# Entropy generation and transport

Size: px
Start display at page:

Transcription

1 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? 2) How is the entropy of a fluid a ected by its motion? We have touched on the latter questions several times in this text but without a rigorous analysis. In this chapter the precise form of the flow properties that represent the sources of entropy will be derived from first principles. In this endeavor we will use the Gibbs equation on a fluid element. T Ds Dt = De Dt + P Dv Dt (7.1) The Navier-Stokes equations are the foundation of the science of fluid mechanics. With the inclusion of an equation of state, virtually all flow solving revolves around finding solutions of the Navier-Stokes equations. Most exceptions involve fluids where the relation between stress and rate-of-strain is nonlinear such as polymers, or where the equation of state is not very well understood (for example supersonic flow in water) or rarefied flows where the Boltzmann equation must be used to explicitly account for particle collisions. The equations can take on many forms depending on what approximations or assumptions may be appropriate to a given flow. In addition, transforming the equations to di erent forms may enable one to gain insight into the nature of the solutions. It is essential to learn the 7-1

2 CHAPTER 7. ENTROPY GENERATION AND TRANSPORT 7-2 many di erent forms of the equations and to become practiced in the manipulations used to transform them. 7.2 The kinetic energy equation To answer these questions we first need to form a conservation equation for the kinetic energy. This can be accomplished by manipulating the momentum and continuity equations. Multiply the momentum equation by (sum over ( U j + P ij ij ) G i = 0 (7.2) First let s look at the temporal and convective terms in this ( ) ( U j ) + U j + U (7.3) The sum of the underlined terms is zero by continuity. ( ) ( U j ) + j (7.4) where, k = /2. Use continuity again in the form, + j + j = 0 (7.5) Add the above two equations to get the temporal-convective part of the kinetic energy ( ) ( U j ) ( k) ( ku j) (7.6) Now let s rearrange the pressure (P ij ) (PU j) (7.7)

3 CHAPTER 7. ENTROPY GENERATION AND TRANSPORT 7-3 and the viscous stress ij ( ij ) (7.8) which we derived in Chapter 2. Finally, the kinetic energy equation ( k) ( U j k + PU j ij )= U j G j + (7.9) 7.3 Internal energy When the kinetic energy equation (7.9) is subtracted from the energy equation (7.2) the result is the conservation equation for internal ( e) ( U j e + Q j )= + (7.10) Note that both of these equations are in the general conservation ( ) + r ( ) = sources (7.11) The series of steps used to generate these equations are pretty standard and illustrate that the equations of motion can be (and often are) manipulated to create a conservation equation for practically any variable we wish. The source term = P r Ū (7.12) appears in both the kinetic and internal energy equations with opposite sign. It can be either positive or negative depending on the whether the fluid is expanding or being compressed. Recall the continuity equation in the form 1 D Dt = r Ū (7.13)

4 CHAPTER 7. ENTROPY GENERATION AND TRANSPORT 7-4 Multiply by the pressure P r Ū = P D Dt = P Dv Dt (7.14) The term in parentheses we recognize as the work term in the Gibbs equation (the whole term is actually the work per second per unit volume). This term represents a reversible exchange between internal and kinetic energy due to the work of compression or expansion of a fluid element Viscous dissipation of kinetic energy Now let s look at the term = (7.15) This needs to be worked a little further. Substitute the expression for the Newtonian stress tensor and decompose the velocity gradient into a symmetric and antisymmetric part. = 2 2µS ij 3 µ µ v ijs kk (S ij + W ij ) (7.16) The sum over the antisymmetric part is zero. Thus =2µ (S ij S ij ) 2 3 µ (S iis kk )+µ v (S ii S kk ) (7.17) We can write the first two terms as a square. 1 1 =2µ S ij 3 ij S kk S ij 3 ij S kk + µ v (S ii S kk ) (7.18) The scalar quantity,, is the viscous dissipation of kinetic energy and it is clear from the fact that (7.19) is a sum of squares that it is always positive. The dissipation term appears as a sink in the kinetic energy equation and as a source in the internal energy equation. It is the irreversible conversion of kinetic energy to internal energy due to viscous friction. We can attach two physical interpretations to Stokes hypothesis; the assumption that µ v = 0. Recall that the mean normal stress is

