Saffman–Taylor instability in yield stress fluids

Abstract

Pushing a fluid with a less viscous one gives rise to the well known Saffman-Taylor instability. This instability is important in a wide variety of applications involving strongly non-Newtonian fluids that often exhibit a yield stress.Here we investigate the Saffmann-Taylor instability in this type of fluid, in longitudinal flows in Hele-Shaw cells. In particular, we study Darcy's law for yield stress fluids. The dispersion equation for the flow is similar to the equations obtained for ordinary viscous fluids but the viscous terms in the dimensionless numbers conditioning the instability now contain the yield stress. This also has repercussions on the wavelength of the instability as it follows from a linear stability analysis. As a consequence of the presence of yield stress, the wavelength of maximum growth is finite even at vanishing velocities. We study Darcy's law and the fingering patterns experimentally for a yield stress fluid in a linear Hele-Shaw cell. The results are in rather good agreement with the theoretical predictions. In addition we observe different regimes that lead to different morphologies of the fingering patterns, in both rectangular and circular Hele-Shaw cells.

Slow flows of yield stress fluids: yielding liquids or flowing solids?

“…This led to detailed instability criteria and descriptions and analysis of various trends (Coussot 1999, Fontana et al 2013, Ebrahimi et al 2016. The validation of these conclusions by confrontation with experimental data remains extremely limited, and when this was done a reasonable agreement for the finger size (Lindner et al 2000, Derks et al 2003, Maleki-Jirsaraei et al 2005, but a strong discrepancy was found concerning the instability criterion (Barral et al 2010), suggesting some important effect is missed.…”

Nonequilibrium Interfacial Tension in Simple and Complex Fluids

“…where r ¼ ½r 2 0 þ _ Vt=ðπbÞ 0.5 is the time-dependent average radius of the interface, Γ e the interfacial tension between the two fluids, _ γ I ¼ 3 _ Vð2πr 0 b 2 Þ −1 the shear rate at the injection hole, and η 2 ð_ γ r Þ the shear rate-dependent viscosity [18,52,53], with _ γ r ¼ ð4r 0 _ γ I =rÞ the shear rate calculated at a distance r from the injecting hole, where the onset of the instability is observed. The shear-rate-dependent viscosity η 2 is measured independently by standard rheometry [54].…”

Viscous fingering in yield stress fluids: a numerical study

“…For ease of analysis, most studies carried out in the past on ST instability of non-Newtonian fluids have focused on the Hele-Shaw cell for modeling homogenous porous media. And, virtually all aspects of non-Newtonian behavior have been addressed in these studies (see, for example, [12][13][14][15] for shear-thinning fluids, [16] for elastic fluids, [17][18][19] for thixotropic fluids, and [20][21][22] for viscoplastic fluids). Having said this, it should be conceded that, due to the complexity of the viscous fingering phenomenon and the diversity of the parameters involved, most studies carried out in the past have been concerned with a linear stability analysis only (see, for example, [20]).…”

“…For instance, the effect of yield stress on the nonlinear growth of fingers still remains unexplored in the theoretical domain. Such nonlinearities (which exhibit themselves by tip-splitting and/or side-branching) have been witnessed in experimental studies by Lindner et al [21], Maleki-Jirsaraei et al [22], and Van Damme et al [23]. A theoretical prediction of such nonlinear effects has not been carried out in the past, perhaps because of its inherent difficulty.…”

“…Of particular importance is to identify the fluid key property giving rise to such complex patterns. The speculation is that the yield stress plays a crucial role in this regard, but because polymeric gels used in the experiments [21,22] were not purely viscoplastic, it is hard to make a concrete conclusion without further scrutiny.…”