Level Set Tumor Growth Model
|
|
- Stewart Fowler
- 6 years ago
- Views:
Transcription
1 Level Set Tumor Growth Model Andrew Nordquist and Rakesh Ranjan, PhD University of Texas, San Antonio July 29, 2013 Andrew Nordquist and Rakesh Ranjan, PhD (University Level Set of Texas, TumorSan Growth Antonio) Model July 29, / 14
2 Level Set vs Mixture Theory The Level Set model is based on Mixture Theory. While Mixture Theory has the concept of Volume Fractions, κ n α = α=1 κ α=1 ρ α = 1, (1) ραr Level Set says, We have volume fractions, two solids and one liquid-dissolved nutrient. The two solids do not mix. The nutrient is consumed by one of the two solids. One solid is Tumor tissue, the other is Host (or normal) tissue. Yes, host tissue consumes nutrient, but at a much lower rate than tumor tissue. Thus, we ignore homeostasis for this model. Nordquist & Ranjan (UTSA) Level Set Tumor Growth Model July 29, / 14
3 Level Set departure from Mixture Theory Mixture Theory concerns itself with Balance of Mass and Momentum equations of Continuum Mechanics: Balance of Mass - constituent s mass rate ˆρ α, and velocity v α n α t + (n α v α ) = ˆρ α ρ αr (2) Balance of Momentum - Cauchy stress tensor T α, body force b α ( ) T α + ρ αr n α b α dvα + ˆp α ˆρ α v α = 0 (3) dt Balance of Energy (1st Law of Thermo) is combined with Entropy Inequality (2nd Law of Thermo) to help construct constitutive equations to go with above equations. Nordquist & Ranjan (UTSA) Level Set Tumor Growth Model July 29, / 14
4 Level Set departure from Mixture Theory, (continued) We temporarily shift to 2 dimensions, without loss of generality. Assume the two solid tissues act as fluids, with only one volume fraction (n [0, 1]), and incompressible Navier-Stokes behavior: v x x + v y y = 0 (4) Here is x-component of momentum balance equations, ( v x t + v v x x x + v v x 2 ) y y + ν v x x v x y 2 = P x + ρg x + kcn (5) and y-component of momentum balance equations, ( v y t + v v y x x + v v y 2 y y + ν v y x 2 ) + 2 v y y 2 = P y + ρg y + kcn (6) Nordquist & Ranjan (UTSA) Level Set Tumor Growth Model July 29, / 14
5 Level Set Methodology: Governing Equations for Nutrient and Interface The Interface is defined by the volume fraction of tumor tissue, n (AKA level set ). Where n = 1, we have tumor tissue, and where n = 0 we have normal host tissue. Equation to move the interface with velocity field v: n t + v n = γ [ ( ɛ n n(1 n) n )] n Evolution equation for nutrient as a dissolved species ( c t + v c x x + v c 2 y y D v x x 2 (7) ) + 2 v y y 2 = ṙ c (c, n) (8) Nordquist & Ranjan (UTSA) Level Set Tumor Growth Model July 29, / 14
6 Level Set Methodology: Description of Symbols n if n = 1, tumor tissue; if n = 0, host tissue v x, v y x- and y- components of tumor tissue velocity field v P pressure ν the viscosity tensor ρ the (tumor) tissue density g x, g y x- and y- components of the gravitational field g (negligible) c concentration of a dissolved nutrient species, such as oxygen k rate of consumption of the nutrient D the diffusivity tensor ɛ thickness of the interface region γ amount of reinitialization or stabilization of the level set function Nordquist & Ranjan (UTSA) Level Set Tumor Growth Model July 29, / 14
7 Level Set - COMSOL Implementation Phase initialization step - GI is reciprocal of initial interface distance Time Dependent step GI GI + σ w GI ( GI ) = (1 + 2σ w )GI 4 (9) I w = 1 GI I ret 1 (10) ρ v [ ] t + ρ(v )v = ρi + µ( v + ( v) T ) + ρg + f (11) v = 0, n t [ + v n = γ ɛ ls n n(1 n) n ] n (12) Nordquist & Ranjan (UTSA) Level Set Tumor Growth Model July 29, / 14
8 Level Set - COMSOL Model Setup Length scale factor: 1 meter in model is 10 microns (10 5 m) in reality Time scale factor: 1 second in model is 1 day (86400 s) in reality Model is 100µm on each side capillary structure is 20µm in diameter, 10µm away from tumor tumor starts out at 20µm in diameter intent is to see tumor grow, and prefer to grow around capillary, forming a tumor cord (... Show Pictures Now...) Nordquist & Ranjan (UTSA) Level Set Tumor Growth Model July 29, / 14