5 CHAPTER 7. ENTROPY GENERATION AND TRANSPORT 7-5 mean =(1/3) ii = P + µ v S kk (7.19) In terms of flow forces, Stokes hypothesis implies that, in a viscous Newtonian fluid, the mean normal stress is everywhere equal to the pressure. We can give a second interpretation to Stokes hypothesis in terms of flow energy by noting that trace (S ij (1/3) ij S kk ) = 0 (7.20) Under Stokes hypothesis the second term in the dissipation expression (7.18) is zero and the only contributor to kinetic energy dissipation is the anisotropic (o diagonal) part of the rate of strain tensor. 7.4 Entropy We are now in a position to say something about the entropy of the system. Recall the Gibbs equation. De Dt Write the internal energy equation, P D Ds = T Dt Dt e ( U j e + Q j )= + (7.22) in terms of the substantial derivative. De Dt + = + (7.23) We have already written down the continuity equation in the form required to replace the second term on the left hand side. P D Dt = (7.24)

6 CHAPTER 7. ENTROPY GENERATION AND TRANSPORT 7-6 The energy equation now becomes De Dt P D Dt = + (7.25) Comparing this form of the energy equation with the Gibbs equation, we see that the left-hand-side corresponds to the entropy term T Ds Dt = + (7.26) Let s put this in the form of a conservation equation for the entropy. The approach is to use the continuity equation yet + j = 1 T + T + j + j =0 (7.27) Add the two equations in (7.28) to s ( U j s)= 1 T + T (7.28) The heat flux term can be rearranged into a divergence term and a squared Qj T = 1 T Q j T (7.29) For a linear conducting material, the heat flux is Q j = (7.30) Let = (7.31)

7 CHAPTER 7. ENTROPY GENERATION AND TRANSPORT 7-7 Finally, the conservation equation for the entropy s U j s = + T T (7.32) The integral form of the entropy transport equation on a general control volume is Z Z D sdv + U j U Aj s Dt V (t) A(t) Z + n j da = T V (t) T dv (7.33) The right hand side of (7.32) is the entropy source term and is always positive. This remarkable result provides an explicit expression (in terms of squared temperature gradients and squared velocity gradients) for the irreversible changes in entropy that occur in a compressible flow. Notice that the body force term contributes nothing to the entropy. Consider a closed adiabatic box containing a viscous, heat conducting fluid subject to the no slip condition on the walls. The fluid is stirred by a fan that is then turned o and the flow is allowed to settle. Figure 7.1: Fluid stirred in an adiabatic box. The entropy of the fluid in the box behaves according to Z Z D sdv + U j s Dt V A Z + n j da = T V T dv (7.34) The surface integral is zero by the no-slip condition and the assumption of adiabaticity. The volume integral of the density times the intensive entropy is the extensive entropy.

8 CHAPTER 7. ENTROPY GENERATION AND TRANSPORT 7-8 Z S = V sdv (7.35) Thus as the closed, adiabatic, system comes to rest, the entropy continues to increase until all the gradients in velocity and temperature have vanished. Z ds + dt = V T dv > 0 (7.36) 7.5 Problems Problem 1 - Show that (7.17) can be expressed in the form (7.18). Problem 2 - Imagine a spherically symmetric flow in the absence of shear. Figure 7.2: Radially directed viscous flow. Let the fluid velocity in the radial direction be f(r). Work out the expression for the kinetic energy dissipation in terms of f and the two viscosities µ and µ v. Suppose the fluid is a monatomic gas for which µ v = 0. Does there exist a function f for which the dissipation is zero? Refer to Appendix B for the dissipation function in spherical polar coordinates. Problem 3 - Suppose the fluid in the box shown in Section 7.5 is Air initially at 300 K and one atmosphere. Let the tip velocity of the propeller be 50 m/sec and the typical boundary layer thickness over the surface of the propeller be 1 millimeter. The surface

9 CHAPTER 7. ENTROPY GENERATION AND TRANSPORT 7-9 area of the propeller is 1 m 2 and the volume of the box is 1 m 3. Roughly estimate the rate at which the temperature of the air in the box increases due to viscous dissipation. Problem 4 - A simple pendulum of mass m = 1.0 kilogram is placed inside a rigid adiabatic container filled with 0.1 kilograms of Helium gas initially at rest (state 1). The gas pressure is one atmosphere and the temperature is 300 K. The acceleration due to gravity is g =9.8 meters/sec 2. Figure 7.3: Pendulum released from rest in a viscous fluid in an adiabatic box. At t = 0 the pendulum is released from an initial displacement height h =0.1 meters and begins to oscillate and stir the gas. Eventually the pendulum and gas all come to rest (state 2). 1) Assume all the potential energy of the pendulum is converted to internal energy of the gas. What is the change in temperature of the Helium in going from state 1 to state 2? 2) What is the change in entropy per unit mass of the Helium in going from state 1 to state 2?

