Viscous fingering in yield stress fluids: a numerical study

The effect of yield stress is numerically investigated on the viscous fingering phenomenon in a rectangular Hele–Shaw cell. It is assumed that the displacing fluid is Newtonian, while the displaced fluid is assumed to obey the bi-viscous Bingham model. The lubrication approximation together with the creeping-flow assumption is used to simplify the governing equations. The equations so obtained are made two-dimensional using the gap-averaged variables. The initially flat interface between the two (immiscible) fluids is perturbed by a waveform perturbation of arbitrary amplitude/wavelength to see how it grows in the course of time. Having treated the interfacial tension like a body force, the governing equations are solved using the finite-volume method to obtain the pressure and velocity fields. The volume-of-fluid method is then used for interface tracking. Separate effects of the Bingham number, the aspect ratio, the perturbation parameters (amplitude/wavelength), and the inlet velocity are examined on the steady finger width and the morphology of the fingers (i.e., tip-splitting and/or side-branching). It is shown that the shape of the fingers is dramatically affected by the fluid’s yield stress. It is also shown that a partial slip has a stabilizing effect on the viscous fingering phenomenon for yield-stress fluids.