9 What needs to happen now to the Level Set model? Calculate scaling parameters for: density, viscosity, velocity, etc. Find a transform to convert viscosity to some kind of tissue hardness unit Make model more realistic by converting real parameters to scaled parameters Nordquist & Ranjan (UTSA) Level Set Tumor Growth Model July 29, / 14
10 That s all, Folks! Thank You! Any Questions? Nordquist & Ranjan (UTSA) Level Set Tumor Growth Model July 29, / 14
11 Thermodynamics: First Law Consider our mixture in a 3D volume Ω, and an increment of that volume, dv. Ω s surface is Ω. This is a widely accepted mathematical form for the First Law of Thermodynamics, applicable to each constituent (α) in our mixture: ρ εα t T α : L α ρ α ˆρ c + q α = 0 (13) ρ α the density of the constituent ε α internal energy of the constituent T α : L α stress power of the constituent ˆρ c reduction of energy due to consumption of c internal to dv q α heat flux through dv T α Cauchy stress tensor L α velocity gradient = D α (symmetric) + W α (skew symmetric) Nordquist & Ranjan (UTSA) Level Set Tumor Growth Model July 29, / 14
12 First law of thermodynamics, continued Since the Cauchy stress tensor is symmetric, the stress power simplifies to T : L = T : D = tr(td) (14) So the First Law simplifies to ρ ε t tr(td) ρˆρc + q = 0 (15) Nordquist & Ranjan (UTSA) Level Set Tumor Growth Model July 29, / 14
13 Thermodynamics: Second Law The mathematical form for the Second Law of Thermodynamics, (AKA Entropy Inequality) is: d ρη dv ρ ˆρc dt Ω Ω θ dv q ˆn da, where (16) Ω θ η the specific entropy of the constituent ˆn the unit vector normal to S. The divergence theorem changes our surface integral to a volume integral. Due to the arbitrary nature of the volumes under consideration, we may then discard the volume integrals after converting the surface integral, leaving the integrands: ρ η ρˆρ c θ + q θ q θ θ 2 0 (17) Nordquist & Ranjan (UTSA) Level Set Tumor Growth Model July 29, / 14
14 Second law of thermodynamics, continued After multiplying Equation (17) through by θ, we subtract Equation (13) from it to get: ρ ηθ ρ ε t q θ + T : D 0 (18) θ We now introduce the Helmholtz Free Energy, defined as ψ = ε ηθ (19) Differentiate with respect to time and rearrange to get, ηθ = ε t η θ ψ t (20) Nordquist & Ranjan (UTSA) Level Set Tumor Growth Model July 29, / 14
15 The Entropy Inequality Replace the ηθ term in Equation (18) by the right-hand side of the last Equation, and we get: ρ ε t ρη θ ρ ψ t ρ ε t q θ + T : D 0 (21) θ The terms involving ε cancel each other, and this equation simplifies to T : D ρη θ ρ ψ t q θ 0 (22) θ This is known as the modified Clausius-Duhem inequality. Nordquist & Ranjan (UTSA) Level Set Tumor Growth Model July 29, / 14
Fundamentals 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 informationChapter 2: Fluid Dynamics Review
7 Chapter 2: Fluid Dynamics Review This chapter serves as a short review of basic fluid mechanics. We derive the relevant transport equations (or conservation equations), state Newton s viscosity law leading
More informationChapter 1. Governing Equations of GFD. 1.1 Mass continuity
Chapter 1 Governing Equations of GFD The fluid dynamical governing equations consist of an equation for mass continuity, one for the momentum budget, and one or more additional equations to account for
More informationOn pore fluid pressure and effective solid stress in the mixture theory of porous media
On pore fluid pressure and effective solid stress in the mixture theory of porous media I-Shih Liu Abstract In this paper we briefly review a typical example of a mixture of elastic materials, in particular,
More informationNavier-Stokes Equation: Principle of Conservation of Momentum
Navier-tokes Equation: Principle of Conservation of Momentum R. hankar ubramanian Department of Chemical and Biomolecular Engineering Clarkson University Newton formulated the principle of conservation
More information1 Introduction to Governing Equations 2 1a Methodology... 2
Contents 1 Introduction to Governing Equations 2 1a Methodology............................ 2 2 Equation of State 2 2a Mean and Turbulent Parts...................... 3 2b Reynolds Averaging.........................
