Andrés Santos Universidad de Extremadura, Badajoz (Spain)
|
|
- Opal Townsend
- 5 years ago
- Views:
Transcription
1 Andrés Santos Universidad de Extremadura, Badajoz (Spain)
2 Outline Moment equations molecules for Maxwell Some solvable states: Planar Fourier flow Planar Fourier flow with gravity Planar Couette flow Force-driven Poiseuille flow Uniform shear and longitudinal flows Conclusions 2
3 ( ) (Cartoon by Bernhard Reischl, University of Vienna) 3
4 streaming external force collisions Differential cross section 4
5 Interaction potential 5
6 Weak form of the Boltzmann equation Density of ψ(v) Flux Source: external force Source: collisions 6
7 Hierarchy of moment equations 7
8 Maxwell molecules On the Dynamical Theory of Gases Phil. Trans. Roy. Soc. (London) 157, (1867) In the present paper I propose to consider the molecules of a gas, not as elastic spheres of definite radius, but as small bodies or groups of smaller molecules repelling one another with a force whose direction always passes very nearly through the centres of gravity of the molecules, and whose magnitude is represented very nearly by some function of the distance of the centres of gravity. ( ) I have made this modification of the theory in consequence of the results of my experiments on the viscosity of air at different temperatures, and I have deduced from these experiments that the repulsion is inversely as the fifth power of the distance. I have found by experiment that the coefficient of viscosity in a given gas is independent of the density, and proportional to the absolute temperature, so that if ET be the viscosity, ET p/ρ. 8
9 Maxwell molecules 9
10 Properties of the collisional moments in Maxwell models 10
11 Properties of the collisional moments in Maxwell models. Eigenfunctions and eigenvalues Eigenfunctions Eigenvalues 11
12 Properties of the eigenvalues in Maxwell models 12
13 Navier-Stokes constitutive equations Claude-Louis Navier ( ) P ij = pδ ij η George Gabriel Stokes ( ) µ i u j + j u i 2 3 uδ ij, η = p λ 02 13
14 1. Steady planar Fourier flow z=l T w ' Hot q z Jean-Baptiste Joseph Fourier ( ) z=0 T w Cold 14
15 Asmolov, Makashev, and Nosik (1979) proved that an exact solution of the (nonlinear) Boltzmann equation for Maxwell molecules exists with 15
16 Dimensionless quantities (Local) Knudsen number: Reduced distribution function: Reduced moments: 16
17 Hierarchy of moment equations 17
18 j k=2r+l 18
19 First few moments Fourier s law 19
20 First few moments 20
21 Benchmark for DSMC simulations Gallis et al., Phys. Fluids 18, (2006) 21
22 BGK description Same qualitative results for moments. Full velocity distribution function φ(c;²). Divergence of the (CE) expansion of φ(c;²) in powers of ². 22
23 2. Steady planar Fourier flow with gravity (Rayleigh-Bénard-like flow) Lord Rayleigh ( ) z=l T w ' g>0 Hot Henri Bénard ( ) q z z=0 T w Cold 23
24 Perturbation analysis Tij, Garzó, and Santos, Phys. Rev. E 56, 6729 (1997) akin to the Rayleigh number Pure Fourier flow 24
25 Corrections to Navier-Stokes 25
26 Corrections to Navier-Stokes Heating from below Heating from above 26
27 BGK description Same qualitative results. Moments up to order γ 6. Divergence of the expansion in powers of γ. 27
28 3. Steady planar Couette flow T=T w ' U w z=+l u x (z) Maurice Couette ( ) z=-l -U w T=T -w 28
29 Makashev and Nosik (1981) proved that an exact solution of the (nonlinear) Boltzmann equation for Maxwell molecules exists with NS: θ(a)=1 29
30 Dimensionless quantities Knudsen numbers: Reduced distribution function: Reduced moments: 30
31 Hierarchy of moment equations 31
32 Corrections to Navier-Stokes 32
33 Perturbation analysis Tij and Santos, Phys. Fluids 7, 2857 (1995) super-burnett Burnett 33
34 Benchmark for DSMC simulations Gallis et al., Phys. Fluids 18, (2006) 34
35 BGK description Same qualitative results for moments. Explicit expressions for κ*(a), η*(a), θ(a), Φ(a), 1 (a), and 2 (a). Full velocity distribution function φ(c;²,a). Divergence of the expansion in powers of a. Influence of gravity analyzed. 35
36 4. Force-driven Poiseuille flow Jean-Louis-Marie Poiseuille ( ) 36
37 Equal normal stresses Newton s law Fourier s law No longitudinal heat flux Temperature is maximal at the central layer (y=0)
38 Do NS predictions agree with computer simulations? (z) DSMC T 0 T max DSMC NS but... NS 2.5 mfp y max 38
39 A Burnett-order effect? DSMC In the slab y< y max, Burnett sgn q y = sgn ØT/Øy Heat flows from the colder to the hotter layers!! 39
