a Newtonian one in a radial Hele-Shaw cell. We execute a mode-coupling approach to the problem and examine the morphology of the fluid-fluid interface in the weak shear limit. A differential equation describing the early nonlinear evolution of the interface modes is derived in detail. Owing to vorticity arising from our modified Darcy's law, we introduce a vector potential for the velocity in contrast to the conventional scalar potential. Our analytical results address how mode-coupling dynamics relates to tip splitting and sidebranching in both shear-thinning and shear-thickening cases. The development of non-Newtonian interfacial patterns in rectangular Hele-Shaw cells is also analyzed.
Constantin
Magdalena
M
Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA.
Widom
Michael
M
Miranda
José A
JA