Hydrodynamic Regimes, Knudsen Layer, Numerical Schemes: Definition of Boundary Fluxes

2011 ◽  
Vol 3 (5) ◽  
pp. 519-561 ◽  
Author(s):  
Christophe Besse ◽  
Saja Borghol ◽  
Thierry Goudon ◽  
Ingrid Lacroix-Violet ◽  
Jean-Paul Dudon
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Conservation Laws ◽  
Numerical Solution ◽  
Boundary Layers ◽  
Mean Free Path ◽  
Euler System ◽  
Knudsen Layer ◽  
Microscopic Modeling ◽  
Definition Of

AbstractWe propose a numerical solution to incorporate in the simulation of a system of conservation laws boundary conditions that come from a microscopic modeling in the small mean free path regime. The typical example we discuss is the derivation of the Euler system from the BGK equation. The boundary condition relies on the analysis of boundary layers formation that accounts from the fact that the incoming kinetic flux might be far from the thermodynamic equilibrium.

scholarly journals Investigating the Effect of Solid Boundaries on the Gas Molecular Mean-Free-Path

2009 ◽  
Author(s):  
Erik J. Arlemark ◽  
Jason M. Reese
Keyword(s):  
Molecular Dynamics ◽  
Boundary Conditions ◽  
Free Path ◽  
Mean Free Path ◽  
Gas Flows ◽  
Standard Temperature ◽  
Binary Collisions ◽  
The Mean ◽  
Temperature And Pressure ◽  
Definition Of

A key parameter for micro-gas-flows, the mean free path, is investigated in this paper. The mean free path is used in various models for predicting micro gas flows, both in the governing equations and their boundary conditions. The conventional definition of the mean free path is based on the assumption that only binary collisions occur and is commonly described using the macroscopic quantities density, viscosity and temperature. In this paper we compare the prediction by this definition of the mean free paths for helium, neon and argon gases under standard temperature and pressure conditions, with the mean free paths achieved by measurements of individual molecules using the numerical simulation technique of molecular dynamics. Our simulation using molecular dynamics consists of a cube with six periodic boundary conditions, allowing us to simulate an unconfined gas “package”. Although, the size of this package is important, since its impact on computational cost is considerable, it is also important to have enough simulated molecules to average data from. We find that the molecular dynamics method using 20520 simulated molecules yields results that are within 1% accuracy from the conventional definition of the mean free paths for neon and argon and within 2.5% for helium. We can also conclude that the normal approximation of only considering binary collisions is seemingly adequate for these gases under standard temperature and pressure conditions. We introduce a single planar wall and two parallel planar walls to the simulated gas of neon and record the mean free paths at various distances to the walls. It is found that the mean free paths affected by molecular collisions with the walls corresponds well with theoretical models up to Knudsen numbers of 0.2.

scholarly journals Slip boundary condition of heat flux in Knudsen layers

2014 ◽  
Vol 470 (2161) ◽  
pp. 20130578 ◽  
Author(s):  
Mingtian Xu
Keyword(s):  
Boundary Condition ◽  
Heat Flux ◽  
Velocity Slip ◽  
Slip Boundary Condition ◽  
Mean Free Path ◽  
Knudsen Layer ◽  
Boltzmann Transport ◽  
Slip Boundary ◽  
The Mean ◽  
Solid Walls

In a Knudsen layer with thickness comparable to the mean free path, collisions between heat carriers and solid walls play an important role in nanoscale heat transports. An interesting question is that whether these collisions also induce the slip of heat flow similar to the velocity slip condition of the rarefied gases on solid walls. In this work based on the discrete Boltzmann transport equation, the slip boundary condition of heat flux on solid walls in the Knudsen layer is established. This result is exemplified by the slip boundary condition of heat flux in nanowires, which has been proposed in a phenomenological way.

The treatment of constraints due to standard boundary conditions in the context of the mixed Web-spline finite element method

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Ouadie Koubaiti ◽  
Said EL Fakkoussi ◽  
Jaouad El-Mekkaoui ◽  
Hassan Moustachir ◽  
Ahmed Elkhalfi ◽  
...  
Keyword(s):  
Finite Element Method ◽  
Boundary Condition ◽  
Finite Element ◽  
Boundary Conditions ◽  
Numerical Solution ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Dirichlet Boundary Conditions ◽  
Content Type ◽  
Element Method