### CHAPTER 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?

### CHAPTER 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,

### Several forms of the equations of motion

Chapter 6 Several forms of the equations of motion 6.1 The Navier-Stokes equations Under the assumption of a Newtonian stress-rate-of-strain constitutive equation and a linear, thermally conductive medium,

### The conservation equations

Chapter 5 The conservation equations 5.1 Leibniz rule for di erentiation of integrals 5.1.1 Di erentiation under the integral sign According to the fundamental theorem of calculus if f is a smooth function

### AA210A Fundamentals of Compressible Flow. Chapter 5 -The conservation equations

AA210A Fundamentals of Compressible Flow Chapter 5 -The conservation equations 1 5.1 Leibniz rule for differentiation of integrals Differentiation under the integral sign. According to the fundamental

### 2 GOVERNING EQUATIONS

2 GOVERNING EQUATIONS 9 2 GOVERNING EQUATIONS For completeness we will take a brief moment to review the governing equations for a turbulent uid. We will present them both in physical space coordinates

### 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

### V (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

### Computational 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

### Review of Fluid Mechanics

Chapter 3 Review of Fluid Mechanics 3.1 Units and Basic Definitions Newton s Second law forms the basis of all units of measurement. For a particle of mass m subjected to a resultant force F the law may

### Tutorial 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

### Fluid 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

### Math 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

### Basic hydrodynamics. David Gurarie. 1 Newtonian fluids: Euler and Navier-Stokes equations

Basic hydrodynamics David Gurarie 1 Newtonian fluids: Euler and Navier-Stokes equations The basic hydrodynamic equations in the Eulerian form consist of conservation of mass, momentum and energy. We denote

### From 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

### 2 Equations of Motion

2 Equations of Motion system. In this section, we will derive the six full equations of motion in a non-rotating, Cartesian coordinate 2.1 Six equations of motion (non-rotating, Cartesian coordinates)

### Chapter 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

### Computer 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

### High Speed Aerodynamics. Copyright 2009 Narayanan Komerath

Welcome to High Speed Aerodynamics 1 Lift, drag and pitching moment? Linearized Potential Flow Transformations Compressible Boundary Layer WHAT IS HIGH SPEED AERODYNAMICS? Airfoil section? Thin airfoil

### Chapter 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

### Revisit 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

### 150A Review Session 2/13/2014 Fluid Statics. Pressure acts in all directions, normal to the surrounding surfaces

Fluid Statics Pressure acts in all directions, normal to the surrounding surfaces or Whenever a pressure difference is the driving force, use gauge pressure o Bernoulli equation o Momentum balance with

### Chapter 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

### Rational 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

### Notes 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.

### Chapter 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

### Conservation 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

### Fluid 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

### Chapter 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

### Fluid 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

### Euler 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

### Fluid Mechanics Abdusselam Altunkaynak

Fluid Mechanics Abdusselam Altunkaynak 1. Unit systems 1.1 Introduction Natural events are independent on units. The unit to be used in a certain variable is related to the advantage that we get from it.

### Getting 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 =

### Quick 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

### Chemical and Biomolecular Engineering 150A Transport Processes Spring Semester 2017

Chemical and Biomolecular Engineering 150A Transport Processes Spring Semester 2017 Objective: Text: To introduce the basic concepts of fluid mechanics and heat transfer necessary for solution of engineering

### AA210A 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

### 7 The Navier-Stokes Equations

18.354/12.27 Spring 214 7 The Navier-Stokes Equations In the previous section, we have seen how one can deduce the general structure of hydrodynamic equations from purely macroscopic considerations and

### ( ) 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

### Differential 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

### meters, 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.

### Continuum Mechanics Fundamentals

Continuum Mechanics Fundamentals James R. Rice, notes for ES 220, 12 November 2009; corrections 9 December 2009 Conserved Quantities Let a conseved quantity have amount F per unit volume. Examples are

### Boundary 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

### Lecture 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

### Navier-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

### The Hopf equation. The Hopf equation A toy model of fluid mechanics

The Hopf equation A toy model of fluid mechanics 1. Main physical features Mathematical description of a continuous medium At the microscopic level, a fluid is a collection of interacting particles (Van

### Soft 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

### Chapter 1: Basic Concepts

What is a fluid? A fluid is a substance in the gaseous or liquid form Distinction between solid and fluid? Solid: can resist an applied shear by deforming. Stress is proportional to strain Fluid: deforms