More informationComputational Modelling of Mechanics and Transport in Growing Tissue
Computational Modelling of Mechanics and Transport in Growing Tissue H. Narayanan, K. Garikipati, E. M. Arruda & K. Grosh University of Michigan Eighth U.S. National Congress on Computational Mechanics
More informationChapter 2. General concepts. 2.1 The Navier-Stokes equations
Chapter 2 General concepts 2.1 The Navier-Stokes equations The Navier-Stokes equations model the fluid mechanics. This set of differential equations describes the motion of a fluid. In the present work
More informationShell Balances in Fluid Mechanics
Shell Balances in Fluid Mechanics R. Shankar Subramanian Department of Chemical and Biomolecular Engineering Clarkson University When fluid flow occurs in a single direction everywhere in a system, shell
More informationThe Navier-Stokes Equations
s University of New Hampshire February 22, 202 and equations describe the non-relativistic time evolution of mass and momentum in fluid substances. mass density field: ρ = ρ(t, x, y, z) velocity field:
More information4 Constitutive Theory
ME338A CONTINUUM MECHANICS lecture notes 13 Tuesday, May 13, 2008 4.1 Closure Problem In the preceding chapter, we derived the fundamental balance equations: Balance of Spatial Material Mass ρ t + ρ t
More informationQuick Recapitulation of Fluid Mechanics
Quick Recapitulation of Fluid Mechanics Amey Joshi 07-Feb-018 1 Equations of ideal fluids onsider a volume element of a fluid of density ρ. If there are no sources or sinks in, the mass in it will change
More informationLecture 8: Tissue Mechanics
Computational Biology Group (CoBi), D-BSSE, ETHZ Lecture 8: Tissue Mechanics Prof Dagmar Iber, PhD DPhil MSc Computational Biology 2015/16 7. Mai 2016 2 / 57 Contents 1 Introduction to Elastic Materials
More informationNotes on Entropy Production in Multicomponent Fluids
Notes on Entropy Production in Multicomponent Fluids Robert F. Sekerka Updated January 2, 2001 from July 1993 Version Introduction We calculate the entropy production in a multicomponent fluid, allowing
More informationSummary of the Equations of Fluid Dynamics
Reference: Summary of the Equations of Fluid Dynamics Fluid Mechanics, L.D. Landau & E.M. Lifshitz 1 Introduction Emission processes give us diagnostics with which to estimate important parameters, such
More informationME338A CONTINUUM MECHANICS
ME338A CONTINUUM MECHANICS lecture notes 10 thursday, february 4th, 2010 Classical continuum mechanics of closed systems in classical closed system continuum mechanics (here), r = 0 and R = 0, such that
More informationA SHORT INTRODUCTION TO TWO-PHASE FLOWS Two-phase flows balance equations
A SHORT INTRODUCTION TO TWO-PHASE FLOWS Two-phase flows balance equations Hervé Lemonnier DM2S/STMF/LIEFT, CEA/Grenoble, 38054 Grenoble Cedex 9 Ph. +33(0)4 38 78 45 40 herve.lemonnier@cea.fr, herve.lemonnier.sci.free.fr/tpf/tpf.htm
More informationElmer :Heat transfert with phase change solid-solid in transient problem Application to silicon properties. SIF file : phasechange solid-solid
Elmer :Heat transfert with phase change solid-solid in transient problem Application to silicon properties 3 6 1. Tb=1750 [K] 2 & 5. q=-10000 [W/m²] 0,1 1 Ω1 4 Ω2 7 3 & 6. α=15 [W/(m²K)] Text=300 [K] 4.
