Michael Junk

MAXIMUM ENTROPY FOR REDUCED MOMENT PROBLEMS

The existence of maximum entropy solutions for a wide class of reduced moment problems on arbitrary open subsets of ℝd is considered. In particular, new results for the case of unbounded domains are obtained. A precise condition is presented under which solvability of the moment problem implies existence of a maximum entropy solution.

Discretizations for the Incompressible Navier--Stokes Equations Based on the Lattice Boltzmann Method

A discrete velocity model with spatial and velocity discretization based on a lattice Boltzmann method is considered in the low Mach number limit. A uniform numerical scheme for this model is investigated. In the limit, the scheme reduces to a finite difference scheme for the incompressible Navier--Stokes equation, which is a projection method with a second order spatial discretization on a regular grid. The discretization is analyzed and the method is compared to Chorins original spatial discretization. Numerical results supporting the analytical statements are presented.

Boundary forces in lattice Boltzmann: Analysis of Momentum Exchange algorithm

When lattice Boltzmann methods are used to simulate fluid-structure interaction problems, they need to be coupled with additional routines to evaluate the boundary forces without destroying the efficiency and accuracy of the original method. We use the asymptotic expansion technique to analyze one such approach, the Momentum Exchange algorithm, investigating its properties in detail, whether it can be improved and in which cases it can be successfully used. A statement regarding the accuracy is presented, together with results of numerical tests which illustrate the theoretical considerations.

A lattice Boltzmann method for immiscible multiphase flow simulations using the level set method

We consider the lattice Boltzmann method for immiscible multiphase flow simulations. Classical lattice Boltzmann methods for this problem, e.g. the colour gradient method or the free energy approach, can only be applied when density and viscosity ratios are small. Moreover, they use additional fields defined on the whole domain to describe the different phases and model phase separation by special interactions at each node. In contrast, our approach simulates the flow using a single field and separates the fluid phases by a free moving interface. The scheme is based on the lattice Boltzmann method and uses the level set method to compute the evolution of the interface. To couple the fluid phases, we develop new boundary conditions which realise the macroscopic jump conditions at the interface and incorporate surface tension in the lattice Boltzmann framework. Various simulations are presented to validate the numerical scheme, e.g. two-phase channel flows, the Young-Laplace law for a bubble and viscous fingering in a Hele-Shaw cell. The results show that the method is feasible over a wide range of density and viscosity differences.

Outflow boundary conditions for the lattice Boltzmann method

In this paper, we propose lattice Boltzmann implementations of several Navier-Stokes outflow boundary conditions like the zero normal-stress, the Neumann, and the do-nothing rule. A consistency analysis using asymptotic methods is given and the algorithms are numerically tested in two space dimensions with respect to accuracy and interaction with the inner flow field.

A Study of Parylene C Polymer Deposition Inside Microscale Gaps

Parylene C is a biocompatible polymer that has been investigated as an encapsulation material for implantable microsystems. Since parylene C is deposited directly on the substrate from the vapor phase it can conform to a wide range of geometries. However, the thickness of the deposited layer tends to decrease in microscale gaps, which might lead to an insulation failure. To ensure more robustness of the coating, the changes in parylene C coating thickness have been investigated experimentally using simple gaps of known dimensions. In an attempt to better understand these experimental findings, two theoretical models have been developed. The first one, which is based on a diffusion approximation, is able to reproduce and extrapolate the experimental results, leading to a useful design rule for practical applications involving parylene C coating. As an example, we present the substrate design of a flexible sieve electrode for a peripheral nerve interface. The second model aims at an appropriate microscopic description of the coating process in terms of kinetic theory of gases.

A QMC approach for high dimensional Fokker-Planck equations modelling polymeric liquids

A classical model used in the study of dynamics of polymeric liquids is the bead-spring chain representation of polymer molecules. The chain typically consists of a large number of beads and thus the state space V of its configuration, which is essentially the position of all the constituent beads, turns out to be high dimensional. The distribution function governing the configuration of a bead-spring chain undergoing shear flow is a Fokker-Planck equation on V. In this article, we present QMC methods for the approximate solution of the Fokker-Planck equation which are based on the time splitting technique to treat convection and diffusion separately. Convection is carried out by moving the particles along the characteristics and we apply the algorithms presented in [G. Venkiteswaran, M. Junk, QMC algorithms for diffusion equations in high dimensions, Math. Comput. Simul. 68 (2005) 23-41.] for diffusion. Altogether, we find that some of the QMC methods show reduced variance and thus slightly outperform standard MC.

Weighted

This article is concerned with the linearized stability of the lattice Boltzmann method both on periodic domains and on bounded domains with the bounce-back rule used at the boundaries. Under a structural hypothesis, we prove that a weighted

\mathbb{L}^2

\mathbb{L}^2

-Stability of the Lattice Boltzmann Method

-norm of the solutions to the linearized lattice Boltzmann method is decreasing with time. Moreover, we show that the structural hypothesis holds true for many lattice Boltzmann models.