Non-Newtonian Hele-Shaw Flow and the Saffman-Taylor Instability

Ljubinko Kondic, Michael J. Shelley, Peter Palffy-Muhoray
1998 Physical Review Letters  
We explore the Saffman-Taylor instability of a gas bubble expanding into a shear thinning liquid in a radial Hele-Shaw cell. Using Darcy's law generalized for non-Newtonian fluids, we perform simulations of the full dynamical problem. The simulations show that shear thinning significantly influences the developing interfacial patterns. Shear thinning can suppress tip splitting, and produce fingers which oscillate during growth and shed side branches. Emergent length scales show reasonable
more » ... ow reasonable agreement with a general linear stability analysis. [S0031-9007 (97) 05226-5] PACS numbers: 47.50. + d, 47.20.Ma, 47.54. + r, 68.10. -m Complex fluids, such as liquid crystals [1], polymer solutions and melts [2], clays [3], and foams [4] , display rich non-Newtonian behavior-viscoelasticity, shear thinning and thickening, boundary or flow induced anisotropywhose nonlinear effect on flow is understood at best phenomenologically. Hele-Shaw (H-S) flow between two closely spaced plates has been used to study such fluids; inertia is negligible, and the resulting description is simplified by the high aspect ratio geometry. Such "thin-gap" flows of non-Newtonian liquids are also relevant to industrial processes such as injection molding [5] and display device design [6] . Interest stems also from the close analogy between Newtonian H-S flow and quasistatic solidification; the Saffman-Taylor (S-T) instability of the driven fluid-fluid interface plays the same role as the Mullins-Sekerka instability of the solidification front [7] . Features usually associated with solidification, such as the growth of dendritic fingers and side branching, have also been observed in Newtonian fluids with imposed anisotropy, say by scoring lines on the cell plates [8] . However, experiments using non-Newtonian or anisotropic fluids have shown that solidification structures, such as snowflake patterns in liquid crystal flows [1], or needle crystals in polymeric solutions [2], can be induced by the bulk properties of the fluid itself, without any imposed anisotropy. In [9], we conjectured that shear thinning-a property of polymeric liquids and effectively of nematic liquid crystals in certain geometries-was a crucial ingredient in suppressing tip splitting, and might lead to the appearance of dendritelike structures in complex fluids. In this scenario, the tip of a finger lies in a region of high shear, and thus lower viscosity, which causes it to advance with higher relative velocity than surrounding portions of the interface, suppressing the spreading of the tip. To study this, we derived from first principles a natural generalization of Darcy's law which takes into account shear thinning (or thickening) in an isotropic fluid. In support of our conjecture, we showed that for a gas bubble expanding into a weakly shear thinning fluid, the S-T instability is modified to give increased length-scale selection. In this Letter, we use the generalized Darcy's law to perform fully nonlinear simulations of a bubble expanding into a strongly shear thinning liquid. The simulations demonstrate that shear thinning significantly modifies the evolution of the interface, and can produce fingers whose tip splitting is suppressed, and which have dendritic appearance. The resulting patterns are often similar to those observed in experiments [1] [2] [3] [4] 10, 11] . Length scales from our linear stability analysis are consistent with simulational results. Finally, we give a morphological phase diagram in terms of flow and fluid parameters. Formulation.-Consider a gas bubble expanding under applied pressure into a non-Newtonian fluid in a radial H-S cell. The fluid domain is an annular region V with inner boundary G i and external boundary G e . Neglecting inertia, we use the Stokes equations with shear-rate dependent viscosity, Here p is pressure, S is the rate-of-strain tensor for the fluid velocity v͑x, y, z͒ ͑u, y, w͒, with z the "short," cross-gap direction, and jSj 2 tr͑S 2 ͒. We follow [12] and use the viscosity model m͑jSj 2 ͒ m 0 f a ͑t 2 jSj 2 ͒, with f a ͑j 2 ͒ ͑1 1 aj 2 ͒͑͞1 1 j 2 ͒. Here t is the longest (Zimm) relaxation time of the fluid, m 0 is its zero shearrate viscosity, and a measures shear dependence: a 1 is Newtonian, a . 1 gives shear thickening, and a , 1 gives shear thinning. In practice, most non-Newtonian fluids are shear thinning. The flow is simplified by the small aspect ratio e b͞L ø 1, where L is a typical lateral length scale, and b is the plate separation. To nondimensionalize, the lateral and vertical distances are scaled by L and b, respectively. The scale for pressure is taken asp͞e, wherep͞e is the driving (gauge) pressure. The natural velocity and time scales are then U c eLp͞m 0 and T c m 0 ͞ep. This is the scaling required for shear thinning of the fluid to be apparent [13] . At leading order in e, in [9] we derived from Eq. (1) a generalized Darcy's law for the gap averaged, lateral 0031-9007͞ 98͞80(7)͞1433(4)$15.00
doi:10.1103/physrevlett.80.1433 fatcat:34bvtqi6jreulady2awd5xtl6y