40 Other Non-Newtonian properties Non-uniform pressure (z) (zz) Normal stress differences Longitudinal component of the heat flux (but no longitudinal thermal gradient!) 40
41 Hierarchy of moment equations 41
42 Perturbation analysis Tij, Sabanne, and Santos, Phys. Fluids 10, 1021 (1998) Equilibrium moments (at y=0) 42
43 Results to order g 2 : Hydrodynamic fields 43
44 Results to order g 2 : Hydrodynamic fluxes 44
45 Numerical values of the coefficients Coefficient Quantity NS Burnett a 13-moment b R13-moment c 19-moment d Exact C p p C T T C zz P zz ? C yy P yy ? C q q z a Uribe & Garcia (1999) b Risso & Cordero (1998) c Taheri, Struchtrup & Torrilhon (2008) d Hess & Malek Mansour (1999) 45
46 T(y)/T NS 11.6g 2 ρ 0 η 2 0 /p y/[η 0 (k B T 0 /m) 1/2 /p 0 ] 46
47 P ii (y)/p zz xx NS 6.48g 2 ρ 0 η 2 0 /p3 0 yy 6.26g 2 ρ 0 η 2 0 /p y/[η 0 (k B T 0 /m) 1/2 /p 0 ] 47
48 BGK description Same qualitative results. Hydrodynamic fields up to order g 5. Velocity distribution function up to order g 3. Divergence of the expansion in powers of g. Exact non-perturbative solution for a special value of g. 48
49 5. Other (quasi-uniform) states 49
50 In both cases, Exact rheological functions for arbitrary values (non-perturbative solution) of the corresponding Knudsen number (a=γ xy /λ 02, a=γ xx /λ 02 ). Divergence of the fourth-degree moments beyond a critical value a=a c. Algebraic high-velocity decay of the velocity distribution function for any a: Finite number of convergent moments. 50
51 Conclusions The hierarchy of moment equations can be recursively solved in the case of Maxwell molecules for some non-trivial states. No need to apply truncation closures. In some cases, a perturbation analysis is required. Exact results are important by themselves and also useful as an assessment of simulation techniques approximate methods kinetic models 51
52 More details about shear flows (including mixtures, BGK model description, ) Acknowlegments (in alphabetical order): J. Javier Brey, Universidad de Sevilla, Spain James. W. Dufty, University of Florida, USA Vicente Garzó, Universidad de Extremadura, Spain Mohamed Tij, Université Moulay Ismaïl, Morocco (Springer, 2003) THANKS FOR YOUR ATTENTION! 52
Andrés Santos* Badajoz (Spain)
Granular Poiseuille flow Andrés Santos* University of Extremadura Badajoz (Spain) *In collaboration with Mohamed Tij, Université Moulay Ismaïl, Meknès (Morocco) Departament de Física, Universitat Autònoma
More informationBOLTZMANN KINETIC THEORY FOR INELASTIC MAXWELL MIXTURES
BOLTZMANN KINETIC THEORY FOR INELASTIC MAXWELL MIXTURES Vicente Garzó Departamento de Física, Universidad de Extremadura Badajoz, SPAIN Collaborations Antonio Astillero, Universidad de Extremadura José
More informationDynamics of sheared gases
Computer Physics Communications 121 122 (1999) 225 230 www.elsevier.nl/locate/cpc Dynamics of sheared gases Patricio Cordero a,1, Dino Risso b a Departamento de Física, Facultad de Ciencias Físicas y Matemáticas,
More informationSurprising i Hydrodynamic d Results Discovered by means of Direct Simulation Monte Carlo
Surprising i Hydrodynamic d Results Discovered by means of Direct Simulation Monte Carlo Alejandro L. Garcia San Jose State University Lawrence Berkeley Nat. Lab. RTO-AVT-VKI Lecture Series, January 24
More informationNumerical study of the influence of gravity on the heat conductivity on the basis of kinetic theory
PHYSICS OF FLUIDS VOLUME 11, NUMBER 11 NOVEMBER 1999 Numerical study of the influence of gravity on the heat conductivity on the basis of kinetic theory Toshiyuki Doi a) Department of Applied Mathematics
More informationHydrodynamics of Inelastic Maxwell Models
Math. Model. Nat. Phenom. Vol. 6, No. 4, 2011, pp. 37-76 DOI: 10.1051/mmnp/20116403 Hydrodynamics of Inelastic Maxwell Models V. Garzó and A. Santos Departamento de Física, Universidad de Extremadura,
More informationNonlinear Poiseuille flow in a gas
PHYSICS OF FLUIDS VOLUME 10 NUMBER 4 APRIL 1998 Nonlinear Poiseuille flow in a gas Mohamed Tij and Mohamed Sabbane Département de Physique Université Moulay Ismaïl Menès Morocco Andrés Santos Departamento
More informationComparison between the Boltzmann and BGK equations for uniform shear flow. Vicente Garz6, Andr6s Santos
ELSEVIER Physica A 213 (1995) 426-434 PHYSICA Comparison between the Boltzmann and BGK equations for uniform shear flow Vicente Garz6, Andr6s Santos Departamento de Ffsica, Universidad de Extremadura,
