Minimum error solutions of the Boltzmann equation for shock structure

‘Best’ solutions for the shock-structure problem are obtained by solving the Boltzmann equation for a rigid sphere gas by applying minimum error criteria on the Mott-Smith ansatz. The use of two such criteria minimizing respectively the local and total errors, as well as independent computations of the remaining error, establish the high accuracy of the solutions, although it is shown that the Mott-Smith distribution is not an exact solution of the Boltzmann equation even at infinite Mach number. The minimum local error method is found to be particularly simple and efficient. Adopting the present solutions as the standard of comparison, it is found that the widely used v2x-moment solutions can be as much as a third in error, but that results based on Rosen's method provide good approximations. Finally, it is shown that if the Maxwell mean free path on the hot side of the shock is chosen as the scaling length, the value of the density-slope shock thickness is relatively insensitive to the intermolecular potential. A comparison is made on this basis of present results with experiment, and very satisfactory quantitative agreement is obtained.

Chapman-Enskog analysis for finite-volume formulation of lattice Boltzmann equation

The classical Chapman-Enskog expansion is performed for the recently proposed finite-volume formulation of lattice Boltzmann equation (LBE) method D.V. Patil, K.N. Lakshmisha, Finite volume TVD formulation of lattice Boltzmann simulation on unstructured mesh, J. Comput. Phys. 228 (2009) 5262-5279]. First, a modified partial differential equation is derived from a numerical approximation of the discrete Boltzmann equation. Then, the multi-scale, small parameter expansion is followed to recover the continuity and the Navier-Stokes (NS) equations with additional error terms. The expression for apparent value of the kinematic viscosity is derived for finite-volume formulation under certain assumptions. The attenuation of a shear wave, Taylor-Green vortex flow and driven channel flow are studied to analyze the apparent viscosity relation.

A lattice Boltzmann equation for waves

We propose a lattice Boltzmann model for the wave equation. Using a lattice Boltzmann equation and the Chapman-Enskog expansion, we get 1D and 2D wave equations with truncation error of order two. The numerical tests show the method can be used to simulate the wave motions.

Solution of the Poisson-Boltzmann equation for a system with four ionic species

The analytical solution of the Poisson-Boltzmann equation in an electrolyte with four ionic species (2:2:1:1), in the presence of a charged planar membrane or surface is presented. The function describing the mean electrical potential provides a convenient description that helps the understanding of electrical processes of biological interest.

INTEGRATION OF THE NONLINEAR POISSON BOLTZMANN-EQUATION FOR CHARGED VESICLES IN ELECTROLYTIC SOLUTIONS

The Poisson-Boltzmann equation (PBE), with specific ion-surface interactions and a cell model, was used to calculate the electrostatic properties of aqueous solutions containing vesicles of ionic amphiphiles. Vesicles are assumed to be water- and ion-permeable hollow spheres and specific ion adsorption at the surfaces was calculated using a Volmer isotherm. We solved the PBE numerically for a range of amphiphile and salt concentrations (up to 0.1 M) and calculated co-ion and counterion distributions in the inside and outside of vesicles as well as the fields and electrical potentials. The calculations yield results that are consistent with measured values for vesicles of synthetic amphiphiles.

Finite volume TVD formulation of lattice Boltzmann simulation on unstructured mesh

A numerical scheme is presented for accurate simulation of fluid flow using the lattice Boltzmann equation (LBE) on unstructured mesh. A finite volume approach is adopted to discretize the LBE on a cell-centered, arbitrary shaped, triangular tessellation. The formulation includes a formal, second order discretization using a Total Variation Diminishing (TVD) scheme for the terms representing advection of the distribution function in physical space, due to microscopic particle motion. The advantage of the LBE approach is exploited by implementing the scheme in a new computer code to run on a parallel computing system. Performance of the new formulation is systematically investigated by simulating four benchmark flows of increasing complexity, namely (1) flow in a plane channel, (2) unsteady Couette flow, (3) flow caused by a moving lid over a 2D square cavity and (4) flow over a circular cylinder. For each of these flows, the present scheme is validated with the results from Navier-Stokes computations as well as lattice Boltzmann simulations on regular mesh. It is shown that the scheme is robust and accurate for the different test problems studied.

Simulation of laminar flow in a three-dimensional lid-driven cavity by lattice Boltzmann method

Purpose - The purpose of this paper is to apply lattice Boltzmann equation method (LBM) with multiple relaxation time (MRT) model, to investigate lid-driven flow in a three-dimensional (3D), rectangular cavity, and compare the results with flow in an equivalent two-dimensional (2D) cavity. Design/methodology/approach - The second-order MRT model is implemented in a 3D LBM code. The flow structure in cavities of different aspect ratios (0.25-4) and Reynolds numbers (0.01-1000) is investigated. The LBM simulation results are compared with those from numerical solution of Navier-Stokes (NS) equations and with available experimental data. Findings - The 3D simulations demonstrate that 2D models may predict the flow structure reasonably well at low Reynolds numbers, but significant differences with experimental data appear at high Reynolds numbers. Such discrepancy between 2D and 3D results are attributed to the effect of boundary layers near the side-walls in transverse direction (in 3D), due to which the vorticity in the core-region is weakened in general. Secondly, owing to the vortex stretching effect present in 3D flow, the vorticity in the transverse plane intensifies whereas that in the lateral plane decays, with increase in Reynolds number. However, on the symmetry-plane, the flow structure variation with respect to cavity aspect ratio is found to be qualitatively consistent with results of 2D simulations. Secondary flow vortices whose axis is in the direction of the lid-motion are observed; these are weak at low. Reynolds numbers, but become quite strong at high Reynolds numbers. Originality/value - The findings will be useful in the study of variety of enclosed fluid flows.

