Moment-based formulation of Navier–Maxwell slip boundary conditions for lattice Boltzmann simulations of rarefied flows in microchannels

We present an implementation of first-order Navier–Maxwell slip boundary conditions for simulating near-continuum rarefied flows in microchannels with the lattice Boltzmann method. Rather than imposing boundary conditions directly on the particle velocity distribution functions, following the existing discrete analogs of the specular and diffuse reflection conditions from continuous kinetic theory, we use a moment-based method to impose the Navier–Maxwell slip boundary conditions that relate the velocity and the strain rate at the boundary. We use these conditions to solve for the unknown distribution functions that propagate into the\ud domain across the boundary. We achieve second-order accuracy by reformulating these conditions for the second set of distribution functions that arise in the derivation of the lattice Boltzmann method by an integration along characteristics. The results are in excellent agreement with asymptotic solutions of the compressible Navier-Stokes equations for microchannel flows in the slip regime. Our moment formalism is also valuable for analysing the existing boundary conditions, and explains the origin of numerical slip in the bounce-back and other common boundary conditions that impose explicit conditions on the higher moments instead of on the local tangential velocity

A random projection method for sharp phase boundaries in lattice Boltzmann simulations

Existing lattice Boltzmann models that have been designed to recover a macroscopic description of immiscible liquids are only able to make predictions that are quantitatively correct when the interface that exists between the fluids is smeared over several nodal points. Attempts to minimise the thickness of this interface generally leads to a phenomenon known as lattice pinning, the precise cause of which is not well understood. This spurious behaviour is remarkably similar to that associated with the numerical simulation of hyperbolic partial differential equations coupled with a stiff source term. Inspired by the seminal work in this field, we derive a lattice Boltzmann implementation of a model equation used to investigate such peculiarities. This implementation is extended to different spacial discretisations in one and two dimensions. We shown that the inclusion of a quasi-random threshold dramatically delays the onset of pinning and facetting

A NOTE ON COMONOTONICITY AND POSITIVITY OF THE CONTROL COMPONENTS OF DECOUPLED QUADRATIC FBSDE

In this small note we are concerned with the solution of Forward-Backward Stochastic Differential Equations (FBSDE) with drivers that grow quadratically in the control component (quadratic growth FBSDE or qgFBSDE). The main theorem is a comparison result that allows comparing componentwise the signs of the control processes of two different qgFBSDE. As a byproduct one obtains conditions that allow establishing the positivity of the control process.Comment: accepted for publicatio

Phases of granular segregation in a binary mixture

We present results from an extensive experimental investigation into granular segregation of a shallow binary mixture in which particles are driven by frictional interactions with the surface of a vibrating horizontal tray. Three distinct phases of the mixture are established viz; binary gas (unsegregated), segregation liquid and segregation crystal. Their ranges of existence are mapped out as a function of the system's primary control parameters using a number of measures based on Voronoi tessellation. We study the associated transitions and show that segregation can be suppressed is the total filling fraction of the granular layer, $C$, is decreased below a critical value, $C_{c}$, or if the dimensionless acceleration of the driving, $\gamma$, is increased above a value $\gamma_{c}$.Comment: 12 pages, 12 figures, submitted to Phys. Rev.

Raquialgias na Criança

As raquialgias são, na criança e no adolescente, menos frequentes do que no adulto, mas traduzem, com maior frequência, a existência de patologia subjacente. Longe de constituírem um gasto excessivo de tempo do clínico, a colheita minuciosa da história clínica e um exame objectivo cuidadoso e sistemático são, indubitavelmente, um dos melhores investimentos do médico que vê crianças com este tipo de sintoma. Permitem abordar correctamente o doente e a sua doença e a solicitação dos exames complementares de diagnóstico mais adequados, em face de uma hipótese diagnóstica correctamente equacionada. Perante uma criança com raquialgias, a atitude depende da idade, da gravidade dos sinais encontrados na observação e da existência de complicações neurológicas ou outros sinais extra-raquidianos; na sua ausência, é prudente seguir a evolução das queixas durante algum tempo, antes de submeter o doente a uma investigação frequentemente onerosa, consumidora de tempo e com baixa probabilidade de êxito

Universal velocity distributions in an experimental granular fluid

We present experimental results on the velocity statistics of a uniformly heated granular fluid, in a quasi-2D configuration. We find the base state, as measured by the single particle velocity distribution $f(c)$, to be universal over a wide range of filling fractions and only weakly dependent on all other system parameters. There is a consistent overpopulation in the distribution's tails, which scale as $f\propto\exp(\mathrm{const.}\times c^{-3/2})$. More importantly, the high probability central region of $f(c)$, at low velocities, deviates from a Maxwell-Boltzmann by a second order Sonine polynomial with a single adjustable parameter, in agreement with recent theoretical analysis of inelastic hard spheres driven by a stochastic thermostat. To our knowledge, this is the first time that Sonine deviations have been measured in an experimental system.Comment: 13 pages, 15 figures, with minor corrections, submitted to Phys. Rev.

Introduction to the Method of Finite Elements by a balance Sheet Problem: A Simplification for an Initial understanding of the Method

The Finite Element method is one of the most widely used methods by Engineers in the various areas of activity, especially Mechanical Engineering, to design or solve problems. However, the understanding of the method is not always easy to perform, since in the literature, when explaining the method, the examples are generic or presented quickly. Thus, this paper presents the solution of a problem involving a rocking beam (set), which is solved analytically and later by the finite element method. The comparison of the solutions found is established as reflection analysis. Elasticity theory, Ordinary Differential Equations and Finite Element Method are used to approximate the reader of the Finite Element Method, in a concise and objective, easy-to-understand reading performed with a reduced explanation. Comparing the method by means of a problem