More informationReceived: 21 January 2003 Accepted: 13 March 2003 Published: 25 February 2004
Nonlinear Processes in Geophysics (2004) 11: 75 82 SRef-ID: 1607-7946/npg/2004-11-75 Nonlinear Processes in Geophysics European Geosciences Union 2004 A mixture theory for geophysical fluids A. C. Eringen
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 informationChapter 1. Continuum mechanics review. 1.1 Definitions and nomenclature
Chapter 1 Continuum mechanics review We will assume some familiarity with continuum mechanics as discussed in the context of an introductory geodynamics course; a good reference for such problems is Turcotte
More informationCH.9. CONSTITUTIVE EQUATIONS IN FLUIDS. Multimedia Course on Continuum Mechanics
CH.9. CONSTITUTIVE EQUATIONS IN FLUIDS Multimedia Course on Continuum Mechanics Overview Introduction Fluid Mechanics What is a Fluid? Pressure and Pascal s Law Constitutive Equations in Fluids Fluid Models
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 informationIntroduction to Fluid Mechanics
Introduction to Fluid Mechanics Tien-Tsan Shieh April 16, 2009 What is a Fluid? The key distinction between a fluid and a solid lies in the mode of resistance to change of shape. The fluid, unlike the
More informationPhysical Conservation and Balance Laws & Thermodynamics
Chapter 4 Physical Conservation and Balance Laws & Thermodynamics The laws of classical physics are, for the most part, expressions of conservation or balances of certain quantities: e.g. mass, momentum,
More informationGetting started: CFD notation
PDE of p-th order Getting started: CFD notation f ( u,x, t, u x 1,..., u x n, u, 2 u x 1 x 2,..., p u p ) = 0 scalar unknowns u = u(x, t), x R n, t R, n = 1,2,3 vector unknowns v = v(x, t), v R m, m =
More informationConservation of Mass. Computational Fluid Dynamics. The Equations Governing Fluid Motion
http://www.nd.edu/~gtryggva/cfd-course/ http://www.nd.edu/~gtryggva/cfd-course/ Computational Fluid Dynamics Lecture 4 January 30, 2017 The Equations Governing Fluid Motion Grétar Tryggvason Outline Derivation
More informationNon-linear Wave Propagation and Non-Equilibrium Thermodynamics - Part 3
Non-linear Wave Propagation and Non-Equilibrium Thermodynamics - Part 3 Tommaso Ruggeri Department of Mathematics and Research Center of Applied Mathematics University of Bologna January 21, 2017 ommaso
More information2. FLUID-FLOW EQUATIONS SPRING 2019
2. FLUID-FLOW EQUATIONS SPRING 2019 2.1 Introduction 2.2 Conservative differential equations 2.3 Non-conservative differential equations 2.4 Non-dimensionalisation Summary Examples 2.1 Introduction Fluid
More informationFree energy concept Free energy approach LBM implementation Parameters
BINARY LIQUID MODEL A. Kuzmin J. Derksen Department of Chemical and Materials Engineering University of Alberta Canada August 22,2011 / LBM Workshop OUTLINE 1 FREE ENERGY CONCEPT 2 FREE ENERGY APPROACH
More information- Marine Hydrodynamics. Lecture 4. Knowns Equations # Unknowns # (conservation of mass) (conservation of momentum)
2.20 - Marine Hydrodynamics, Spring 2005 Lecture 4 2.20 - Marine Hydrodynamics Lecture 4 Introduction Governing Equations so far: Knowns Equations # Unknowns # density ρ( x, t) Continuity 1 velocities
More informationElectromagnetic energy and momentum
Electromagnetic energy and momentum Conservation of energy: the Poynting vector In previous chapters of Jackson we have seen that the energy density of the electric eq. 4.89 in Jackson and magnetic eq.
More information6.2 Governing Equations for Natural Convection
6. Governing Equations for Natural Convection 6..1 Generalized Governing Equations The governing equations for natural convection are special cases of the generalized governing equations that were discussed
More informationLecture: Wave-induced Momentum Fluxes: Radiation Stresses
Chapter 4 Lecture: Wave-induced Momentum Fluxes: Radiation Stresses Here we derive the wave-induced depth-integrated momentum fluxes, otherwise known as the radiation stress tensor S. These are the 2nd-order
More informationHydrodynamics. Stefan Flörchinger (Heidelberg) Heidelberg, 3 May 2010
Hydrodynamics Stefan Flörchinger (Heidelberg) Heidelberg, 3 May 2010 What is Hydrodynamics? Describes the evolution of physical systems (classical or quantum particles, fluids or fields) close to thermal
More informationMath background. Physics. Simulation. Related phenomena. Frontiers in graphics. Rigid fluids
Fluid dynamics Math background Physics Simulation Related phenomena Frontiers in graphics Rigid fluids Fields Domain Ω R2 Scalar field f :Ω R Vector field f : Ω R2 Types of derivatives Derivatives measure
More informationExercise 5: Exact Solutions to the Navier-Stokes Equations I
Fluid Mechanics, SG4, HT009 September 5, 009 Exercise 5: Exact Solutions to the Navier-Stokes Equations I Example : Plane Couette Flow Consider the flow of a viscous Newtonian fluid between two parallel
More informationSimulation of Thermomechanical Couplings of Viscoelastic Materials
Simulation of Thermomechanical Couplings of Viscoelastic Materials Frank Neff 1, Thomas Miquel 2, Michael Johlitz 1, Alexander Lion 1 1 Institute of Mechanics Faculty for Aerospace Engineering Universität
More informationTutorial School on Fluid Dynamics: Aspects of Turbulence Session I: Refresher Material Instructor: James Wallace
Tutorial School on Fluid Dynamics: Aspects of Turbulence Session I: Refresher Material Instructor: James Wallace Adapted from Publisher: John S. Wiley & Sons 2002 Center for Scientific Computation and
More informationmeters, we can re-arrange this expression to give
Turbulence When the Reynolds number becomes sufficiently large, the non-linear term (u ) u in the momentum equation inevitably becomes comparable to other important terms and the flow becomes more complicated.