More informationGeneralized Gas Dynamic Equations
47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exposition 5-8 January 2009 Orlando Florida AIAA 2009-672 Generalized Gas Dynamic Equations Kun Xu Mathematics Department
More informationThe velocity boundary condition at solid walls in rarefied gas simulations. Abstract
APS/123-QED The velocity boundary condition at solid walls in rarefied gas simulations Duncan A. Lockerby Department of Mechanical Engineering, King s College London, London WC2R 2LS, UK Jason M. Reese
More informationChapter 9: Differential Analysis of Fluid Flow
of Fluid Flow Objectives 1. Understand how the differential equations of mass and momentum conservation are derived. 2. Calculate the stream function and pressure field, and plot streamlines for a known
More informationREGULARIZATION AND BOUNDARY CONDITIONS FOR THE 13 MOMENT EQUATIONS
1 REGULARIZATION AND BOUNDARY CONDITIONS FOR THE 13 MOMENT EQUATIONS HENNING STRUCHTRUP ETH Zürich, Department of Materials, Polymer Physics, CH-8093 Zürich, Switzerland (on leave from University of Victoria,
More informationChapter 9: Differential Analysis
9-1 Introduction 9-2 Conservation of Mass 9-3 The Stream Function 9-4 Conservation of Linear Momentum 9-5 Navier Stokes Equation 9-6 Differential Analysis Problems Recall 9-1 Introduction (1) Chap 5: Control
More informationDivergence of the gradient expansion and the applicability of fluid dynamics Gabriel S. Denicol (IF-UFF)
Divergence of the gradient expansion and the applicability of fluid dynamics Gabriel S. Denicol (IF-UFF) arxiv:1608.07869, arxiv:1711.01657, arxiv:1709.06644 Frankfurt University 1.February.2018 Preview
More informationFluid Equations for Rarefied Gases
1 Fluid Equations for Rarefied Gases Jean-Luc Thiffeault Department of Applied Physics and Applied Mathematics Columbia University http://plasma.ap.columbia.edu/~jeanluc 23 March 2001 with E. A. Spiegel
More informationAndrés Santos Universidad de Extremadura, Badajoz, Spain (in collaboration with A. Lasanta, F. Vega Reyes, and A. Prados)
Andrés Santos Universidad de Extremadura, Badajoz, Spain (in collaboration with A. Lasanta, F. Vega Reyes, and A. Prados) º When I first met Luis (1970?) Imagine we organize a party to celebrate Luis birthday
More informationBulk equations and Knudsen layers for the regularized 13 moment equations
Continuum Mech. Thermon. 7 9: 77 89 DOI.7/s6-7-5- ORIGINAL ARTICLE Henning Struchtrup Toby Thatcher Bulk equations and Knudsen layers for the regularized 3 moment equations Received: 9 December 6 / Accepted:
More informationarxiv: v3 [cond-mat.stat-mech] 2 Dec 2015
Inelastic Maxwell models for monodisperse gas-solid flows Aleksander Kubicki Departamento de Física, Universidad de Extremadura, Universidad de Extremadura, E-0607 Badajoz, Spain arxiv:4.345v3 [cond-mat.stat-mech]
More informationFluid dynamics for a vapor-gas mixture derived from kinetic theory
IPAM Workshop II The Boltzmann Equation: DiPerna-Lions Plus 20 Years (IPAM-UCLA, April 15-17, 2009) Fluid dynamics for a vapor-gas mixture derived from kinetic theory Kazuo Aoki Department of Mechanical
More informationFluid Equations for Rarefied Gases
1 Fluid Equations for Rarefied Gases Jean-Luc Thiffeault Department of Applied Physics and Applied Mathematics Columbia University http://plasma.ap.columbia.edu/~jeanluc 21 May 2001 with E. A. Spiegel
More informationCHAPTER 9. Microscopic Approach: from Boltzmann to Navier-Stokes. In the previous chapter we derived the closed Boltzmann equation:
CHAPTER 9 Microscopic Approach: from Boltzmann to Navier-Stokes In the previous chapter we derived the closed Boltzmann equation: df dt = f +{f,h} = I[f] where I[f] is the collision integral, and we have
More informationAnisotropic fluid dynamics. Thomas Schaefer, North Carolina State University
Anisotropic fluid dynamics Thomas Schaefer, North Carolina State University Outline We wish to extract the properties of nearly perfect (low viscosity) fluids from experiments with trapped gases, colliding
More informationA wall-function approach to incorporating Knudsen-layer effects in gas micro flow simulations
A wall-function approach to incorporating Knudsen-layer effects in gas micro flow simulations D. A. Lockerby 1, J. M. Reese 2 and M. A. Gallis 3 1 Department of Mechanical Engineering, King s College London,