### Fundamentals of Fluid Dynamics: Ideal Flow Theory & Basic Aerodynamics

Fundamentals of Fluid Dynamics: Ideal Flow Theory & Basic Aerodynamics Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI (after: D.J. ACHESON s Elementary Fluid Dynamics ) bluebox.ippt.pan.pl/

### III.3 Momentum balance: Euler and Navier Stokes equations

32 Fundamental equations of non-relativistic fluid dynamics.3 Momentum balance: Euler and Navier tokes equations For a closed system with total linear momentum P ~ with respect to a given reference frame

### - 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

### Chapter 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

### ONSAGER 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

### Game 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

### Basic 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

### CHAPTER 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

### Plasmas as fluids. S.M.Lea. January 2007

Plasmas as fluids S.M.Lea January 2007 So far we have considered a plasma as a set of non intereacting particles, each following its own path in the electric and magnetic fields. Now we want to consider

### 2. 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

### Interpreting Differential Equations of Transport Phenomena

Interpreting Differential Equations of Transport Phenomena There are a number of techniques generally useful in interpreting and simplifying the mathematical description of physical problems. Here we introduce

### AE/ME 339. K. M. Isaac Professor of Aerospace Engineering. 12/21/01 topic7_ns_equations 1

AE/ME 339 Professor of Aerospace Engineering 12/21/01 topic7_ns_equations 1 Continuity equation Governing equation summary Non-conservation form D Dt. V 0.(2.29) Conservation form ( V ) 0...(2.33) t 12/21/01

### Accretion Disks I. High Energy Astrophysics: Accretion Disks I 1/60

Accretion Disks I References: Accretion Power in Astrophysics, J. Frank, A. King and D. Raine. High Energy Astrophysics, Vol. 2, M.S. Longair, Cambridge University Press Active Galactic Nuclei, J.H. Krolik,

### APPENDIX B. The primitive equations

APPENDIX B The primitive equations The physical and mathematical basis of all methods of dynamical atmospheric prediction lies in the principles of conservation of momentum, mass, and energy. Applied to

### Contents. 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.........................

### MA3D1 Fluid Dynamics Support Class 5 - Shear Flows and Blunt Bodies

MA3D1 Fluid Dynamics Support Class 5 - Shear Flows and Blunt Bodies 13th February 2015 Jorge Lindley email: J.V.M.Lindley@warwick.ac.uk 1 2D Flows - Shear flows Example 1. Flow over an inclined plane A

### Fluid Mechanics Prof. S. K. Som Department of Mechanical Engineering Indian Institute of Technology, Kharagpur

Fluid Mechanics Prof. S. K. Som Department of Mechanical Engineering Indian Institute of Technology, Kharagpur Lecture - 15 Conservation Equations in Fluid Flow Part III Good afternoon. I welcome you all

### cos(θ)sin(θ) Alternative Exercise Correct Correct θ = 0 skiladæmi 10 Part A Part B Part C Due: 11:59pm on Wednesday, November 11, 2015

skiladæmi 10 Due: 11:59pm on Wednesday, November 11, 015 You will receive no credit for items you complete after the assignment is due Grading Policy Alternative Exercise 1115 A bar with cross sectional

### EXPERIENCE COLLEGE BEFORE COLLEGE

Mechanics, Heat, and Sound (PHY302K) College Unit Week Dates Big Ideas Subject Learning Outcomes Assessments Apply algebra, vectors, and trigonometry in context. Employ units in problems. Course Mathematics

### FUNDAMENTALS OF CHEMISTRY Vol. II - Irreversible Processes: Phenomenological and Statistical Approach - Carlo Cercignani

IRREVERSIBLE PROCESSES: PHENOMENOLOGICAL AND STATISTICAL APPROACH Carlo Dipartimento di Matematica, Politecnico di Milano, Milano, Italy Keywords: Kinetic theory, thermodynamics, Boltzmann equation, Macroscopic

### PHYSICS. Course Structure. Unit Topics Marks. Physical World and Measurement. 1 Physical World. 2 Units and Measurements.

