Obtaining Self-similar Scalings in Focusing Flows

The surface structure of converging thin fluid films displays self-similar behavior, as was shown in the work by Diez et al [Q. Appl. Math 210, 155, 1990]. Extracting the related similarity scaling exponents from either numerical or experimental data is non-trivial. Here we provide two such methods. We apply them to experimental and numerical data on converging fluid films driven by both surface tension and gravitational forcing. In the limit of pure gravitational driving, we recover Diez' semi-analytic result, but our methods also allow us to explore the entire regime of mixed capillary and gravitational driving, up to entirely surface tension driven flows. We find scaling forms of smoothly varying exponents up to surprisingly small Bond numbers. Our experimental results are in reasonable agreement with our numerical simulations, which confirm theoretically obtained relations between the scaling exponents.Comment: 11 pages, 11 figures, accepted for Phys Rev

Examination of optimizing information flow in networks

The central role of the Internet and the World-Wide-Web in global communications has refocused much attention on problems involving optimizing information flow through networks. The most basic formulation of the question is called the "max flow" optimization problem: given a set of channels with prescribed capacities that connect a set of nodes in a network, how should the materials or information be distributed among the various routes to maximize the total flow rate from the source to the destination. Theory in linear programming has been well developed to solve the classic max flow problem. Modern contexts have demanded the examination of more complicated variations of the max flow problem to take new factors or constraints into consideration; these changes lead to more difficult problems where linear programming is insufficient. 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

Maximizing Minimum Pressure in Fluid Dynamic Bearings of Hard Disk Drives

We focus on the central spindle which supports the rotating magnetic platters which hold all of the data. The spindle must operate with great precision and stability at high rotational speeds. Design practice has converged on oil-lubricated hydrodynamic journal bearings as the most common choice for spindles. That is, a layer of viscous oil separates a rotating shaft (the bearing) from the fixed outer sleeve (the journal). In hard drives, it is very important for the shaft to be centered within the sleeve. Plain journal bearings (i.e. both surfaces are circular cylinders) are unstable to perturbations that push the shaft off-center. It was found that this stability problem can be overcome by cutting diagonal grooves into the journal in a pattern called a herring-bone. Another consequence of this design is that very high pressures are generated by the grooves as they drive the oil to the middle of the bearing, away from the top/bottom ends of the spindle. This pumping action generally works to oppose leakage out of the bearing. We examine how choices for the groove pattern can influence the key properties of the bearing. The focus is to understand the effect of the groove geometry on the pumping action. In particular the undesirable behavior caused by the low pressures created near the top/bottom ends of the bearing which, under many conditions, may result in the pressure becoming negative, relative to atmospheric pressure. Negative pressure can result in cavitation or, when it occurs near an air-oil interface, can cause air to be ingested and hence create bubbles. Any bubbles in the oil can corrupt the lubricating layer in the bearing and, as they are created and collapse, can cause significant undesirable vibrations. The negative pressures have therefore been identified as one of the key problems in design of hard disk drive bearings. We will use numerical computations and some analysis to show that by modifying the groove geometry we can reduce the negative pressure while retaining good stability characteristics

Flow and fouling in a pleated membrane filter

Pleated membrane filters are widely used in many applications, and offer significantly better surface area to volume ratios than equal area unpleated membrane filters. However, their filtration characteristics are markedly inferior to those of equivalent unpleated membrane filters in dead-end filtration. While several hypotheses have been advanced for this poor performance, one possibility is that the flow field induced by the pleating leads to spatially nonuniform fouling of the filter, which in turn affects performance. In this paper we investigate this hypothesis by developing a simplified model for the flow and fouling within a pleated membrane filter. Our model accounts for the pleated membrane geometry (which affects the flow), for porous support layers surrounding the membrane, and for two membrane fouling mechanisms: (i) adsorption of very small particles within membrane pores; and (ii) blocking of entire pores by large particles. We use asymptotic techniques based on the small pleat aspect ratio to solve the model, and we compare solutions to those for the closest-equivalent unpleated filter