# Applied/ACMS/absF12

## Contents

# 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.