Viscous fingering, or the Saffman-Taylor instability, has become one of the canonical systems in the study of pattern formation. It is a two-part story. First, a fluid confined between two closely-spaced plates is governed by a reduced form of the Navier-Stokes equation, known as Darcy's Law - coming from the lubrication approximation for the viscous term. Second, for an interface between two fluids, one forced into the other, an instability occurs when the less viscous fluid pushes the more viscous fluid. And this instability is viscous fingering.
We are investigating a more unusual instability: the image above shows a fingering pattern between two fluids of identical viscosity. The instability occurs because there is a sort of reaction occurring at the interface between the two fluids, a reaction in which spherical micelles of a surfactant become long, wormlike tubes when exposed to an organic salt solution. These are the two components of a viscoelastic material known as a wormlike micellar fluid. This instability we have found also occurs, but with a different morphology, if the two fluids reverse roles.
In another area of collaboration, we are investigating the role of explicit time-dependence in this system. The most recent two publications of our collaboration (see below) give an overview of this - controlling the mode of the instability, and deriving new models in the complex plane. We plan to expand these efforts once funding is obtained.

People

Faculty

  • Andrew Belmonte,  Associate Professor, Dept of Mathematics, Penn State
  • Shuwang Li,  Assistant Professor, Dept of Applied Mathematics, Illinois Institute of Technology
  • John Lowengrub,  Professor, Dept of Mathematics, UC Irvine

Graduate students

Former members involved
  • Andong He, PhD student, Dept of Mathematics, Penn State
  • Peng Song,  Postdoctoral Associate, Dept of Mathematics, UC Irvine

Publications

  • A. He and A. Belmonte, ``Inertial effects on viscous fingering in the complex plane."  Journal of Fluid Mechanics 668, 436-445 (2011)  pdf

  • S. Li, J. S. Lowengrub, J. Fontana, and P. Palffy-Muhoray, ``Control of Viscous Fingering Patterns in a Radial Hele-Shaw Cell."  Physical Review Letters 102, 174501 (2009)  pdf

  • T. Podgorski, M. C. Sostarecz, S. Zorman, and A. Belmonte, ``Fingering instabilities of a reactive micellar interface''  Physical Review E 76, 016202[1--6] (2007)  pdf

  • T. Grumstrup and A. Belmonte, ``Elastic splash of two Newtonian liquids"  Physics of Fluids 19, 091109 (2007)  pdf   (Winning entry,  * 2007 Gallery of Fluid Motion * )

  • S. Li, J. S. Lowengrub, P. Leo, ``A rescaling scheme with application to the long time simulation of viscous fingering in a Hele-Shaw cell."  J. Comput. Phys. 225, 554-567 (2007)  pdf


Dept of Applied Mathematics             Dept of Mathematics             Dept of Mathematics