Among the many instabilities observed in fluid dynamics, is it possible that there are some driven by material complexity occurring only at interfaces? The answer is Yes! For instance, when two reacting micellar liquids are brought into contact, the reaction may produce a growing gel-like phase at the interface, which significantly stiffens the boundary between the fluids. Another example arises from the presence of nanoscale colloidal particles which accumulate at interfaces - intercolloid forces also endow the interface with stiffness, and may even jam the interface at sufficiently high volume fraction. Interfacial instabilities occur when driving forces compete with resistive forces - a consequence may be the formation of complex patterns. Examples beyond our studies are seen in diverse systems such as filamentary microorganisms, growing biofilms, smoldering flame fronts, and lava flows.
Viscous fingering, or the Saffman-Taylor instability, has become one of the canonical systems in the study of pattern formation. Mathematically 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 leads to finger patterns.
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.

This work is supported in part by the National Science Foundation (NSF Award DMS-1217177).


  • Andrew Belmonte,  Professor, Dept of Mathematics / Materials Science and Engineering, Pennsylvania State University
  • Shuwang Li,  Associate Professor, Dept of Applied Mathematics, Illinois Institute of Technology
  • John Lowengrub,  Professor, Dept of Mathematics, University of California Irvine

  • Zahra Niroobakhsh,  Graduate Student, Dept of Materials Science and Engineering, Pennsylvania State University


  • Modeling an elastic fingering instability in a reactive Hele-Shaw flow.
    A. He, J. S. Lowengrub, and A. Belmonte
    SIAM J Applied Math 72 , 842-856 (2012)  pdf

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

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

  • Fingering instabilities of a reactive micellar interface.
    T. Podgorski, M. C. Sostarecz, S. Zorman, and A. Belmonte
    Physical Review E 76, 016202[1--6] (2007)  pdf

  • Elastic splash of two Newtonian liquids.
    T. Grumstrup and A. Belmonte
    Physics of Fluids 19, 091109 (2007)  pdf

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