[The slurry may be thought of as a mixture of clay (or other polymers), water, and other binders. There are many ways to model this mixture, some of which will be discussed in this report. For instance, the slurry can be modeled as a non-Newtonian fluid, a two-phase flow with liquid and solids, or a viscoelastic fluid. One can also model the mixture as an elongated particle suspension in water, where changes in the orientation of the particles could affect the flow.]

After the extrusion process is complete, one finds that ‘dead zones’ of dry paste accumulate in two areas. Most prominently, they occur at the lower corners of the tank. They also occur on the floor of the extrusion chamber near the slots, both near and away from the walls. Since we will consider wall effects in the tank, for the extrusion chamber we consider only flow cells sufficiently far away from the walls. Then we may exploit the periodic nature of the device and consider only a single flow cell.

The aim of this project is to determine the formation mechanisms of these dead zones, and see how they affect the overall flow.

For the case of longitudinal roughness, we derived a one-dimensional lubrication-type equation for the leading behavior of the pressure in the direction parallel to the velocity of the disk. The coefficients of the equation are determined by solving linear elliptic equations on a domain bounded by the gap height in the vertical direction and the period of the roughness in the span-wise direction.

For the case of transverse roughness the unsteady lubrication equations were reduced, following a multiple scale homogenization analysis, to a steady equation for the leading behavior of the pressure in the gap. The reduced equation involves certain averages of the gap height, but retains the same form of the usual steady, compressible lubrication equations.

Numerical calculations were performed for both cases, and the solution for the case of transverse roughness was shown be in excellent agreement with a corresponding numerical calculation of the original unsteady equations.

In the workshop we examined models for information flow on networks that considered trade-offs between the overall network utility (or flow rate) and path diversity to ensure balanced usage of all parts of the network (and to ensure stability and robustness against local disruptions in parts of the network).

While the linear programming solution of the basic max flow problem cannot handle the current problem, the approaches primal/dual formulation for describing the constrained optimization problem can be applied to the current generation of problems, called network utility maximization (NUM) problems. In particular, primal/dual formulations have been used extensively in studies of such networks.

A key feature of the traffic-routing model we are considering is its formulation as an economic system, governed by principles of supply and demand. Considering channel capacities as a commodity of limited supply, we might suspect that a system that regulates traffic via a pricing scheme would assign prices to channels in a manner inversely proportional to their respective capacities.

Once an appropriate network optimization problem has been formulated, it remains to solve the optimization problem; this will need to be done numerically, but the process can greatly benefit from simplifications and reductions that follow from analysis of the problem. Ideally the form of the numerical solution scheme can give insight on the design of a distributed algorithm for a Transmission Control Protocol (TCP) that can be directly implemented on the network.

At the workshop we considered the optimization problems for two small prototype network topologies: the two-link network and the diamond network. These examples are small enough to be tractable during the workshop, but retain some of the key features relevant to larger networks (competing routes with different capacities from the source to the destination, and routes with overlapping channels, respectively). We have studied a gradient descent method for solving obtaining the optimal solution via the dual problem. The numerical method was implemented in MATLAB and further analysis of the dual problem and properties of the gradient method were carried out. Another thrust of the group's work was in direct simulations of information flow in these small networks via Monte Carlo simulations as a means of directly testing the efficiencies of various allocation strategies.

In this study group, we proposed two mathematical models to describe plaque growth and rupture.

The first model is a mechanical one that approximately treats the plaque as an inflating elastic balloon. In this model, the pressure inside the core increases and then decreases suggesting that plaque stabilization and prevention of rupture is possible.

The second model is a biochemical one that focuses on the role of MMPs in degrading the fibrous plaque cap. The cap stress, MMP concentration, plaque volume and cap thickness are coupled together in a system of phenomenological equations. The equations always predict an eventual rupture since the volume, stresses and MMP concentrations generally grow without bound. The main weakness of the model is that many of the important parameters that control the behavior of the plaque are unknown.

The two simple models suggested by this group could serve as a springboard for more realistic theoretical studies. But most importantly, we hope they will motivate more experimental work to quantify some of the important mechanical and biochemical properties of vulnerable plaques.