More informationGeneralized hydrodynamics for a Poiseuille flow: Theory and simulations
PHYSICAL REVIEW E VOLUME 5, NUMBER 1 JULY 199 Generalized hydrodynamics for a Poiseuille flow: Theory and simulations Dino Risso Departamento de Física, Facultad de Ciencias, Universidad del Bío-Bío, Casilla
More informationRevisit to Grad s Closure and Development of Physically Motivated Closure for Phenomenological High-Order Moment Model
Revisit to Grad s Closure and Development of Physically Motivated Closure for Phenomenological High-Order Moment Model R. S. Myong a and S. P. Nagdewe a a Dept. of Mechanical and Aerospace Engineering
More informationDSMC-Based Shear-Stress/Velocity-Slip Boundary Condition for Navier-Stokes Couette-Flow Simulations
DSMC-Based Shear-Stress/Velocity-Slip Boundary Condition for Navier-Stokes Couette-Flow Simulations J. R. Torczynski and M. A. Gallis Engineering Sciences Center, Sandia National Laboratories, P. O. Box
More informationComparison of the DSMC Method with an Exact Solution of the Boltzmann Equation
Comparison of the DSMC Method with an Exact Solution of the Boltzmann Equation J M Montanero and A Santos Departamento de Fisica, Universidad de Extremadura, Spain Abstract Results obtained from the Direct
More informationStructure formation. Yvonne Y. Y. Wong Max-Planck-Institut für Physik, München
Structure formation Yvonne Y. Y. Wong Max-Planck-Institut für Physik, München Structure formation... Random density fluctuations, grow via gravitational instability galaxies, clusters, etc. Initial perturbations
More informationFree cooling of a binary granular fluid of inelastic rough hard spheres Andrés Santos Universidad de Extremadura, Badajoz (Spain) In collaboration with Gilberto M. Kremer and Vicente Garzó Minimal model
More informationEl fluido de esferas penetrables. Teoría y simulación Andrés Santos Universidad de Extremadura, Badajoz, España Colaboradores: Luis Acedo (Universidad
Influence of particle roughness on some properties of granular gases Andrés Santos Universidad de Extremadura, Badajoz (Spain) In collaboration with Gilberto M. Kremer and Vicente Garzó Instituto de Ciencias
More informationBoundary Conditions for Regularized 13-Moment-Equations for Micro-Channel-Flows
Boundary Conditions for Regularized 13-Moment-Equations for Micro-Channel-Flows Manuel Torrilhon and Henning Struchtrup (007) Abstract Boundary conditions are the major obstacle in simulations based on
More informationKinetic Models for Granular Flow
Journal of Statistical Physics, Vol. 97, Nos. 12, 1999 Kinetic Models for Granular Flow J. Javier Brey, 1 James W. Dufty, 2 and Andre s Santos 3 Received September 28, 1998; final March 29, 1999 The generalization
More informationBehaviour of microscale gas flows based on a power-law free path distribution function
Behaviour of microscale gas flows based on a power-law free path distribution function Nishanth Dongari, Yonghao Zhang and Jason M Reese Department of Mechanical Engineering, University of Strathclyde,
More informationKinetic Theory for Binary Granular Mixtures at Low Density
10 Kinetic Theory for Binary Granular Mixtures at Low Density V. Garzó Departamento de Física, Universidad de Extremadura, E-06071 Badajoz, Spain vicenteg@unex.es Many features of granular media can be
More informationThe Regularized 13 Moment Equations for Rarefied and Vacuum Gas Flows
xxxx The Regularized 13 Moment Equations for Rarefied and Vacuum Gas Flows Henning Struchtrup Peyman Taheri Anirudh Rana University of Victoria, Canada Manuel Torrilhon RWTH Aachen Goal: Accurate and efficient
More informationAnalytical and numerical solutions of the regularized 13 moment equations for rarefied gases
xxxx Analytical and numerical solutions of the regularized 13 moment equations for rarefied gases Henning Struchtrup University of Victoria, Canada Manuel Torrilhon ETH Zürich Peyman Taheri University
More informationOn the modelling of isothermal gas flows at the microscale
University of Warwick institutional repository: http://go.warwick.ac.uk/wrap This paper is made available online in accordance with publisher policies. Please scroll down to view the document itself. Please
More information(Super) Fluid Dynamics. Thomas Schaefer, North Carolina State University
(Super) Fluid Dynamics Thomas Schaefer, North Carolina State University Hydrodynamics Hydrodynamics (undergraduate version): Newton s law for continuous, deformable media. Fluids: Gases, liquids, plasmas,...
