*Research Projects*

### I. Microfluidic Flows

A. In recent work, we have studied the flow of a polymeric fluid through the stagnation point of a cross-channel flow, where the polymer molecules are stretched. This results in macroscopic changes in the rheological properties of the fluid, and macroscopic flow instabilities. These instabilities have been characterized by particle image velocimetry.

B. Studies of surface tension phenomena in polymeric fluids are being studied in collaboration with Paulo Arratia and Douglas Durian at University of Pennsylvania. Droplet breakoff phenomena induced by elongation are substantially affected by viscoelasticity.

C. The presence of particles in Newtonian fluids can substantially change their flow properties. We are currently studying the flow of such suspensions in microfluidic channels, in collaboration with Durian's and Yodh's groups at Penn.

### II. Granular Materials

Extensive studies are being made of the *dynamics of granular materials*,
by means of sensitive force measurements and rapid tracking of large numbers
of particles. We are addressing questions of fundamental and practical importance
that have been raised by recent theories, in which adaptive inhomogeneous force
networks are thought to be important in governing the behavior of granular materials.
This work includes the following:

A. We are studying thresholds for motion and the dynamics of horizontally oscillating fluid layers, and are comparing oscillating flows to avalanches in inclined layers. Oscillating flows can reveal aspects of the dynamics that differ from what happens in steady flow.

B. We have studied he internal dynamics of slowly sheared granular material immersed in an index matched fluid, using fluorescent imaging to track the internal particle motion. In this way, we have been able to relate the microscopic dynamics to the macroscopic behavior of the material. We discovered that applied shear can cause a granular packing to crystallize, and we have studied this interesting transititon.

C. We have studied the dynamics of "chiral" (or handed) particles that lack inversion symmetry when they are subjected to vibration. We find that these particles begin to rotate spontaneously and can extract angular momentum from the container.

D. We investigated the dynamical behavior of granular matter that has been simultaneously sheared by rotation and fluidized by a controlled upward air flow. We find that the confinement of the flow to a narrow shear layer can be understood on a hydrodynamic basis as resulting from a strongly density-dependent viscosity.

We compare this granular flow behavior with analogous conventional solid and fluid phenomena, to elucidate the special properties of particulates, and to determine the usefulness of various theoretical paradigms. Some of these experiments have geophysical motivations.

### III. Mixing in Fluids

We are also studying *fluid mixing phenomena* in thin layers.
The process by which an impurity is dispersed in a moving fluid is an
old one that is providing new challenges. It has interesting
applications in oceanography and in industry. Some laminar
time-periodic hydrodynamic flows, even in two dimensions, can advect
material in such a way that nearby fluid elements diverge
exponentially in time. This process of Lagrangian turbulence or
chaotic advection causes an initially inhomogeneous scalar field
(e.g. passive impurity) to acquire complex spatial structure.

In some cases, for example periodic flows, the concepts of nonlinear dynamics provide a deep theoretical basis for understanding mixing. Unfortunately, the building blocks of this theory, i.e. the fixed points and invariant manifolds of the associated Poincaré map, have remained inaccessible to direct experimental study, thus limiting the insight that could be obtained. Using precision measurements of tracer particle trajectories in a two-dimensional fluid flow producing chaotic mixing, we directly measure the time-dependent stretching and compression fields. These quantities, previously available only numerically, attain local maxima along lines coinciding with the stable and unstable manifolds, thus revealing the dynamical structures that control mixing. Contours or level sets of a passive impurity field are found to be aligned parallel to the lines of large compression (unstable manifolds) at each instant. This connection appears to persist as the onset of turbulence is approached.

In recent work, we extended these studies of mixing to the case of elastic fluids containing polymers. We have discovered that the shear-thinning property of certain polymer solutions can substantially affect the mixing process. Finally, we have studied chemically reacting flows, and have demonstrated that the progress of a chemical reaction can be predicted by measuring the ing rate of the mixing process. This work was highlighed in Physics Today in February 2006.

### IV. Shear Flow Phenomena

A. We are studying the instabilities of shear layers in which the flow parallel to one axis varies periodically in an orthoganal direction; these are known as Kolmogorov flows. At a critical point, such flows become unstable to the formation of a triangular array of vortices. We are studying this transition, and also the effect of this instability on thermal transport.

B. In collaboration with David Pine at NYU, we have studied and are continuing to study the time reversibility of viscous sheared suspensions. We showed that such flows become irreversible if sheared beyond a threshold that depends on the concentration of particles. An initial report appeared in Nature in 2005.

### V. Past Projects

A. Dynamic Ordering in Particulates: We have discovered remarkable ordering phenomena in a partial monolayer of particles suspended in a fluid layer that is subjected to an external periodic acceleration, so that the particles oscillate slightly relative to the fluid. An attractive interaction at large particle separations is induced by a mean streaming flow. The attraction leads to clustering of particles into groups, and coarsening on larger scales. However, a repulsive interaction also occurs at large accelerations. The relative importance of attraction and repulsion can be varied by adjusting the frequency and amplitude of the applied acceleration. As a consequence, a variety of ordered and disordered states can be produced, including hexagonally ordered crystallites, clusters that have perfect micro-crystallites within them, and time-dependent patterns showing complex particle dynamics. Quick-time videos showing the dynamics of these remarkable states may be viewed at our lab web site.