More informationINDIAN INSTITUTE OF TECHNOOGY, KHARAGPUR Date: -- AN No. of Students: 5 Sub. No.: ME64/ME64 Time: Hours Full Marks: 6 Mid Autumn Semester Examination Sub. Name: Convective Heat and Mass Transfer Instructions:
More informationComputational Fluid Dynamics 2
Seite 1 Introduction Computational Fluid Dynamics 11.07.2016 Computational Fluid Dynamics 2 Turbulence effects and Particle transport Martin Pietsch Computational Biomechanics Summer Term 2016 Seite 2
More informationA Thermo-Hydro-Mechanical Damage Model for Unsaturated Geomaterials
A Thermo-Hydro-Mechanical Damage Model for Unsaturated Geomaterials Chloé Arson ALERT PhD Prize PhD Advisor : Behrouz Gatmiri Paris-Est University, U.R. Navier, geotechnical group (CERMES) This research
More informationThe Shallow Water Equations
The Shallow Water Equations Clint Dawson and Christopher M. Mirabito Institute for Computational Engineering and Sciences University of Texas at Austin clint@ices.utexas.edu September 29, 2008 The Shallow
More informationRANS Equations in Curvilinear Coordinates
Appendix C RANS Equations in Curvilinear Coordinates To begin with, the Reynolds-averaged Navier-Stokes RANS equations are presented in the familiar vector and Cartesian tensor forms. Each term in the
More informationA solid-fluid mixture theory of porous media
A solid-luid mixture theory o porous media I-Shih Liu Instituto de Matemática, Universidade Federal do Rio de Janeiro Abstract The theories o mixtures in the ramework o continuum mechanics have been developed
More informationEULERIAN DERIVATIONS OF NON-INERTIAL NAVIER-STOKES EQUATIONS
EULERIAN DERIVATIONS OF NON-INERTIAL NAVIER-STOKES EQUATIONS ML Combrinck, LN Dala Flamengro, a div of Armscor SOC Ltd & University of Pretoria, Council of Scientific and Industrial Research & University
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 informationCandidates must show on each answer book the type of calculator used. Log Tables, Statistical Tables and Graph Paper are available on request.
