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ACMS Abstracts: Fall 2012

David Saintillan (U. Illinois)

Living fluids: modeling and simulation of biologically active suspensions

Active particle suspensions, of which a bath of swimming bacteria is a paradigmatic example, are characterized by complex dynamics involving strong fluctuations and large-scale correlated motions. These motions, which result from the many-body interactions between particles, are biologically relevant as they impact mean particle transport, mixing and diffusion, with possible consequences for nutrient uptake and the spreading of bacterial infections. To analyze these effects, a kinetic theory is presented and applied to elucidate the dynamics and pattern formation arising from mean-field interactions. Based on this model, the stability of both aligned and isotropic suspensions is investigated. In isotropic suspensions, a new instability for the active particle stress is found to exist, in which shear stresses are eigenmodes and grow exponentially at low wavenumbers, resulting in large-scale fluctuations in suspensions of rear-actuated swimmers, or pushers, when the product of the linear system size with the suspension volume fraction exceeds a given threshold; no such instability is predicted for head-actuated swimmers, or pullers. The predictions from the kinetic model are also tested using direct numerical simulations based on a slender-body model for hydrodynamically interacting self-propelled particles. These simulations confirm the existence of a transition to large-scale correlated motions in suspensions of pushers above a critical volume fraction and system size, which is seen most clearly in particle velocity and passive tracer statistics. Extensions of this work to model chemotactic interactions with an external oxygen field as well as steric interactions in concentrated suspensions are also discussed.

Sebastien Roch (UW)

Assembling the tree of life: theory beyond the substitution-only model of sequence evolution

Recent advances in DNA sequencing technology have led to new mathematical challenges in the analysis of the massive datasets produced in current evolutionary studies. In particular, much progress has been made in the design and analysis of computationally efficient algorithms for assembling the Tree of Life from present-day molecular sequences. In the first half of the talk, I will briefly review some of the mathematical techniques that have led to our current understanding of large-scale tree-building algorithms. Prior theoretical results, however, often rely on statistical models of molecular evolution that are too simplistic. In the second half, I will discuss recent work on the probabilistic modeling and analysis of more complex settings, including insertion-deletion events and lateral genetic transfer. No biology background will be assumed.

Shane Keating (NYU)

Models and measures of turbulent mixing in the ocean

Ocean eddies play a critical role in an wide range of natural processes, from plankton dynamics to climate change. This reinforces the need for a detailed understanding of eddies and their role in transporting heat, carbon, and nutrients throughout the world's oceans. The challenges are significant, however: ocean turbulence is difficult to observe, and numerical models must parameterize subgrid transport, a notoriously difficult problem in inhomogeneous, anisotropic flows dominated by coherent structures such as jets and vortices.

In this talk, I will describe some mathematical approaches to modeling and measuring turbulent mixing in the ocean. First I will outline attempts to quantify uncertainty in satellite estimates of ocean mixing. Next I will describe inexpensive new data assimilation methods for estimating ocean transport that exploit the effect of aliasing to derive "superresolved" velocity fields with a nominal resolution increase of double or more. Finally, I will discuss efforts to develop parameterization schemes for ocean mixing for use in numerical ocean models. These include rigorous approaches based on homogenization theory, as well as adaptive stochastic schemes that efficiently parameterize unresolved scales with a model that can be learned adaptively from observations.

Ellen Zweibel (UW)

The fluid dynamics of stellar interiors

In the simplest approximation, stars are just gravitationally bound balls of hot gas that remain in equilibrium for eons. Going deeper, rotation, thermal convection, and large scale circulation in stellar interiors are vital in stellar evolution. These flows destroy and regenerate magnetic fields, mix chemical elements, and affect the coupling stars to their environments via winds. In principle, the most accurate equations for modeling the dynamics of stellar interiors are the Navier-Stoke equations in the magnetohydrodynamic approximation. However, when one allows for compressibility effects, there is a vast disparity of timescales in the problem: acoustic waves propagate throughout the Sun in about 20 minutes, but the solar rotation period is about a month, and the solar cycle period is 22 years. This problem has been addressed by introducing approximations to the fluid equations that suppress the sound waves. It has long been known that these approximations wreak havoc on energy conservation, but the stellar astronomy community has not systematically studied the consequences of this. I will present recent and ongoing work which analyzes the physical inconsistencies and spurious features of these "soundproof" treatments and proposes some ways of resolving them.

Cyrill Muratov (NJIT)

Gamma-convergence for pattern forming systems with competing interactions

I will discuss a problem of energy-driven pattern formation, in which the appearance of two distinct phases caused by short-range attractive forces is frustrated by a long-range repulsive force. I will focus on the regime of strong compositional asymmetry, in which one of the phases has very small volume fraction, thus creating small ?droplets? of the minority phase in a ?sea? of the majority phase. I will present a setting for the study of Gamma-convergence of the governing energy functional in the regime leading to many droplets. The Gamma-limit and the properties of almost minimizers with prescribed limit density will then be established in the important physical case when the long-range repulsive force is Coulombic in two space dimensions. This is joint work with D. Goldman and S. Serfaty.

Gautam Iyer (CMU)

Time discrete approximations to the Navier–Stokes equations: Existence, stability and coercivity

The numerical study of the incompressible Navier-Stokes equations in the presence of spatial boundaries is faced with many challenging issues. To address some of these issues, pressure-Poisson schemes replace the incompressibility constraint with an explicit Poisson equation for the pressure. I will talk about the formal time-continuous limit arising from a recent scheme of Jonston and Liu. This gives a system that ``extendends the dynamics of incompressible fluids to compressible vector fields.

