# Applied/ACMS/absS17

## Contents

### Chung-Nan Tzou (UW)

*Optimal mixing of buoyant jets and plumes in stratified fluids: theory and experiments*

We present results from an experimental and theoretical study of the influence of ambient fluid stratification on buoyant miscible jets and plumes. Given a fixed set of jet/plume parameters, and an ambient fluid stratification sandwiched between top and bottom homogenous densities, a theoretical criterion is identified showing how step-like density profiles constitute the most effective mixers within a broad class of stable density transitions. This is assessed both analytically and experimentally, respectively by establishing rigorous a priori estimates on generalized Morton-Taylor-Turner (MTT) models, and by studying a critical phenomenon determined by the distance between the jet/plume release heights with respect to the depth of the ambient density transition. For fluid released sufficiently close to the background density transition, the buoyant jet fluid escapes and rises indefinitely. For fluid released at locations lower than a critical depth, the buoyant fluid stops rising and is trapped indefinitely. We develop a mathematical formulation providing rigorous estimates on MTT models, by establishing nonlinear jump conditions and an exact critical-depth formula in good quantitative agreement with the experiments. Our mathematical analysis provides rigorous justification for the critical trapping/escaping criteria, first presented in Caulfied and Woods (1998), within a class of algebraic density decay rates. Further, the analysis uncovers surprising differences between the Gaussian and Top-hat profile turbulent entrainment closures concerning initial mixing of the jet and ambient fluid. Laboratory experimental results and comparisons with the theory will be discussed.

### Molei Tao (GaTech)

*Numerical methods for identifying hyperbolic periodic orbits and characterizing rare events in nongradient systems*

We consider differential equations perturbed by small noises. The goal is to quantify what noises can do and possibly also utilize them. More specifically, noise-induced dynamics are understood by maximizing transition probability characterized by Freidlin-Wentzell large deviation theory. In gradient systems (i.e., reversible thermodynamics), metastable transitions were known to cross separatrices at saddle points. We investigate nongradient systems (which may no longer be reversible), and show a very different type of transitions that cross hyperbolic periodic orbits. Numerical tools for both identifying such periodic orbits and computing transition paths are described. If time permits, I will also discuss how these results may help design control strategies.

### Benoit Perthame (University of Paris VI)

*Models for neural networks; analysis, simulations and behaviour*

Neurons exchange informations via discharges, propagated by membrane potential, which trigger firing of the many connected neurons. How to describe large networks of such neurons? What are the properties of these mean-field equations? How can such a network generate a spontaneous activity? Such questions can be tackled using nonlinear integro-differential equations. These are now classically used in the neuroscience community to describe neuronal networks or neural assemblies. Among them, the best known is certainly Wilson-Cowan's equation which describe spiking rates arising in different brain locations.

Another classical model is the integrate-and-fire equation that describes neurons through their voltage using a particular type of Fokker-Planck equations. Several mathematical results will be presented concerning existence, blow-up, convergence to steady state, for the excitatory and inhibitory neurons, with or without refractory states. Conditions for the transition to spontaneous activity (periodic solutions) will be discussed.

One can also describe directly the spike time distribution which seems to encode more directly the neuronal information. This leads to a structured population equation that describes at time $t$ the probability to find a neuron with time $s$ elapsed since its last discharge. Here, we can show that small or large connectivity leads to desynchronization. For intermediate regimes, sustained periodic activity occurs. A common mathematical tool is the use of the relative entropy method.

This talk is based on works with K. Pakdaman and D. Salort, M. Caceres, J. A. Carrillo and D. Smets.

### Jeffrey Guasto (Tufts)

*Two problems in porous media flows: From swimming cells to complex fluids*

Fluid and particulate transport in porous environments regulates processes ranging from remediation in soils to the spread of infection in human tissues. In this talk, we will address two important aspects of porous media flows using microfluidic devices to precisely prescribe the microstructure and flow within a model porous medium. First, we will examine the physical mechanisms underlying the transport of swimming cells in porous fluid environments. We show that such confined flows generate significant heterogeneity in the spatial distribution of motile bacteria and significantly modify the transport coefficients of active cells. As a consequence, the chemotactic ability of cells is suppressed, while surface attachment is enhanced. Second, we will briefly discuss recent measurements on the transport of yield stress fluid flow in random porous media, where we demonstrate that surface interactions play a significant role in the development of flow topology.

### Roger Temam (Indiana)

*On the mathematical modeling of the humid atmosphere*

The humid atmosphere is a multi-phase system, made of air, water vapor, cloud-condensate, and rain water (and possibly ice / snow, aerosols and other components). The possible changes of phase due to evaporation and condensation make the equations nonlinear, non-continuous (and non-monotone) in the framework of nonlinear partial differential equations. We will discuss some modeling aspects, and some issues of existence, uniqueness and regularity for the solutions of the considered problems, making use of convex analysis, variational inequalities, and quasi-variational inequalities.

### Christian Klingenberg (Wuerzburg)

*The compressible Euler equations with gravity: well-balanced schemes and all Mach number solvers*

We consider astrophysical systems that are modeled by the multidimensional Euler equations with gravity.

First for the homogeneous Euler equations we look at flow in the low Mach number regime. Here for conventional finite volume discretizations one has excessive dissipation in this regime. We identify inconsistent scaling for low Mach numbers of the numerical fux function as the origin of this problem. Based on the Roe solver a technique that allows to correctly represent low Mach number flows with a discretization of the compressible Euler equations is proposed. We analyze properties of this scheme and demonstrate that its limit yields a discretization of the incompressible limit system.

Next for the Euler equations with gravity we seek well-balanced methods. We describe a numerical discretization of the compressible Euler equations with a gravitational potential. A pertinent feature of the solutions to these inhomogeneous equations is the special case of stationary solutions with zero velocity, described by a nonlinear PDE, whose solutions are called hydrostatic equilibria. We present well-balanced methods, for which we can ensure robustness, accuracy and stability, since it satisfies discrete entropy inequalities.

We will then present work in progress where we combine the two methods above.

### Seung-Yeal Ha (Seoul National Univ)

*Emergent dynamics of classical and quantum oscillators*

Synchronization of weakly coupled oscillators is ubiquitous in biological, chemical and physical complex systems. Recently, research on collective dynamics of many-body systems has been received much attention due to their possible applications in engineering. In this survey talk, we mainly focus on the large-time dynamics of several synchronization models and review state-of-art results on the collective behaviors for synchronization models. Following a chronological order, we begin our discussion with two classical phase models (Winfree and Kuramoto models), and two quantum synchronization models (Lohe and Schrodinger-Lohe models). For these models, we present several sufficient conditions for the emergence of synchronization using mathematical tools from dynamical systems theory, kinetic theory and partial differential equations in a unified framework.

### Lili Ju (South Carolina)

*A conservative nonlocal convection-diffusion model and asymptotically compatible finite difference discretization*

In this talk, we first propose a nonlocal convection-diffusion model, in which the convection term is constructed in a special upwind manner so that mass conservation and maximum principle are maintained. The well-posedness of the proposed nonlocal model and its convergence to the classical local convection-diffusion model are established. A quadrature-based finite difference discretization is then developed to numerically solve the nonlocal problem and it is shown to be consistent and unconditionally stable. We also demonstrate that the numerical scheme is asymptotically compatible, that is, the approximate solutions converge to the exact solution of the corresponding local problem when the mesh size and the horizon parameter decrease zero. Numerical experiments are performed to complement the theoretical analysis.