Pattern formation of the three-layer Saffman-Taylor problem in a radial Hele-Shaw cell

M. Zhao, Pedro H. A. Anjos, J. Lowengrub, and Shuwang Li
Phys. Rev. Fluids 5, 124005 – Published 30 December 2020

Abstract

The three-layer Saffman-Taylor problem introduces two coupled moving interfaces separating the three fluids. A very recent weakly nonlinear analysis of this problem in a radial Hele-Shaw cell setup has shown that the morphologies of the emerging fingering patterns strongly depend on the initial thickness of the intermediate layer connecting the two interfaces. Here we go beyond the weakly nonlinear regime and explore full nonlinear interfacial dynamics using a spectrally accurate boundary integral method. We quantify the nonlinear instability of both interfaces as the relevant physical parameters (e.g., viscosities and surface tensions) are varied and show that our nonlinear computations are in good agreement with the experimental observations and the weakly nonlinear analysis. Nonlinear simulations reveal that due to the existence of a second interface, the classical highly branched morphologies are replaced by less unstable structures in which finger-tip-splitting and finger competition phenomena are evidently restrained as the initial annulus's thickness is reduced. In addition, these patterns develop fingers with a series of low-amplitude bumps at their tips, associated with the enhanced growth of high-frequency modes promoted by the increasing coupling strength of interfaces.

  • Figure
  • Figure
  • Figure
  • Figure
  • Figure
  • Figure
  • Figure
3 More
  • Received 5 June 2020
  • Accepted 17 December 2020

DOI:https://doi.org/10.1103/PhysRevFluids.5.124005

©2020 American Physical Society

Physics Subject Headings (PhySH)

Fluid Dynamics

Authors & Affiliations

M. Zhao1,*, Pedro H. A. Anjos2,†, J. Lowengrub1,‡, and Shuwang Li2,§

  • 1Department of Mathematics, University of California at Irvine, Irvine, California 92697, USA
  • 2Department of Applied Mathematics, Illinois Institute of Technology, Chicago, Illinois 60616, USA

  • *mzhao9@uci.edu
  • pamorimanjos@iit.edu
  • lowengrb@math.uci.edu
  • §sli@math.iit.edu

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand
Issue

Vol. 5, Iss. 12 — December 2020

Reuse & Permissions
Access Options
Author publication services for translation and copyediting assistance advertisement

Authorization Required


×
×

Images

×

Sign up to receive regular email alerts from Physical Review Fluids

Log In

Cancel
×

Search


Article Lookup

Paste a citation or DOI

Enter a citation
×