UNIVERSITY OF EAST ANGLIA School of Mathematics Spring Semester Examination 2004 FLUID DYNAMICS Time allowed: 3 hours Attempt Question 1 and FOUR other questions. Candidates must show on each answer book
More informationChristel Hohenegger A simple model for ketchup-like liquid, its numerical challenges and limitations April 7, 2011
Notes by: Andy Thaler Christel Hohenegger A simple model for ketchup-like liquid, its numerical challenges and limitations April 7, 2011 Many complex fluids are shear-thinning. Such a fluid has a shear
More informationONSAGER S VARIATIONAL PRINCIPLE AND ITS APPLICATIONS. Abstract
ONSAGER S VARIAIONAL PRINCIPLE AND IS APPLICAIONS iezheng Qian Department of Mathematics, Hong Kong University of Science and echnology, Clear Water Bay, Kowloon, Hong Kong (Dated: April 30, 2016 Abstract
More informationDifferential relations for fluid flow
Differential relations for fluid flow In this approach, we apply basic conservation laws to an infinitesimally small control volume. The differential approach provides point by point details of a flow
More informationTumour angiogenesis as a chemo-mechanical surface instability. Supplementary Information Chiara Giverso 1, Pasquale Ciarletta 2
Tumour angiogenesis as a chemo-mechanical surface instability 1 Theoretical Derivation Supplementary Information Chiara Giverso 1, Pasquale Ciarletta 2 In order to close the description of the process
More informationSupplement to Molecular Gas TitleBirkhäuser, Boston, 007 Dynamic Version Authors Sone, Yoshio Citation Yoshio Sone. 008 Issue Date 008-09-0 URL http://hdl.handle.net/433/66098 Right Type Book Textversion
More informationA Study on Numerical Solution to the Incompressible Navier-Stokes Equation
A Study on Numerical Solution to the Incompressible Navier-Stokes Equation Zipeng Zhao May 2014 1 Introduction 1.1 Motivation One of the most important applications of finite differences lies in the field
More informationBasic concepts in viscous flow
Élisabeth Guazzelli and Jeffrey F. Morris with illustrations by Sylvie Pic Adapted from Chapter 1 of Cambridge Texts in Applied Mathematics 1 The fluid dynamic equations Navier-Stokes equations Dimensionless
More informationFluid Dynamics Exercises and questions for the course
Fluid Dynamics Exercises and questions for the course January 15, 2014 A two dimensional flow field characterised by the following velocity components in polar coordinates is called a free vortex: u r
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 informationViscous Fluids. Amanda Meier. December 14th, 2011
Viscous Fluids Amanda Meier December 14th, 2011 Abstract Fluids are represented by continuous media described by mass density, velocity and pressure. An Eulerian description of uids focuses on the transport
More informationReview of fluid dynamics
Chapter 2 Review of fluid dynamics 2.1 Preliminaries ome basic concepts: A fluid is a substance that deforms continuously under stress. A Material olume is a tagged region that moves with the fluid. Hence
More informationCHAPTER 8 ENTROPY GENERATION AND TRANSPORT
CHAPTER 8 ENTROPY GENERATION AND TRANSPORT 8.1 CONVECTIVE FORM OF THE GIBBS EQUATION In this chapter we will address two questions. 1) How is Gibbs equation related to the energy conservation equation?
More informationEuler equation and Navier-Stokes equation
Euler equation and Navier-Stokes equation WeiHan Hsiao a a Department of Physics, The University of Chicago E-mail: weihanhsiao@uchicago.edu ABSTRACT: This is the note prepared for the Kadanoff center
More information( ) Notes. Fluid mechanics. Inviscid Euler model. Lagrangian viewpoint. " = " x,t,#, #
Notes Assignment 4 due today (when I check email tomorrow morning) Don t be afraid to make assumptions, approximate quantities, In particular, method for computing time step bound (look at max eigenvalue
More informationLecture 3: 1. Lecture 3.
Lecture 3: 1 Lecture 3. Lecture 3: 2 Plan for today Summary of the key points of the last lecture. Review of vector and tensor products : the dot product (or inner product ) and the cross product (or vector
More informationTopic 5.9: Divergence and The Divergence Theorem
Math 275 Notes (Ultman) Topic 5.9: Divergence and The Divergence Theorem Textbook ection: 16.9 From the Toolbox (what you need from previous classes): Computing partial derivatives. Computing the dot product.
More informationPEAT SEISMOLOGY Lecture 3: The elastic wave equation
PEAT8002 - SEISMOLOGY Lecture 3: The elastic wave equation Nick Rawlinson Research School of Earth Sciences Australian National University Equation of motion The equation of motion can be derived by considering
More informationEntropy generation and transport
Chapter 7 Entropy generation and transport 7.1 Convective form of the Gibbs equation In this chapter we will address two questions. 1) How is Gibbs equation related to the energy conservation equation?