More informationChapter 5. The Differential Forms of the Fundamental Laws
Chapter 5 The Differential Forms of the Fundamental Laws 1 5.1 Introduction Two primary methods in deriving the differential forms of fundamental laws: Gauss s Theorem: Allows area integrals of the equations
More informationFundamentals of Fluid Dynamics: Elementary Viscous Flow
Fundamentals of Fluid Dynamics: Elementary Viscous Flow Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pan.pl/ tzielins/ Institute of Fundamental Technological Research
More informationRegularization of the Chapman-Enskog Expansion and Its Description of Shock Structure
NASA/CR-2001-211268 ICASE Report No. 2001-39 Regularization of the Chapman-Enskog Expansion and Its Description of Shock Structure Kun Xu Hong Kong University, Kowloon, Hong Kong ICASE NASA Langley Research
More informationFluctuating Hydrodynamics and Direct Simulation Monte Carlo
Fluctuating Hydrodynamics and Direct Simulation Monte Carlo Kaushi Balarishnan Lawrence Bereley Lab John B. Bell Lawrence Bereley Lab Alesandar Donev New Yor University Alejandro L. Garcia San Jose State
More informationExplaining and modelling the rheology of polymeric fluids with the kinetic theory
Explaining and modelling the rheology of polymeric fluids with the kinetic theory Dmitry Shogin University of Stavanger The National IOR Centre of Norway IOR Norway 2016 Workshop April 25, 2016 Overview
More informationRheology of two- and three-dimensional granular mixtures under uniform shear flow: Enskog kinetic theory versus molecular dynamics simulations
Rheology of two- and three-dimensional granular mixtures under uniform shear flow: Enskog kinetic theory versus molecular dynamics simulations José María Montanero Departamento de Electrónica e Ingeniería
More informationKinetic model for the hard-sphere fluid and solid
PHYSICAL REVIEW E VOLUME 57, NUMBER FEBRUARY 1998 Kinetic model for the hard-sphere fluid and solid Andrés Santos Departamento de Física, Universidad de Extremadura, E-671 Badajoz, Spain José M. Montanero
More information1 Continuum Models Some History and Background Added Value of Continuum Equations... 6
Re g u la riz a tio n o f Gra d s 1 3 -Mo m e nt-equ a tio n s in K in e tic Ga s Th e o ry Manuel Torrilhon Department of Mathematics & Center for Computational Engineering Science RWTH Aachen University
More informationBoundary Conditions for Regularized 13-Moment-Equations for Micro-Channel-Flows
Eidgenössische Technische Hochschule Zürich Ecole polytechnique fédérale de Zurich Politecnico federale di Zurigo Swiss Federal Institute of Technology Zurich Boundary Conditions for Regularized 13-Moment-Equations
More informationFluid equations, magnetohydrodynamics
Fluid equations, magnetohydrodynamics Multi-fluid theory Equation of state Single-fluid theory Generalised Ohm s law Magnetic tension and plasma beta Stationarity and equilibria Validity of magnetohydrodynamics
More informationComparison of Kinetic Theory and Hydrodynamics for Poiseuille Flow
Journal of Statistical Phsics, Vol. 109, Nos. 3/4, November 2002 ( 2002) Comparison of Kinetic Theor and Hdrodnamics for Poiseuille Flow Yihao Zheng, 1 Alejandro L. Garcia, 2, 3 and Berni J. Alder 1, 3
More informationCHAPTER 4. Basics of Fluid Dynamics
CHAPTER 4 Basics of Fluid Dynamics What is a fluid? A fluid is a substance that can flow, has no fixed shape, and offers little resistance to an external stress In a fluid the constituent particles (atoms,
More informationHeat transfer in micro devices packaged in partial vacuum
Heat transfer in micro devices packaged in partial vacuum AnirudhRana 1, Manuel Torrilhon 2, HenningStruchtrup 1 1 DepartmentofMechanicalEngineering, University of Victoria, Victoria BC, Canada 2 DepartmentofMathematics,
More informationBoundary-Layer Theory
Hermann Schlichting Klaus Gersten Boundary-Layer Theory With contributions from Egon Krause and Herbert Oertel Jr. Translated by Katherine Mayes 8th Revised and Enlarged Edition With 287 Figures and 22
More informationEnergy production rates in fluid mixtures of inelastic rough hard spheres Andrés Santos, 1 Gilberto M. Kremer, 2 and Vicente Garzó 1 1 Universidad de Extremadura, Badajoz (Spain) 2 Universidade id d Federal
More informationMicrochannel flow in the slip regime: gas-kinetic BGK Burnett solutions
J. Fluid Mech. (2004), vol. 513, pp. 87 110. c 2004 Cambridge University Press DOI: 10.1017/S0022112004009826 Printed in the United Kingdom 87 Microchannel flow in the slip regime: gas-kinetic BGK Burnett
More informationKinetic relaxation models for reacting gas mixtures
Kinetic relaxation models for reacting gas mixtures M. Groppi Department of Mathematics and Computer Science University of Parma - ITALY Main collaborators: Giampiero Spiga, Giuseppe Stracquadanio, Univ.