Purpose This paper aims to propose a new boundary condition and a web-spline basis of finite element space approximation to remedy the problems of constraints due to homogeneous and non-homogeneous; Dirichlet boundary conditions. This paper considered the two-dimensional linear elasticity equation of Navier–Lamé with the condition CAB. The latter allows to have a total insertion of the essential boundary condition in the linear system obtained; without using a numerical method as Lagrange multiplier. This study have developed mixed finite element; method using the B-splines Web-spline space. These provide an exact implementation of the homogeneous; Dirichlet boundary conditions, which removes the constraints caused by the standard; conditions. This paper showed the existence and the uniqueness of the weak solution, as well as the convergence of the numerical solution for the quadratic case are proved. The weighted extended B-spline; approach have become a much more workmanlike solution. Design/methodology/approach In this paper, this study used the implementation of weighted finite element methods to solve the Navier–Lamé system with a new boundary condition CA, B (Koubaiti et al., 2020), that generalises the well-known basis, especially the Dirichlet and the Neumann conditions. The novel proposed boundary condition permits to use a single Matlab code, which summarises all kind of boundary conditions encountered in the system. By using this model is possible to save time and programming recourses while reap several programs in a single directory. Findings The results have shown that the Web-spline-based quadratic-linear finite elements satisfy the inf–sup condition, which is necessary for existence and uniqueness of the solution. It was demonstrated by the existence of the discrete solution. A full convergence was established using the numerical solution for the quadratic case. Due to limited regularity of the Navier–Lamé problem, it will not change by increasing the degree of the Web-spline. The computed relative errors and their rates indicate that they are of order 1/H. Thus, it was provided their theoretical validity for the numerical solution stability. The advantage of this problem that uses the CA, B boundary condition is associated to reduce Matlab programming complexity. Originality/value The mixed finite element method is a robust technique to solve difficult challenges from engineering and physical sciences using the partial differential equations. Some of the important applications include structural mechanics, fluid flow, thermodynamics and electromagnetic fields (Zienkiewicz and Taylor, 2000) that are mainly based on the approximation of Lagrange. However, this type of approximation has experienced a great restriction in the level of domain modelling, especially in the case of complicated boundaries such as that in the form of curvilinear graphs. Recently, the research community tried to develop a new way of approximation based on the so-called B-spline that seems to have superior results in solving the engineering problems.

Surface Energies Arising in Microscopic Modeling of Martensitic Transformations in Shape-Memory Alloys

2015 ◽  
Vol 651-653 ◽  
pp. 941-943
Author(s):  
Georgy Kitavtsev ◽  
Stephan Luckhaus ◽  
Angkana Rüland
Keyword(s):  
Surface Energy ◽  
Boundary Conditions ◽  
Boundary Layers ◽  
Continuum Limit ◽  
Vertical Direction ◽  
Martensitic Transformations ◽  
Surface Energies ◽  
Atomic Chains ◽  
Rank One ◽  
Microscopic Modeling

In this study we construct and analyze a two-well Hamiltonian on a 2D atomic lattice.The two wells of the Hamiltonian are prescribed by two rank-one connected martensitic twins,respectively. By constraining the deformed con gurations to special 1D atomic chains withposition-dependent elongation vectors for the vertical direction, we show that the structure ofground states under appropriate boundary conditions is close to the macroscopically expectedtwinned con gurations with additional boundary layers localized near the twinning interfaces.In addition, we proceed to a continuum limit, show asymptotic piecewise rigidity of minimizingsequences and rigorously derive the corresponding limiting form of the surface energy.

scholarly journals Classifying boundary conditions in JT gravity: from energy-branes to α-branes

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Akash Goel ◽  
Luca V. Iliesiu ◽  
Jorrit Kruthoff ◽  
Zhenbin Yang
Keyword(s):  
Boundary Condition ◽  
Black Holes ◽  
Boundary Conditions ◽  
Two Dimensional ◽  
Dimensional Model ◽  
Ensemble Averaging ◽  
Matrix Integral ◽  
Two Dimensional Model ◽  
Definition Of ◽  
Extremal Black Holes

Abstract We classify the possible boundary conditions in JT gravity and discuss their exact quantization. Each boundary condition that we study will reveal new features in JT gravity related to its matrix integral interpretation, its factorization properties and ensemble averaging interpretation, the definition of the theory at finite cutoff, its relation to the physics of near-extremal black holes and, finally, its role as a two-dimensional model of cosmology.

scholarly journals Boundary Conditions that Remove Certain Ultraviolet Divergences

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 577
Author(s):  
Roderich Tumulka
Keyword(s):  
Boundary Condition ◽  
Wave Function ◽  
Boundary Conditions ◽  
Particle Creation ◽  
Quantum Field ◽  
Interior Boundary ◽  
Direct Definition ◽  
Definition Of ◽  
Particle Creation And Annihilation ◽  
Interior Boundary Conditions

In quantum field theory, Hamiltonians contain particle creation and annihilation terms that are usually ultraviolet (UV) divergent. It is well known that these divergences can sometimes be removed by adding counter-terms and by taking limits in which a UV cutoff tends toward infinity. Here, I review a novel way of removing UV divergences: by imposing a type of boundary condition on the wave function. These conditions, called interior-boundary conditions (IBCs), relate the values of the wave function at two configurations linked by the creation or annihilation of a particle. They allow for a direct definition of the Hamiltonian without renormalization or limiting procedures. In the last section, I review another boundary condition that serves to determine the probability distribution of detection times and places on a time-like 3-surface.

