This project focuses on the dynamics of thin layers of fluid, and on avalanche flow of granular materials.

Thin Liquid Films

The flow of thin layers is modeled using the lubrication approximation, leading to a PDE for the height of the free surface. Various forces can be incorporated into the model, such surface tension, Marangoni forces (generated by surface tension gradients, for example due to temperature variations), van der Waals forces, and force due to gravity.

In recent years, we have been exploring the effect of surfactant on the shape of the free surface. Surfactant modifies the surface tension in a complicated way, since motion of the free surface also affects the distribution of the surfactant. Consequently, in place of a single PDE, the model includes an additional equation to capture the distribution of surfactant on the free surface. If the surfactant is soluble, then additional equations are required for the bulk concentration. We have investigated properties of the PDE system for flow down an incline, with insoluble surfactant. Specifically, the system supports shock wave solutions in which the free surface has a jump in height (smoothed by surface tension and other forces), and the surfactant concentration is continuous but has a jump in the first derivative. These solutions occur in combination as a wave with a single speed, and three jumps. This new wave is overcompressive in two senses:

1. Small disturbances ahead of or behind the wave are transported towards the wave;
2. For each choice of upstream and downstream height (the far field conditions), there is a one-parameter family of these waves.
   

Incidentally, the parameter can be the total amount of surfactant in the wave. These waves occur only when the upstream and downstream heights are sufficiently well separated.  In the limit of zero surfactant mass, stability of the waves in one and two dimensions can be analyzed with asymptotics and the Evans function. The stability of the traveling wave with a small amount of surfactant is found to be the same as it would be for the gravity wave with no surfactant.

Another intriguing property of this sytem of PDE is the evolution of a small perturbation of an initially flat free surface when a small amount of surfactant is placed on it. A combination of asymptotics and numerical simulations reveal the subsequent behavior, consisting of a pair of waves traveling at different speeds.

In conjunction with experiments in the Daniels physics lab, Ellen Peterson is designing simulations of flow on a horizontal substrate. The innovation in the experiment is the visualization of surfactant using fluorescence. As a first step, we are establishing the rate of growth of a circular droplet as it spreads across a thin horizontal layer of glycerol, distorting the free surface through Marangoni stress induced by surfactant concentration gradients. The simulations and experiments both appear to support the prediction of self-similar PDE solutions.

Granular Flow

The dynamics of granular materials pose many unresolved scientific issues. For example, there is no widely agreed upon set of constitutive laws or equations of motion that model flow and deformation of granular materials. In this context, it is difficult for mathematicians to make meaningful progress in understanding phenomena associated with granular materials.

At present, we are investigating behavior associated with segregation, in which particles with similar characteristics (shape, size, density, etc.) tend to cluster together, separate from particles with dissimilar characteristics. Models derived by Savage and Lun (1988) and, using a different approach, by Gray and Thornton (2005) describe size segregation in a bidisperse mixture flowing in an avalanche (i.e., down an inclined plane or chute). Interfaces between layers consisting of mostly large particles above layers with mostly small particles are dynamically stable. However, under shear flow, typical of avalanches, the interface may lose stability due to becoming vertical at a point. Subsequently, a mixing zone develops. In an idealized setting, the mixing zone can be analyzed explicitly for short time, but in general, the structure of the evolution is unkown, and is the subject of our analysis and simulation.
Another development is a complete analysis of shock formation under linear shear. This is achieved through analysis of a vector field that captures the evolution of the gradient of the dependent variable, the volume fraction of small particles.

In a related project, we have modified the Gray-Thornton model to apply to shear flow induced by a moving boundary, for which the segregation rate is not uniform across the sample, as it is (approximately) in avalanches. This context is explored experimentally using a Couette cell, in which a mixture of small and large glass beads fill the annular region between two fixed concentric cylinders. Shear is induced by rotating a lower confining plate at a fixed level. The upper confing plate is allowed to move obnly vertically; it's varying height is measured during the experiment. The motion of the particles is filmed using a high speed movie camera, yielding  data on the  position of particles, and hence the velocity distribution needed as data for the model. Since a mixture of large and small particles packs more efficiently than a collection of beads of the same size, the height of the top plate gives a measure of the level of segregation that is compared to predictions of the model through a packing fraction density function. Analysis of the model takes place in two contexts: A: a shear rate depending smoothly and monotonically on depth (specifically, an exponential fit to experimental data), and B: a piecewise constant shear rate, motivated by the notion that the lower part of the sample behaves as a shear band, with a much larger shear rate. In case B, classical theories of existence and uniqueness are not available; more recent analysis suggests there need not be a solution, but in the context of the experiment, the solution can be characterized with a combination of shocks and simple waves. The solution is completed with the numerical solution of an ODE.