More informationAdaptive C1 Macroelements for Fourth Order and Divergence-Free Problems
Adaptive C1 Macroelements for Fourth Order and Divergence-Free Problems Roy Stogner Computational Fluid Dynamics Lab Institute for Computational Engineering and Sciences University of Texas at Austin March
More informationNeeds work : define boundary conditions and fluxes before, change slides Useful definitions and conservation equations
Needs work : define boundary conditions and fluxes before, change slides 1-2-3 Useful definitions and conservation equations Turbulent Kinetic energy The fluxes are crucial to define our boundary conditions,
More informationA REDUCED-ORDER METHANE-AIR COMBUSTION MECHANISM THAT SATISFIES THE DIFFERENTIAL ENTROPY INEQUALITY
THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Special Issue/2018, pp. 285 290 A REDUCED-ORDER METHANE-AIR COMBUSTION MECHANISM THAT SATISFIES THE DIFFERENTIAL
More informationSoft Bodies. Good approximation for hard ones. approximation breaks when objects break, or deform. Generalization: soft (deformable) bodies
Soft-Body Physics Soft Bodies Realistic objects are not purely rigid. Good approximation for hard ones. approximation breaks when objects break, or deform. Generalization: soft (deformable) bodies Deformed
More informationContinuum mechanics V. Constitutive equations. 1. Constitutive equation: definition and basic axioms
Continuum mechanics office Math 0.107 ales.janka@unifr.ch http://perso.unifr.ch/ales.janka/mechanics Mars 16, 2011, Université de Fribourg 1. Constitutive equation: definition and basic axioms Constitutive
More informationV (r,t) = i ˆ u( x, y,z,t) + ˆ j v( x, y,z,t) + k ˆ w( x, y, z,t)
IV. DIFFERENTIAL RELATIONS FOR A FLUID PARTICLE This chapter presents the development and application of the basic differential equations of fluid motion. Simplifications in the general equations and common
More informationn v molecules will pass per unit time through the area from left to
3 iscosity and Heat Conduction in Gas Dynamics Equations of One-Dimensional Gas Flow The dissipative processes - viscosity (internal friction) and heat conduction - are connected with existence of molecular
More informationTABLE OF CONTENTS CHAPTER TITLE PAGE
v TABLE OF CONTENTS CHAPTER TITLE PAGE TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES LIST OF SYMBOLS LIST OF APPENDICES v viii ix xii xiv CHAPTER 1 INTRODUCTION 1.1 Introduction 1 1.2 Literature Review
More informationModeling as a tool for understanding the MEA. Henrik Ekström Utö Summer School, June 22 nd 2010
Modeling as a tool for understanding the MEA Henrik Ekström Utö Summer School, June 22 nd 2010 COMSOL Multiphysics and Electrochemistry Modeling The software is based on the finite element method A number
More informationWhere does Bernoulli's Equation come from?
Where does Bernoulli's Equation come from? Introduction By now, you have seen the following equation many times, using it to solve simple fluid problems. P ρ + v + gz = constant (along a streamline) This
More informationNonlinear elasticity and gels
Nonlinear elasticity and gels M. Carme Calderer School of Mathematics University of Minnesota New Mexico Analysis Seminar New Mexico State University April 4-6, 2008 1 / 23 Outline Balance laws for gels
More informationFluid Dynamics and Balance Equations for Reacting Flows
Fluid Dynamics and Balance Equations for Reacting Flows Combustion Summer School 2018 Prof. Dr.-Ing. Heinz Pitsch Balance Equations Basics: equations of continuum mechanics balance equations for mass and
More informationDiffusive Transport Enhanced by Thermal Velocity Fluctuations
Diffusive Transport Enhanced by Thermal Velocity Fluctuations Aleksandar Donev 1 Courant Institute, New York University & Alejandro L. Garcia, San Jose State University John B. Bell, Lawrence Berkeley
More informationFrom the last time, we ended with an expression for the energy equation. u = ρg u + (τ u) q (9.1)
Lecture 9 9. Administration None. 9. Continuation of energy equation From the last time, we ended with an expression for the energy equation ρ D (e + ) u = ρg u + (τ u) q (9.) Where ρg u changes in potential
More informationUNIVERSITY of LIMERICK
UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH Faculty of Science and Engineering END OF SEMESTER ASSESSMENT PAPER MODULE CODE: MA4607 SEMESTER: Autumn 2012-13 MODULE TITLE: Introduction to Fluids DURATION OF
More informationA Thermomechanical Model of Gels
A Thermomechanical Model of Gels A DISSERTATION SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA BY Minsu Kim IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
More informationDynamics of the Mantle and Lithosphere ETH Zürich Continuum Mechanics in Geodynamics: Equation cheat sheet
Dynamics of the Mantle and Lithosphere ETH Zürich Continuum Mechanics in Geodynamics: Equation cheat sheet or all equations you will probably ever need Definitions 1. Coordinate system. x,y,z or x 1,x
More informationMAE 101A. Homework 7 - Solutions 3/12/2018
MAE 101A Homework 7 - Solutions 3/12/2018 Munson 6.31: The stream function for a two-dimensional, nonviscous, incompressible flow field is given by the expression ψ = 2(x y) where the stream function has
More informationA phase field model for the coupling between Navier-Stokes and e
A phase field model for the coupling between Navier-Stokes and electrokinetic equations Instituto de Matemáticas, CSIC Collaborators: C. Eck, G. Grün, F. Klingbeil (Erlangen Univertsität), O. Vantzos (Bonn)
More informationP = 1 3 (σ xx + σ yy + σ zz ) = F A. It is created by the bombardment of the surface by molecules of fluid.