scholarly journals Modeling of Knudsen Layer Effects in the Micro-Scale Backward-Facing Step in the Slip Flow Regime

Micromachines ◽  
2019 ◽  
Vol 10 (2) ◽  
pp. 118 ◽  
Author(s):  
Apurva Bhagat ◽  
Harshal Gijare ◽  
Nishanth Dongari
Keyword(s):  
Boundary Conditions ◽  
Flow Regime ◽  
Free Path ◽  
Temperature Jump ◽  
Velocity Slip ◽  
Mean Free Path ◽  
Knudsen Layer ◽  
Backward Facing Step ◽  
Micro Scale ◽  
Non Equilibrium

The effect of the Knudsen layer in the thermal micro-scale gas flows has been investigated. The effective mean free path model has been implemented in the open source computational fluid dynamics (CFD) code, to extend its applicability up to slip and early transition flow regime. The conventional Navier-Stokes constitutive relations and the first-order non-equilibrium boundary conditions are modified based on the effective mean free path, which depends on the distance from the solid surface. The predictive capability of the standard `Maxwell velocity slip—Smoluchwoski temperature jump’ and hybrid boundary conditions `Langmuir Maxwell velocity slip—Langmuir Smoluchwoski temperature jump’ in conjunction with the Knudsen layer formulation has been evaluated in the present work. Simulations are carried out over a nano-/micro-scale backward facing step geometry in which flow experiences adverse pressure gradient, separation and re-attachment. Results are validated against the direct simulation Monte Carlo (DSMC) data, and have shown significant improvement over the existing CFD solvers. Non-equilibrium effects on the velocity and temperature of gas on the surface of the backward facing step channel are studied by varying the flow Knudsen number, inlet flow temperature, and wall temperature. Results show that the modified solver with hybrid Langmuir based boundary conditions gives the best predictions when the Knudsen layer is incorporated, and the standard Maxwell-Smoluchowski can accurately capture momentum and the thermal Knudsen layer when the temperature of the wall is higher than the fluid flow.

Analysis of the Boundary Problem with the Preference of Mass Flow

2016 ◽  
Vol 821 ◽  
pp. 70-78 ◽  
Author(s):  
Martin Kyncl ◽  
Jaroslav Pelant
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Numerical Solution ◽  
Boundary Problem ◽  
Mass Flow ◽  
Gas Flow ◽  
Stationary Flow ◽  
Hyperbolic Partial Differential Equations ◽  
Compressible Gas Flow ◽  
Compressible Gas

We work with the numerical solution of the turbulent compressible gas flow, and we focus on the numerical solution of these equations, and on the boundary conditions, particularly on the outlet boundary condition with the preference of given mass flow. Usually, the boundary problem is being linearized, or roughly approximated. The inaccuracies implied by these simplifications may be small, but it has a huge impact on the solution in the whole studied area, especially for the non-stationary flow. The boundary condition with the preference of mass flow is sometimes being implemented with the use of some iterative process, guessing the correct values (for the pressure, density, velocity) in order to match the given mass flow through the boundary. In our approach we try to be as exact as possible, using our own original procedures. We follow the exact solution of the initial-value problem for the system of hyperbolic partial differential equations. This complicated problem is modified at the close vicinity of boundary, where the conservation laws are supplied with the additional boundary conditions. We complement the boundary problem suitably, and we show the analysis of the resulting uniquely-solvable modified Riemann problem.The resulting algorithm was coded and used within our own developed code for the solution of the compressible gas flow (the Euler, NS, and RANS equations). The examples show good behaviour of the analyzed boundary condition.

scholarly journals Acoustic boundary conditions at an impedance lining in inviscid shear flow

2016 ◽  
Vol 796 ◽  
pp. 386-416 ◽  
Author(s):  
Doran Khamis ◽  
Edward James Brambley
Keyword(s):  
Boundary Condition ◽  
Shear Flow ◽  
Boundary Conditions ◽  
Boundary Layers ◽  
Temporal Stability ◽  
Impedance Boundary ◽  
Impedance Boundary Conditions ◽  
Ricatti Equation ◽  
Approximate Boundary Conditions ◽  
Lined Ducts

The accuracy of existing impedance boundary conditions is investigated, and new impedance boundary conditions are derived, for lined ducts with inviscid shear flow. The accuracy of the Ingard–Myers boundary condition is found to be poor. Matched asymptotic expansions are used to derive a boundary condition accurate to second order in the boundary layer thickness, which shows substantially increased accuracy for thin boundary layers when compared with both the Ingard–Myers boundary condition and its recent first-order correction. Closed-form approximate boundary conditions are also derived using a single Runge–Kutta step to solve an impedance Ricatti equation, leading to a boundary condition that performs reasonably even for thicker boundary layers. Surface modes and temporal stability are also investigated.

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