# Poster Session

Thursday, July 25, 2013 - 4:30pm - 6:00pm

Lind 400

**A New Phenomenological RANS Model for the Prediction of Transitional Boundary Layers**

Maurin Lopez Varilla (Mississippi State University)

We present a new model concept for pre-diction of boundary layer transition using a linear eddy-viscosity RANS approach. It is a single-point, phys-ics-based method that adopts an alternative to the Laminar Kinetic Energy (LKE) framework. The model is based on a description of the transition process previously discussed by Walters (2009). The version of the model presented here uses the k-omega SST model as the baseline, and includes the effects of transition through one additional transport equation for v2. Here v2 is interpreted as the energy of fully turbulent, 3D velocity fluctuations, while k represents the energy of both fully turbulent and pre-transitional velocity fluctuations. This modeling approach leads to slow growth of fluctuating energy in the pre-transitional region and relaxation towards a fully turbulent model result downstream of transition. Simplicity of the formulation and ease of extension to other baseline models are two potential advantages of the new method. An initial version of the model has been implemented as a UDF subroutine in the commercial CFD code FLUENT and tested for canonical flat plate boundary layer test cases with different freestream turbulence conditions.**Poisson-Nernst-Planck Systems for Ion Flow with a Local Hard-Sphere Potential for Ion Sizes**

In this work, we analyze a one-dimensional steady-state Poisson-Nernst-Planck type model for ionic flow through a membrane channel with fixed boundary ion concentrations (charges) and electric potentials. We consider two ion species, one positively charged and one negatively charged, and assume zero permanent charge. A local hard-sphere potential that depends pointwise on ion concentra- tions is included in the model to account for ion size effects on the ionic flow. The model problem is treated as a boundary value problem of a singularly perturbed differential system. Our analysis is based on the geometric singular perturba- tion theory but, most importantly, on specific structures of this concrete model. The existence of solutions to the boundary value problem for small ion sizes is established and, treating the ion sizes as small parameters, we also derive an approximation of the I-V (current-voltage) relation and identify two critical po- tentials or voltages for ion size effects. Under electroneutrality (zero net charge) boundary conditions, each of these two critical potentials separates the poten- tial into two regions over which the ion size effects are qualitatively opposite to each other. On the other hand, without electroneutrality boundary conditions, the qualitative effects of ion sizes will depend not only on the critical potentials but also on boundary concentrations. Important scaling laws of I-V relations and critical potentials in boundary concentrations are obtained. Similar results about ion size effects on the flow of matter are also discussed. Under electroneu- trality boundary conditions, the results on the first order approximation in ion diameters of solutions, I-V relations and critical potentials agree with those with a nonlocal hard-sphere potential examined by Ji and Liu [J. Dynam. Differential Equations 24 (2012), 955-983].**Boundary-Roughness Effects in Nematic Liquid Crystals**

Robert Foldes (University of Minnesota, Twin Cities)

Paolo Biscari and Stefano Turzi considered a plate with an undulatory pattern. They replace the corrugation with sinusoidal boundary conditions, and use formal asymptotics for the analysis. I would like to use the method of gamma convergence to determine the effective energy and its minimizers for this problem.**Analysis of Reaction Diffusion System with Mass Transport Type Boundary Conditions**

Vandana Sharma (University of Houston)

We consider coupled reaction-diffusion models, where some components react and diffuse on the boundary of a region, while other components diffuse in the interior and react with those on the boundary through mass transport. Classical potential theory and estimates for linear initial boundary value problems are used to prove local well-posedness and global existence. This type of system arises in mathematical models for cell processes.

Joint Work with Dr. Jeff Morgan**Nonlocal Interaction Equations with Singular Kernels: Wasserstein Gradient Flow vs. Entropy Solutions**

Giovanni Bonaschi (Technische Universiteit Eindhoven)

We prove the equivalence between the notion of Wasserstein gradient flow for a one-dimensional nonlocal transport PDE with attractive/repulsive Newtonian potential on one side, and the notion of Entropy Solution of a Burgers' type scalar conservation law on the other. The solution of the former is obtained by $x$-differentiating the solution of the latter. The proof uses an intermediate step, namely the $L^2$ gradient flow of the pseudo-inverse distribution function of the gradient-flow solution. We use this equivalence to provide a rigorous particle-system approximation to the Wasserstein gradient flow. The abstract particle method relies on the so-called wave-front-tracking algorithm for scalar conservation laws.**Discrete Element Modeling of Fracture in Brittle Polycrystalline Materials**

Katerine Saleme Ruiz (Mississippi State University)

The objective of this work is the development of a fracture model for brittle polycrystalline materials. The model is based on the discrete element method, and the digital representation of the microstructure composed of a sheet of hexagonal, columnar Voroni polyhedra that represents slab of equal sized close packed grains. Here, we present the initial development of a coarse-grained particle dynamics method to model crack propagation in brittle ceramics at the mesoscale where the effects of grain size and orientation strongly influence the fracture mechanics.**Mixing by Microorganisms**

Peter Mueller (University of Wisconsin, Madison)

We begin by analyzing some simple models for Chlamydomonas

reinhardtii, a microscopic organism that swims in the Stokes regime.

Chlamydomonas beats its flagellum at high frequency, creating

oscillations as it moves. One of our goals is to determine how

important these oscillations are to the effect diffusivity caused by

swimming. Later on we will explore larger organisms, like copepods,

that escape the Stokes regime and additionally have more complex fluid

dynamics.**Lyapunov Exponents, Linear Stability Domains, and the Time-dependent Stability of Numerical Methods for Solving ODE**

The nonautonomous (time-dependent) stability theory for numerical methods is an important and underdeveloped subject. Current strategies for stability analysis of numerical methods applied to nonautonomous ODE make assumptions that reduce the problem to the autonomous case. Many ODE do not satisfy the hypotheses of this theory that we would still expect to be stable. We apply ideas and techniques from the Lyapunov stability theory of ODE to study the stability of numerical methods applied to time-dependent ODE. Specifically we study the linear multistep methods applied to scalar linear nonautonomous ODE with a slow-varying assumption. We aim to classify the nonautonomous stability of these relatively easy to analyze methods and develop a fully nonautonomous theory of stability for numerical ODE solvers.**Energy Preserving Schemes for a Nonlinear Variational Wave Equation**

We present energy preserving as well as energy stable schemes for first

order formulations of the nonlinear variational wave equation $ u_{tt} -

c(u)left( c(u) u_xright)_x =0$ in $(x,t)in mathbb{R}times (0,T]$

and test their performance on several numerical examples.