Surprisingly(!), in this extended situation, the linear terms in these equations fail to be coercive, and lead to many interesting issues. I will mainly talk about existence, regularity and well-posedness results, and also briefly address stability of an associated time discrete scheme.

Thomas Yu (Drexel)

Subdivision methods in scientific computing'

Subdivision method is a multiscale method for generating smooth functions from coarse samples. With an interesting twist, it can be used to generate surfaces of arbitrary topology. This methodology and the underlying theory are well-known in the geometric modeling community as the method enjoys box office success in Disney/Pixar animations -- but much less so in the scientific computing community.

In this talk, we demonstrate a scientific application of free-form subdivision surfaces, namely in the computation of geometric biomembranes governed by the minimization of bending elasticity (essentially the Willmore energy) subject to certain geometic constraints (Helfrich model). Since a subdivision method guarantees that the generated surface has a well-defined Willmore energy (as opposed to non-smooth piecewise linear or quadratic surfaces typically used in finite element methods), our method appears to be much simpler and more robust when compared to other existing finite-element methods. A technical contribution in our work is an algorithm for computing the gradient of the Willmore energy with respect to the control data of the subdivision surface.

We show computational results reproducing the shape of a red blood cell and various lipid vesicles of different topologies. (No background in subdivision methods will be assumed. Joint work with Jingmin Chen, Sara Grundel and Andrew Zigerelli.)

Andrew Majda (NYU)

Data driven methods for complex turbulent systems

An important contemporary research topic is the development of physics constrained data driven methods for complex, large-dimensional turbulent systems such as the equations for climate change science. Three new approaches to various aspects of this topic are emphasized here. 1) The systematic development of physics constrained quadratic regression models with memory for low-frequency components of complex systems; 2) Novel dynamic stochastic superresolution algorithms for real time filtering of turbulent systems; 3) New nonlinear Laplacian spectral analysis (NLSA) for large-dimensional time series which capture both intermittency and low-frequency variability unlike conventional EOF or principal component analysis. This is joint work with John Harlim (1,2), Michal Branicki (2), and Dimitri Giannakis (3).

Douglas Weibel (UW)

Insight into the mechanism(s) of Proteus mirabilis community structure

Proteus mirabilis is an opportunistic pathogen that is frequently associated with urinary tract infections. Cells of P. mirabilis move on a variety of different surfaces (including catheters) and produce a characteristic community structure over length scales that can approach tens of centimeters. Although reaction diffusion models can recapitulate patterns that are similar to swarming colonies of P. mirabilis, we currently lack insight into biochemical and biophysical mechanisms that coordinate cell behavior and control the organization of cells in communities.

The onset of swarming involves the differentiation of planktonic cells of P. mirabilis into swarmer cells, which is accompanied by dramatic changes in cell length and the density of flagella. We have quantified these changes among cells at different locations in the population and have found large differences in cell motility in viscous fluids. Cells with a high density of flagella, such as swarmers are capable of translating through fluids with a viscosity approaching 10 Pa s (~10,000 times the viscosity of water). These cells actively colonize surfaces during swarming. Periodically these cells de-differentiate into consolidated cells that have a characteristically low density of flagella, and are unable to move through viscous fluids. These cells actively replicate during swarming and increase the population. The differential motility of these two cell types in the viscous swarming fluid is capable of producing the characteristic spatial patterns of P. mirabilis swarms. In this presentation we describe data that supports a model for community structure and may explain the underlying stratification of cells in different multicellular structures, including swarms and biofilms that have implications in human health.

Alexander Kurganov (Tulane)

Central-Upwind Schemes for Shallow Water Models

I will first give a brief review on simple and robust central-upwind schemes for hyperbolic systems of conservation laws. I will then discuss their application to the Saint-Venant system of shallow water equations. This can be done in a straightforward manner, but then the resulting scheme may suffer from the lack of balance between the fluxes and (possibly singular) geometric source term, which may lead to a so-called numerical storm, and from appearance of negative values of the water height, which may destroy the entire computed solution. To circumvent these difficulties, we have developed a special technique, which guarantees that the designed second-order central-upwind scheme is both well-balanced and positivity preserving.

Finally, I will show how the scheme can be extended to the Ripa system, which is a modification of the Saint-Venant system that takes into account temperature variations, which effect the pressure term. In the Ripa system, the temperature is transported by the fluid, and if one of the velocity components vanishes, the characteristic speed becomes zero and the system exhibits a ?nonlinear resonance? in the sense that wave speeds from different families of waves coincide. Moreover, there are no Riemann invariants for this system and therefore it is very hard to design upwind schemes for the Ripa system, since they are based on (approximate) Riemann problem solvers.

In general, designing a well-balanced scheme for the Ripa system is a highly nontrivial task since steady states at rest are characterized by zero velocity and the differential (not integrable!) form. The developed central-upwind scheme is capable of exactly preserving two special types of steady states at rest. To preserve steady states of the first type, we have implemented the same technique as for the original Saint-Venant system. However, steady states of the second type are of different nature since they correspond to steady contact waves with the pressure remaining constant across them. These waves are similar to the contact waves appearing in compressible multi-fluids. As in the multi-fluid case, a good scheme must be able to preserve constant pressure and velocity at contact waves to avoid appearance of spurious pressure and velocity oscillations. To achieve this goal, we have extended the interface tracking method, which we previously developed for compressible multi-fluids, to the Ripa system. We demonstrate the performance of the new scheme on a number of one- and two-dimensional examples.