CEE 3310 Thermodynamic Properties, Aug. 27, 2010 11 1.4 Review A fluid is a substance that can not support a shear stress. Liquids differ from gasses in that liquids that do not completely fill a container
More informationGame Physics. Game and Media Technology Master Program - Utrecht University. Dr. Nicolas Pronost
Game and Media Technology Master Program - Utrecht University Dr. Nicolas Pronost Soft body physics Soft bodies In reality, objects are not purely rigid for some it is a good approximation but if you hit
More informationCH.5. BALANCE PRINCIPLES. Multimedia Course on Continuum Mechanics
CH.5. BALANCE PRINCIPLES Multimedia Course on Continuum Mechanics Overview Balance Principles Convective Flux or Flux by Mass Transport Local and Material Derivative of a olume Integral Conservation of
More informationAA210A Fundamentals of Compressible Flow. Chapter 1 - Introduction to fluid flow
AA210A Fundamentals of Compressible Flow Chapter 1 - Introduction to fluid flow 1 1.2 Conservation of mass Mass flux in the x-direction [ ρu ] = M L 3 L T = M L 2 T Momentum per unit volume Mass per unit
More informationChapter 0. Preliminaries. 0.1 Things you should already know
Chapter 0 Preliminaries These notes cover the course MATH45061 (Continuum Mechanics) and are intended to supplement the lectures. The course does not follow any particular text, so you do not need to buy
More informationModeling of a centrifuge experiment: the Lamm equations
Modeling of a centrifuge experiment: the Lamm equations O. Gonzalez and J. Li UT-Austin 29 October 2008 Problem statement Setup Consider an experiment in which a 2-component solution (solute + solvent)
More information[variable] = units (or dimension) of variable.
Dimensional Analysis Zoe Wyatt wyatt.zoe@gmail.com with help from Emanuel Malek Understanding units usually makes physics much easier to understand. It also gives a good method of checking if an answer
More informationBoundary Conditions in Fluid Mechanics
Boundary Conditions in Fluid Mechanics R. Shankar Subramanian Department of Chemical and Biomolecular Engineering Clarkson University The governing equations for the velocity and pressure fields are partial
More informationHomework #4 Solution. μ 1. μ 2
Homework #4 Solution 4.20 in Middleman We have two viscous liquids that are immiscible (e.g. water and oil), layered between two solid surfaces, where the top boundary is translating: y = B y = kb y =
More informationPeter Hertel. University of Osnabrück, Germany. Lecture presented at APS, Nankai University, China.
Balance University of Osnabrück, Germany Lecture presented at APS, Nankai University, China http://www.home.uni-osnabrueck.de/phertel Spring 2012 Linear and angular momentum and First and Second Law point
More informationOn Entropy Flux Heat Flux Relation in Thermodynamics with Lagrange Multipliers
Continuum Mech. Thermodyn. (1996) 8: 247 256 On Entropy Flux Heat Flux Relation in Thermodynamics with Lagrange Multipliers I-Shih Liu Instituto de Matemática Universidade do rio de Janeiro, Caixa Postal
More informationComputer Fluid Dynamics E181107
Computer Fluid Dynamics E181107 2181106 Transport equations, Navier Stokes equations Remark: foils with black background could be skipped, they are aimed to the more advanced courses Rudolf Žitný, Ústav
More informationChapter 3: Newtonian Fluids
Chapter 3: Newtonian Fluids CM4650 Michigan Tech Navier-Stokes Equation v vv p t 2 v g 1 Chapter 3: Newtonian Fluid TWO GOALS Derive governing equations (mass and momentum balances Solve governing equations
More informationRational derivation of the Boussinesq approximation
Rational derivation of the Boussinesq approximation Kiyoshi Maruyama Department of Earth and Ocean Sciences, National Defense Academy, Yokosuka, Kanagawa 239-8686, Japan February 22, 2019 Abstract This
More information