Matthew M. Hopkins

A computational model of the collective fluid dynamics of motile micro-organisms

A mathematical model and numerical method for studying the collective dynamics of geotactic, gyrotactic and chemotactic micro-organisms immersed in a viscous fluid is presented. The Navier-Stokes equations of fluid dynamics are solved in the presence of a discrete collection of micro-organisms. These microbes act as point sources of gravitational force in the fluid equations, and thus affect the fluid flow. Physical factors, e.g. vorticity and gravity, as well as sensory factors affect swimming speed and direction. In the case of chemotactic microbes, the swimming orientation is a function of a molecular field. In the model considered here, the molecules are a nutrient whose consumption results in an upward gradient of concentration that drives its downward diffusion. The resultant upward chemotactically induced accumulation of cells results in (Rayleigh-Taylor) instability and eventually in steady or chaotic convection that transports molecules and affects the translocation of organisms. Computational results that examine the long-time behaviour of the full nonlinear system are presented. The actual dynamical system consisting of fluid and suspended swimming organisms is obviously three-dimensional, as are the basic modelling equations. While the computations presented in this paper are two-dimensional, they provide results that match remarkably well the spatial patterns and long-time temporal dynamics of actual experiments; various physically applicable assumptions yield steady states, chaotic states, and bottom-standing plumes. The simplified representation of microbes as point particles allows the variation of input parameters and modelling details, while performing calculations with very large numbers of particles ( 10 4 -10 5 ), enough so that realistic cell concentrations and macroscopic fluid effects can be modelled with one particle representing one microbe, rather than some collection of microbes. It is demonstrated that this modelling framework can be used to test hypotheses concerning the coupled effects of microbial behaviour, fluid dynamics and molecular mixing. Thus, not only are insights provided into the differing dynamics concerning purely geotactic and gyrotactic microbes, the dynamics of competing strategies for chemotaxis, but it is demonstrated that relatively economical explorations in two dimensions can deliver striking insights and distinguish among hypotheses.

Discretization Error Estimates Using Exact Solutions to Nearby Problems

A methodology is presented for generating exact solutions to equations that are “near” the Navier-Stokes equations. First, a highly accurate numerical solution to the Navier-Stokes equations is computed. Second, an analytic function is fit to the numerical solution by least squares optimization. Next, this analytic solution is operated on by the Navier-Stokes equations (including auxiliary relations) to obtain a small analytic source term. When the Navier-Stokes equations are perturbed by adding this source term, the analytic function is recovered as the exact solution. Approaches are presented which address the “goodness” of the curve-fitting procedure and the “nearness” of the modified set of equations to the Navier-Stokes equations. Two examples are given for compressible fluid flow: fully developed flow in a channel, and lid-driven cavity flow. The channel flow is fully captured by a third-order polynomial fit, while the driven-cavity solution is not adequately represented by polynomial curve fits up to fourth order. The generation of an exact solution to a set of equations near the Navier-Stokes equations allows for the evaluation of various discretization error estimators, without reverting to simplification of the governing equations or use of a highly refined “truth” mesh. Preliminary results for a number of extrapolation-based error estimators are also presented.

Theory of the electron sheath and presheath

Electron sheaths are commonly found near Langmuir probes collecting the electron saturation current. The common assumption is that the probe collects the random flux of electrons incident on the sheath, which tacitly implies that there is no electron presheath and that the flux collected is due to a velocity space truncation of the electron velocity distribution function (EVDF). This work provides a dedicated theory of electron sheaths, which suggests that they are not so simple. Motivated by EVDFs observed in particle-in-cell (PIC) simulations, a 1D model for the electron sheath and presheath is developed. In the model, under low temperature plasma conditions ( Te≫Ti), an electron pressure gradient accelerates electrons in the presheath to a flow velocity that exceeds the electron thermal speed at the sheath edge. This pressure gradient generates large flow velocities compared to what would be generated by ballistic motion in response to the electric field. It is found that in many situations, under comm...

Aria 1.5 : user manual.