More informationThe Boltzmann Equation and Its Applications
Carlo Cercignani The Boltzmann Equation and Its Applications With 42 Illustrations Springer-Verlag New York Berlin Heidelberg London Paris Tokyo CONTENTS PREFACE vii I. BASIC PRINCIPLES OF THE KINETIC
More informationRheology of two- and three-dimensional granular mixtures under uniform shear flow: Enskog kinetic theory versus molecular dynamics simulations
Granular Matter (2006) 8: 103 115 DOI 10.1007/s10035-006-0001-7 ORIGINAL PAPER José María Montanero Vicente Garzó MeheboobAlam Stefan Luding Rheology of two- and three-dimensional granular mixtures under
More informationChapter 1 Direct Modeling for Computational Fluid Dynamics
Chapter 1 Direct Modeling for Computational Fluid Dynamics Computational fluid dynamics (CFD) is a scientific discipline, which aims to capture fluid motion in a discretized space. The description of the
More informationNumerical Heat and Mass Transfer
Master Degree in Mechanical Engineering Numerical Heat and Mass Transfer 15-Convective Heat Transfer Fausto Arpino f.arpino@unicas.it Introduction In conduction problems the convection entered the analysis
More informationComputational Astrophysics
Computational Astrophysics Lecture 1: Introduction to numerical methods Lecture 2:The SPH formulation Lecture 3: Construction of SPH smoothing functions Lecture 4: SPH for general dynamic flow Lecture
More informationarxiv:cond-mat/ v2 [cond-mat.stat-mech] 21 Nov 2003
Diffusion of impurities in a granular gas Vicente Garzó Departamento de Física, Universidad de Extremadura, E-0607 Badajoz, Spain arxiv:cond-mat/0309639v [cond-mat.stat-mech] Nov 003 José María Montanero
More informationGeneralized Local Equilibrium in the Cascaded Lattice Boltzmann Method. Abstract
Accepted for publication on Physical Review E (R), code ERR1034 Generalized Local Equilibrium in the Cascaded Lattice Boltzmann Method Pietro Asinari Department of Energetics, Politecnico di Torino, Corso
More informationCHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION
CHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION 7.1 THE NAVIER-STOKES EQUATIONS Under the assumption of a Newtonian stress-rate-of-strain constitutive equation and a linear, thermally conductive medium,
More informationIntroduction. Statement of Problem. The governing equations for porous materials with Darcy s law can be written in dimensionless form as:
Symbolic Calculation of Free Convection for Porous Material of Quadratic Heat Generation in a Circular Cavity Kamyar Mansour Amirkabir University of technology, Tehran, Iran, 15875-4413 mansour@aut.ac.ir
More informationModelling and numerical methods for the diffusion of impurities in a gas
INERNAIONAL JOURNAL FOR NUMERICAL MEHODS IN FLUIDS Int. J. Numer. Meth. Fluids 6; : 6 [Version: /9/8 v.] Modelling and numerical methods for the diffusion of impurities in a gas E. Ferrari, L. Pareschi
More informationAsymptotic solution of the Boltzmann equation for the shear ow of smooth inelastic disks
Physica A 275 (2000) 483 504 www.elsevier.com/locate/physa Asymptotic solution of the Boltzmann equation for the shear ow of smooth inelastic disks V. Kumaran Department of Chemical Engineering, Indian
More informationLATTICE BOLTZMANN MODELLING OF PULSATILE FLOW USING MOMENT BOUNDARY CONDITIONS
6th European Conference on Computational Mechanics (ECCM 6) 7th European Conference on Computational Fluid Dynamics (ECFD 7) 5 June 28, Glasgow, UK LATTICE BOLTZMANN MODELLING OF PULSATILE FLOW USING MOMENT
More informationA Unified Gas-kinetic Scheme for Continuum and Rarefied Flows
A Unified Gas-kinetic Scheme for Continuum and Rarefied Flows K. Xu and J.C. Huang Mathematics Department, Hong Kong University of Science and Technology, Hong Kong Department of Merchant Marine, National
More informationSTEADY VISCOUS FLOW THROUGH A VENTURI TUBE
CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 12, Number 2, Summer 2004 STEADY VISCOUS FLOW THROUGH A VENTURI TUBE K. B. RANGER ABSTRACT. Steady viscous flow through an axisymmetric convergent-divergent
More informationContents. I Introduction 1. Preface. xiii
Contents Preface xiii I Introduction 1 1 Continuous matter 3 1.1 Molecules................................ 4 1.2 The continuum approximation.................... 6 1.3 Newtonian mechanics.........................
