# Viscous fingering in yield stress fluids: a numerical study

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## Abstract

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.

## Keywords

Bi-viscous Bingham model Saffman–Taylor instability Slip boundary condition Viscous fingering## Notes

### Acknowledgments

Kayvan Sadeghy would like to express his thanks to Iran National Science Foundation (INSF) for supporting this work under contract number 93034827. Special thanks are also due to the respectful reviewers for their constructive comments.

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