The Boltzmann collision integrals for a combination of Maxwellians

The gain and loss integrals in the Boltzmann equation for a rigid sphere gas are evaluated in closed form for a distribution which can be expressed as a linear combination of Maxwellians. Application to the Mott-Smith bimodal distribution shows that the gain is also bimodal, but the two modes in the gain are less pronounced than in the distribution. Implications of these results for simple collision models in non-equilibrium flow are discussed.

Two-dimensional flow past circular cylinders using finite volume lattice Boltzmann formulation

In this article, an extension to the total variation diminishing finite volume formulation of the lattice Boltzmann equation method on unstructured meshes was presented. The quadratic least squares procedure is used for the estimation of first-order and second-order spatial gradients of the particle distribution functions. The distribution functions were extrapolated quadratically to the virtual upwind node. The time integration was performed using the fourth-order RungeKutta procedure. A grid convergence study was performed in order to demonstrate the order of accuracy of the present scheme. The formulation was validated for the benchmark two-dimensional, laminar, and unsteady flow past a single circular cylinder. These computations were then investigated for the low Mach number simulations. Further validation was performed for flow past two circular cylinders arranged in tandem and side-by-side. Results of these simulations were extensively compared with the previous numerical data. Copyright (C) 2011 John Wiley & Sons, Ltd.

Recovery of the solitons using a lattice Boltzmann model

We formulate a lattice Boltzmann model which simulates Korteweg-de Vries equation by using a method of higher moments of lattice Boltzmann equation. Using a series of lattice Boltzmann equations in different time scales and the conservation law in time scale to, we obtain equilibrium distribution function. The numerical examples show that the method can be used to simulate soliton.

Aplicação da equação de Poisson-Boltzmann modificada em sistemas biológicos: análise da partição iônica em um eritrócito

Neste trabalho, a partição iônica e o potencial de membrana em um eritrócito são analisados via equação de Poisson-Boltzmann modificada, considerando as interações não eletrostáticas presentes entre os íons e macromoléculas, assim como, o potencial β. Este potencial é atribuído à diferença de potencial químico de referência entre os meios intracelular e extracelular e ao transporte ativo de íons. O potencial de Gibbs-Donnan via equação de Poisson-Boltzmann na presença de carga fixa em um sistema contendo uma membrana semipermeável também é estudado. O método de aproximação paraboloide em elementos finitos em um sistema estacionário e unidimensionalé aplicado para resolver a equação de Poisson-Boltzmann em coordenadas cartesianas e esféricas. O parâmetro de dispersão relativo às interações não eletrostáticas écalculado via teoria de Lifshitz. Os resultados em relação ao potencial de Gibbs-Donnan mostram-se adequados, podendo ser calculado pela equação de Poisson-Boltzmann. No sistema contendo um eritrócito, quando o potencial β é considerado igual a zero, não se verifica a diferença iônica observada experimentalmente entre os meios intracelular e extracelular. Dessa forma, os potenciais não eletrostáticos calculados via teoria de Lifshitz têm apenas uma pequena influência no que se refere à alta concentração de íon K+ no meio intracelular em relação ao íon Na+

Measurement of aqueous film thickness between charged mercury and mica surfaces : a direct experimental probe of the Poisson-Boltzmann distribution

A stable aqueous electrolyte film is formed between a mercury drop and a flat mica surface due to electrical double-layer repulsion when a negative potential is applied to the mercury. Film thickness has been measured as a function of applied potential while keeping the film pressure constant. By making measurements in this way, it is possible to map the data directly according to the Poisson-Boltzmann equation. An excellent fit to the data is obtained, providing direct evidence for this classical equation and its use as the basis of the Gouy-Chapman model of the diffuse double layer in electrolyte solutions.

Modelos cinéticos da equação linearizada de Boltzmann e um problema de transferência de calor em microescala

Neste trabalho, um problema de transferência de calor da dinâmica de gases rarefeitos, causado pela diferença de temperaturas nas superfícies de um canal, é abordado. O problema é formulado através dos modelos cinéticos BGK, S e Gross-Jackson da equação linearizada de Boltzmann e resolvido, de forma unificada, pelo método analítico de ordenadas discretas (método ADO). Resultados numéricos para as perturbações de densidade e temperatura e também para o fluxo de calor são apresentados e comparados, mostrando que não se pode dizer que algum dos três modelos seja uma melhor aproximação da solução aos resultados da equação linearizada de Boltzmann.