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Saverio Spagnolie, Brown

Hydrodynamics of Self-Propulsion Near a Boundary: Construction of a Numerical and Asymptotic Toolbox

The swimming kinematics and trajectories of many microorganisms are altered by the presence of nearby boundaries, be they solid or deformable, and often in perplexing fashion. When an organism's swimming dynamics vary near such boundaries a question arises naturally: is the change in behavior fluid mechanical, biological, or perhaps due to other physical laws? We isolate the first possibility by exploring a far-field description of swimming organisms, providing a general framework for studying the fluid-mediated modifications to swimming trajectories. Using the simplified model we consider trapped/escape trajectories and equilibria for model organisms of varying shape and propulsive activity. This framework may help to explain surprising behaviors observed in the swimming of many microorganisms and synthetic micro-swimmers. Along the way, we will discuss the numerical tools constructed to analyze the problem of current interest, but which have considerable potential for more general applicability.

Ari Stern, UC San Diego

Numerical analysis beyond Flatland: semilinear PDEs and problems on manifolds


Shamgar Gurevich, UW-Madison

Channel Estimation in Wireless Communication in Linear Time

We will present the model of mobile communication, and will discuss the problem of channel estimation -- finding time-frequency shifts which a waveform undergoes while transmitted in the presence of a white noise. The digital model of the problem involves signals of length N (complex-valued vectors of length N). The current method of solving digital channel estimation problem uses O(N^2 log(N)) arithmetic operations. Using ideas from representation theory, we will present a new method of solving channel estimation problem of complexity O(N log(N)). The applications of the new method to mobile communication and GPS system will be discussed.

This is a joint work with A. Fish (Math, UW-Madison), R. Hadani (Math, UT-Austin), A. Sayeed (ECE, UW-Madison), O. Schwartz (EECS, UC Berkeley).

Mike Cullen, Met. Office, UK

Applications of optimal transport to geophysical problems

The optimal transport method can be applied to a number of important problems in geophysical fluid dynamics, including large-scale flows in the atmosphere and ocean, equatorial waves, and the one-dimensional convective adjustment problem. In this talk I go through the basic procedures. The problem to be solved has to conserve mass and energy. The mass distribution is then regarded as a probability measure, and a metric on the space of probability measures defined (usually the Wasserstein distance). This is used via the 'Otto calculus' to define derivatives of the energy with respect to changes in the mass distribution. In the problems listed above, the governing equations can be solved by finding energy minimisers in this sense. Optimal transport theory can then be used to prove existence of minimisers. If the flow evolution can now be expressed as transport of mass by a rotated gradient, then the general theory of Ambrosio and Gangbo proves that solutions exist for all times.

Ricardo Cortez, Tulane

Introduction to the method of regularized Stokeslets for fluid flow and applications to microorganism swimming

Biological flows, such as those surrounding swimming microorganisms, can be properly modeled using the Stokes equations for fluid motion with external forcing. The organism surfaces can be viewed as flexible interfaces imparting force or torque on the fluid. Interesting flows have been observed when the organism swims near a solid wall due to the hydrodynamic interaction of rotating flagella with a neighboring solid surface. I will introduce the method of regularized Stokeslets and some extensions of it that are used to compute these flows. The method is based on fundamental solutions of linear PDEs, leading to integral representations of the solution. I will present the idea of the method, some of the known results and applications to flows generated by swimming flagella.

Meyer Jackson, UW-Madison Neuroscience

Membrane elasticity and mechanics during fusion


Michael Graham, UW-Madison Engineering

Drag reduction and the nonlinear dynamics of turbulence in simple and

complex fluids

At low speed, flow in a pipe or over an aircraft is smooth and steady. At higher speeds, it becomes turbulent -- the smooth motion gives way to fluctuating eddies that sap the fluid's energy and make it more difficult to pump the fluid through the tube or to propel the aircraft through the air. For flowing liquids, adding a small amount of very large polymer molecules or micelle-forming surfactants can dramatically affect the turbulent eddies, reducing their deleterious effects on energy efficiency. This phenomenon is widely used, for example in the Alaska pipeline, but it is not well-understood, and no comparable technology exists to reduce turbulent energy consumption in flows of gases, in which polymers or surfactants cannot be dissolved. The most striking feature of this phenomenon is the existence of a so-called maximum drag reduction (MDR) asymptote: for a given geometry and driving force, there is a maximum level of drag reduction that can be achieved through addition of polymers. Changing the concentration, molecular weight or even the chemical structure of the additives has no effect on this asymptotic value. This universality is the major puzzle of drag reduction.

We describe direct numerical simulations of turbulent channel flow of Newtonian fluids and viscoelastic polymer solutions. Even in the absence of polymers, we show that there are intervals of ?hibernating? turbulence that display very low drag as well as many other features of the MDR asymptote observed in polymer solutions. As viscoelasticity increases, the frequency of these intervals also increases, while the intervals themselves are unchanged, leading to flows that increasingly resemble MDR. A simple theory captures key features of the intermittent dynamics observed in the simulations. Additionally, simulations of ?edge states?, dynamical trajectories that lie on the boundary between turbulent and laminar flow, display characteristics that are similar to those of hibernating turbulence and thus to the MDR asymptote, again even in the absence of polymer additives. Based on these observations, we propose a tentative unified description of rheological drag reduction. The existence of ?MDR-like? intervals even in the absence of additives sheds light on the observed universality of MDR and may ultimately lead to new flow control approaches for improving energy efficiency in a wide range of processes.

Lukas Szpruch, Oxford

Antithetic multilevel Monte Carlo method

We introduce a new multilevel Monte Carlo (MLMC) estimator for multidimensional SDEs driven by Brownian motion. Giles has previously shown that if we combine a numerical approximation with strong order of convergence $O(\D t)$ with MLMC we can reduce the computational complexity to estimate expected values of functionals of SDE solutions with a root-mean-square error of $\eps$ from $O(\eps^{-3})$ to $O(\eps^{-2})$. However, in general, to obtain a rate of strong convergence higher than $O(\D t^{1/2})$ requires simulation, or approximation, of \Levy areas. Through the construction of a suitable antithetic multilevel correction estimator, we are able to avoid the simulation of \Levy areas and still achieve an $O(\D t2)$ variance for smooth payoffs, and almost an $O(\D t^{3/2})$ variance for piecewise smooth payoffs, even though there is only $O(\D t^{1/2})$ strong convergence. This results in an $O(\eps^{-2})$ complexity for estimating the value of European and Asian put and call options. We also comment on the extension of the antithetic approach to pricing Asian and barrier options.

Dwight Barkley, Warwick

The onset of turbulence in pipe flow

More than a century ago Osborne Reynolds launched the quantitative study of turbulent transition as he sought to understand the conditions under which fluid flowing through a pipe would be laminar or turbulent. Since laminar and turbulent flow have vastly different drag laws, this question is as important now as it was in Reynolds' day. Reynolds understood how one should define "the real critical value" for the fluid velocity beyond which turbulence can persist indefinitely. He also appreciated the difficulty in obtaining this value. For years this critical Reynolds number, as we now call it, has been the subject of study, controversy, and uncertainty. I will discuss recent developments in experiments, simulations, and modeling that show a deep connection both to statistical phase transitions (directed percolation) and to the dynamics of action potentials in a nerve axons. From these insights, we at last have an accurate estimate of the real critical Reynolds number for the onset of turbulence in pipe flow, and with it, an understanding of the nature of transitional turbulence.

This work is joint with: K. Avila, D. Moxey, M. Avila, A. de Lozar, and B. Hof.

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