More informationLattice Boltzmann Method for Moving Boundaries
Lattice Boltzmann Method for Moving Boundaries Hans Groot March 18, 2009 Outline 1 Introduction 2 Moving Boundary Conditions 3 Cylinder in Transient Couette Flow 4 Collision-Advection Process for Moving
More informationStability of Shear Flow
Stability of Shear Flow notes by Zhan Wang and Sam Potter Revised by FW WHOI GFD Lecture 3 June, 011 A look at energy stability, valid for all amplitudes, and linear stability for shear flows. 1 Nonlinear
More informationShear Force in Radiometric Flows
Shear Force in Radiometric Flows Natalia E. Gimelshein, Sergey F. Gimelshein, Andrew D. Ketsdever and Nathaniel P. Selden ERC, Inc, Edwards AFB, CA 93524 University of Colorado at Colorado Springs, Colorado
More informationResearch Article. Slip flow and heat transfer through a rarefied nitrogen gas between two coaxial cylinders
Available online wwwjocprcom Journal of Chemical and Pharmaceutical Research, 216, 8(8):495-51 Research Article ISSN : 975-7384 CODEN(USA) : JCPRC5 Slip flow and heat transfer through a rarefied nitrogen
More informationTWO-DIMENSIONAL MAGMA FLOW *
Iranian Journal of Science & Technology, Transaction A, Vol. 34, No. A2 Printed in the Islamic Republic of Iran, 2010 Shiraz University TWO-DIMENSIONAL MAGMA FLOW * A. MEHMOOD 1** AND A. ALI 2 1 Department
More informationSummary We develop an unconditionally stable explicit particle CFD scheme: Boltzmann Particle Hydrodynamics (BPH)
Summary Steps in BPH Space is divided into Cartesian cells Finite number of particles ~10 6-10 7 Particles fly freely between t n and t n+1 Mass, momentum and total energy are conserved Relax into a (LTE)
More informationNotes 4: Differential Form of the Conservation Equations
Low Speed Aerodynamics Notes 4: Differential Form of the Conservation Equations Deriving Conservation Equations From the Laws of Physics Physical Laws Fluids, being matter, must obey the laws of Physics.
More informationFailures of the Burnett and super-burnett equations in steady state processes
Continuum Mech. Thermodyn. (2005) 7: 43 50 Digital Object Identifier (DOI) 0.007/s006-004-086-0 Original article Failures of the Burnett and super-burnett equations in steady state processes H. Struchtrup
More informationBurnett description for plane Poiseuille flow
PHYSICAL REVIEW E VOLUME 60, NUMBER 4 OCTOBER 1999 Burnett description for plane Poiseuille flow F. J. Uribe Physics Department, Universidad Autónoma Metropolitana Iztapalapa, P.O. Box 55-534, 09340 México
More informationKinetic Theory of a Confined Quasi-Two-Dimensional Gas of Hard Spheres. J. Javier Brey *, Vicente Buzón, Maria Isabel García de Soria and Pablo Maynar
entropy Article Kinetic Theory of a Confined Quasi-Two-Dimensional Gas of Hard Spheres J. Javier Brey *, Vicente Buzón, Maria Isabel García de Soria and Pablo Maynar Física Teórica, Universidad de Sevilla,
More informationEvaporation-driven vapour microflows: analytical solutions from moment methods
J. Fluid Mech. (218), vol. 841, pp. 962 988. c Cambridge University Press 218 This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4./),
More informationA Comparative Study Between Cubic and Ellipsoidal Fokker-Planck Kinetic Models
A Comparative Study Between Cubic and Ellipsoidal Fokker-Planck Kinetic Models Eunji Jun German Aerospace Center (DLR), 3773 Göttingen, Germany M. Hossein Gorji Computational Mathematics and Simulation
More informationLattice Boltzmann Method for Fluid Simulations
Lattice Boltzmann Method for Fluid Simulations Yuanxun Bill Bao & Justin Meskas April 14, 2011 1 Introduction In the last two decades, the Lattice Boltzmann method (LBM) has emerged as a promising tool
More informationOscillatory shear-driven gas flows in the transition and free-molecular-flow regimes
PHYSICS OF FLUIDS 17, 100611 005 Oscillatory shear-driven gas flows in the transition and free-molecular-flow regimes Nicolas G. Hadjiconstantinou Department of Mechanical Engineering, Massachusetts Institute
More informationHilbert Sixth Problem
Academia Sinica, Taiwan Stanford University Newton Institute, September 28, 2010 : Mathematical Treatment of the Axioms of Physics. The investigations on the foundations of geometry suggest the problem:
More informationDSMC simulations of thermal escape
DSMC simulations of thermal escape Alexey N. Volkov, R.E. Johnson, O.J. Tucker, J.T. Erwin Materials Science & Engineering University of Virginia, USA Financial support is provided by NASA through Planetary
More informationLes Houches School of Foam: Rheology of Complex Fluids
Les Houches School of Foam: Rheology of Complex Fluids Andrew Belmonte The W. G. Pritchard Laboratories Department of Mathematics, Penn State University 1 Fluid Dynamics (tossing a coin) Les Houches Winter
More informationarxiv:comp-gas/ v1 28 Apr 1993
Lattice Boltzmann Thermohydrodynamics arxiv:comp-gas/9304006v1 28 Apr 1993 F. J. Alexander, S. Chen and J. D. Sterling Center for Nonlinear Studies and Theoretical Division Los Alamos National Laboratory
More informationAccurate representation of velocity space using truncated Hermite expansions.