Aria is a Galerkin finite element based program for solving coupled-physics problems described by systems of PDEs and is capable of solving nonlinear, implicit, transient and direct-to-steady state problems in two and three dimensions on parallel architectures. The suite of physics currently supported by Aria includes the incompressible Navier-Stokes equations, energy transport equation, species transport equations, nonlinear elastic solid mechanics, and electrostatics as well as generalized scalar, vector and tensor transport equations. Additionally, Aria includes support for arbitrary Lagrangian-Eulerian (ALE) and level set based free and moving boundary tracking. Coupled physics problems are solved in several ways including fully-coupled Newtons method with analytic or numerical sensitivities, fully-coupled Newton-Krylov methods, fully-coupled Picards method, and a loosely-coupled nonlinear iteration about subsets of the system that are solved using combinations of the aforementioned methods. Error estimation, uniform and dynamic h-adaptivity and dynamic load balancing are some of Arias more advanced capabilities. Aria is based on the Sierra Framework.

Final Report on LDRD Project: Coupling Strategies for Multi-Physics Applications

Many current and future modeling applications at Sandia including ASC milestones will critically depend on the simultaneous solution of vastly different physical phenomena. Issues due to code coupling are often not addressed, understood, or even recognized. The objectives of the LDRD has been both in theory and in code development. We will show that we have provided a fundamental analysis of coupling, i.e., when strong coupling vs. a successive substitution strategy is needed. We have enabled the implementation of tighter coupling strategies through additions to the NOX and Sierra code suites to make coupling strategies available now. We have leveraged existing functionality to do this. Specifically, we have built into NOX the capability to handle fully coupled simulations from multiple codes, and we have also built into NOX the capability to handle Jacobi Free Newton Krylov simulations that link multiple applications. We show how this capability may be accessed from within the Sierra Framework as well as from outside of Sierra. The critical impact from this LDRD is that we have shown how and have delivered strategies for enabling strong Newton-based coupling while respecting the modularity of existing codes. This will facilitate the use of these codes in a coupled manner to solve multi-physic applications.

Ion flow and sheath structure near positively biased electrodes

What effect does a dielectric material surrounding a small positively biased electrode have on the ion flow and sheath structure near the electrode? Measurements of the ion velocity distribution function and plasma potential near positively biased electrodes were made using laser-induced fluorescence and an emissive probe. The results were compared with 2D particle-in-cell simulations. Both measurements and simulations showed that when the positive electrode was surrounded by the dielectric material, ions were accelerated toward the electrode to approximately 0.5 times the ion sound speed before being deflected radially by the electron sheath potential barrier of the electrode. The axial potential profile in this case contained a virtual cathode. In comparison, when the dielectric material was removed from around the electrode, both the ion flow and virtual cathode depth near the electrode were dramatically reduced. These measurements suggest that the ion presheath from the dielectric material surrounding...

Particle-in-cell study of the ion-to-electron sheath transition

The form of a sheath near a small electrode, with bias changing from below to above the plasma potential, is studied using 2D particle-in-cell simulations. When the electrode is biased within Te/2e below the plasma potential, the electron velocity distribution functions (EVDFs) exhibit a loss-cone type truncation due to fast electrons overcoming the small potential difference between the electrode and plasma. No sheath is present in this regime, and the plasma remains quasineutral up to the electrode. The EVDF truncation leads to a presheath-like density and flow velocity gradients. Once the bias exceeds the plasma potential, an electron sheath is present. In this case, the truncation driven behavior persists, but is accompanied by a shift in the maximum value of the EVDF that is not present in the negative bias cases. The flow moment has significant contributions from both the flow shift of the EVDF maximum, and the loss-cone truncation.

Electron presheaths: the outsized influence of positive boundaries on plasmas

Electron sheaths form near the surface of objects biased more positive than the plasma potential, such as in the electron saturation region of a Langmuir probe trace. Generally, the formation of electron sheaths requires that the electron-collecting area be sufficiently smaller (

The onset of plasma potential locking

\sqrt{2.3m_e/M}

Solution-Verified Reliability Analysis and Design of Compliant Micro-Electro-Mechanical Systems

times) than the ion-collecting area. They are commonly thought to be local phenomena that collect the random thermal electron current, but do not otherwise perturb a plasma. Here, using experiments on an electrode embedded in a wall, particle-in-cell simulations and theory, it is shown that under low temperature plasma conditions (