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
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
Dept of Mathematics / Materials Science and Engineering,
Pennsylvania State University
Dept of Applied Mathematics,
Illinois Institute of Technology
Dept of Mathematics,
University of California Irvine
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)
Inertial effects on viscous fingering in the complex plane.
A. He and A. Belmonte
Journal of Fluid Mechanics 668, 436-445 (2011)
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)
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)
Elastic splash of two Newtonian liquids.
T. Grumstrup and A. Belmonte
Physics of Fluids 19, 091109 (2007)
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)