Accurate representation of velocity space using truncated Hermite expansions. Joseph Parker Oxford Centre for Collaborative Applied Mathematics Mathematical Institute, University of Oxford Wolfgang Pauli
More informationKINETIC THEORY OF GRANULAR SOME APPLICATIONS
KINETIC THEORY OF GRANULAR BINARY MIXTURES: SOME APPLICATIONS Vicente Garzó Departamento t de Física, Universidad id dde Extremadura, Badajoz, Spain OUTLINE 1. Introduction 2. Granular binary mixtures.
More information1.3 Molecular Level Presentation
1.3.1 Introduction A molecule is the smallest chemical unit of a substance that is capable of stable, independent existence. Not all substances are composed of molecules. Some substances are composed of
More informationPhysical Modeling of Multiphase flow. Boltzmann method
with lattice Boltzmann method Exa Corp., Burlington, MA, USA Feburary, 2011 Scope Scope Re-examine the non-ideal gas model in [Shan & Chen, Phys. Rev. E, (1993)] from the perspective of kinetic theory
More informationThe Polymers Tug Back
Tugging at Polymers in Turbulent Flow The Polymers Tug Back Jean-Luc Thiffeault http://plasma.ap.columbia.edu/ jeanluc Department of Applied Physics and Applied Mathematics Columbia University Tugging
More informationLattice Boltzmann Method
3 Lattice Boltzmann Method 3.1 Introduction The lattice Boltzmann method is a discrete computational method based upon the lattice gas automata - a simplified, fictitious molecular model. It consists of
More informationCHAPTER 2 THERMAL EFFECTS IN STOKES SECOND PROBLEM FOR UNSTEADY MICROPOLAR FLUID FLOW THROUGH A POROUS
CHAPTER THERMAL EFFECTS IN STOKES SECOND PROBLEM FOR UNSTEADY MICROPOLAR FLUID FLOW THROUGH A POROUS MEDIUM. Introduction The theory of micropolar fluids introduced by Eringen [34,35], deals with a class
More informationEPJ E. Soft Matter and Biological Physics. EPJ.org. Segregation by thermal diffusion in moderately dense granular mixtures
EPJ E Soft Matter and Biological Physics EPJ.org your physics journal Eur. Phys. J. E 9, 61 74 009) DOI: 10.1140/epje/i009-10488-4 Segregation by thermal diffusion in moderately dense granular mixtures
More informationLow Variance Particle Simulations of the Boltzmann Transport Equation for the Variable Hard Sphere Collision Model
Low Variance Particle Simulations of the Boltzmann Transport Equation for the Variable Hard Sphere Collision Model G. A. Radtke, N. G. Hadjiconstantinou and W. Wagner Massachusetts Institute of Technology,
More informationLecture 5: Kinetic theory of fluids
Lecture 5: Kinetic theory of fluids September 21, 2015 1 Goal 2 From atoms to probabilities Fluid dynamics descrines fluids as continnum media (fields); however under conditions of strong inhomogeneities
More informationStochastic Particle Methods for Rarefied Gases
CCES Seminar WS 2/3 Stochastic Particle Methods for Rarefied Gases Julian Köllermeier RWTH Aachen University Supervisor: Prof. Dr. Manuel Torrilhon Center for Computational Engineering Science Mathematics
More informationScaling Parameters in Rarefied Flow and the Breakdown of the Navier-Stokes Equations Mechanical Engineering Research Report No: 2004/09
Scaling Parameters in Rarefied Flow and the Breakdown of the Navier-Stokes Equations Mechanical Engineering Research Report No: 2004/09 Michael Macrossan, Centre for Hypersonics, University of Queensland
More information