Pattern formation in non-Newtonian Hele-Shaw flow

被引:80
作者
Fast, P
Kondic, L
Shelley, MJ
Palffy-Muhoray, P
机构
[1] NYU, Courant Inst Math Sci, New York, NY 10012 USA
[2] New Jersey Inst Technol, Dept Math Sci, Newark, NJ 07102 USA
[3] New Jersey Inst Technol, Ctr Appl Math & Stat, Newark, NJ 07102 USA
[4] Kent State Univ, Inst Liquid Crystal, Kent, OH 44242 USA
关键词
D O I
10.1063/1.1359417
中图分类号
O3 [力学];
学科分类号
08 ; 0801 ;
摘要
We study theoretically the Saffman-Taylor instability of an air bubble expanding into a non-Newtonian fluid in a Hele-Shaw cell, with the motivation of understanding suppression of tip-splitting and the formation of dendritic structures observed in the flow of complex fluids, such as polymeric liquids or liquid crystals. A standard visco-elastic flow model is simplified in the case of flow in a thin gap, and it is found that there is a distinguished limit where shear thinning and normal stress differences are apparent, but elastic response is negligible. This observation allows formulation of a generalized Darcy's law, where the pressure satisfies a nonlinear elliptic boundary value problem. Numerical simulation shows that shear-thinning alone modifies considerably the pattern formation and can produce fingers whose tip-splitting is suppressed, in agreement with experimental results. These fingers grow in an oscillating fashion, shedding "side-branches" from their tips, closely resembling solidification patterns. A careful analysis of the parametric dependencies of the system provides an understanding of the conditions required to suppress tip-splitting, and an interpretation of experimental observations, such as emerging length-scales. (C) 2001 American Institute of Physics.
引用
收藏
页码:1191 / 1212
页数:22
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