PHYSICS Course Structure Unit Topics Marks I Physical World and Measurement 1 Physical World 2 Units and Measurements II Kinematics 3 Motion in a Straight Line 23 4 Motion in a Plane III Laws of Motion

### 1. The Properties of Fluids

1. The Properties of Fluids [This material relates predominantly to modules ELP034, ELP035] 1.1 Fluids 1.1 Fluids 1.2 Newton s Law of Viscosity 1.3 Fluids Vs Solids 1.4 Liquids Vs Gases 1.5 Causes of viscosity

### 3. FORMS OF GOVERNING EQUATIONS IN CFD

3. FORMS OF GOVERNING EQUATIONS IN CFD 3.1. Governing and model equations in CFD Fluid flows are governed by the Navier-Stokes equations (N-S), which simpler, inviscid, form is the Euler equations. For

### Fundamentals of compressible and viscous flow analysis - Part II

Fundamentals of compressible and viscous flow analysis - Part II Lectures 3, 4, 5 Instantaneous and averaged temperature contours in a shock-boundary layer interaction. Taken from (Pasquariello et al.,

### CHAPTER 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

### An Overview of Fluid Animation. Christopher Batty March 11, 2014

An Overview of Fluid Animation Christopher Batty March 11, 2014 What distinguishes fluids? What distinguishes fluids? No preferred shape. Always flows when force is applied. Deforms to fit its container.

### 1. Introduction, tensors, kinematics

1. Introduction, tensors, kinematics Content: Introduction to fluids, Cartesian tensors, vector algebra using tensor notation, operators in tensor form, Eulerian and Lagrangian description of scalar and

### 2. Conservation Equations for Turbulent Flows

2. Conservation Equations for Turbulent Flows Coverage of this section: Review of Tensor Notation Review of Navier-Stokes Equations for Incompressible and Compressible Flows Reynolds & Favre Averaging

### Math 575-Lecture Viscous Newtonian fluid and the Navier-Stokes equations

Math 575-Lecture 13 In 1845, tokes extended Newton s original idea to find a constitutive law which relates the Cauchy stress tensor to the velocity gradient, and then derived a system of equations. The

### Summary 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

### Measurement p. 1 What Is Physics? p. 2 Measuring Things p. 2 The International System of Units p. 2 Changing Units p. 3 Length p. 4 Time p. 5 Mass p.

Measurement p. 1 What Is Physics? p. 2 Measuring Things p. 2 The International System of Units p. 2 Changing Units p. 3 Length p. 4 Time p. 5 Mass p. 7 Review & Summary p. 8 Problems p. 8 Motion Along

### 2.29 Numerical Fluid Mechanics Fall 2011 Lecture 5

.9 Numerical Fluid Mechanics Fall 011 Lecture 5 REVIEW Lecture 4 Roots of nonlinear equations: Open Methods Fixed-point Iteration (General method or Picard Iteration), with examples Iteration rule: x g(

### Turbulence Modeling. Cuong Nguyen November 05, The incompressible Navier-Stokes equations in conservation form are u i x i

Turbulence Modeling Cuong Nguyen November 05, 2005 1 Incompressible Case 1.1 Reynolds-averaged Navier-Stokes equations The incompressible Navier-Stokes equations in conservation form are u i x i = 0 (1)

### Module 2 : Convection. Lecture 12 : Derivation of conservation of energy

Module 2 : Convection Lecture 12 : Derivation of conservation of energy Objectives In this class: Start the derivation of conservation of energy. Utilize earlier derived mass and momentum equations for

### Microscopic Momentum Balance Equation (Navier-Stokes)

CM3110 Transport I Part I: Fluid Mechanics Microscopic Momentum Balance Equation (Navier-Stokes) Professor Faith Morrison Department of Chemical Engineering Michigan Technological University 1 Microscopic

### The Euler Equation of Gas-Dynamics

The Euler Equation of Gas-Dynamics A. Mignone October 24, 217 In this lecture we study some properties of the Euler equations of gasdynamics, + (u) = ( ) u + u u + p = a p + u p + γp u = where, p and u

### A 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

### DIVIDED SYLLABUS ( ) - CLASS XI PHYSICS (CODE 042) COURSE STRUCTURE APRIL

DIVIDED SYLLABUS (2015-16 ) - CLASS XI PHYSICS (CODE 042) COURSE STRUCTURE APRIL Unit I: Physical World and Measurement Physics Need for measurement: Units of measurement; systems of units; SI units, fundamental

### Shell 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

### Course Syllabus: Continuum Mechanics - ME 212A

Course Syllabus: Continuum Mechanics - ME 212A Division Course Number Course Title Academic Semester Physical Science and Engineering Division ME 212A Continuum Mechanics Fall Academic Year 2017/2018 Semester

### 2.29 Numerical Fluid Mechanics Fall 2011 Lecture 7

Numerical Fluid Mechanics Fall 2011 Lecture 7 REVIEW of Lecture 6 Material covered in class: Differential forms of conservation laws Material Derivative (substantial/total derivative) Conservation of Mass

### EULERIAN 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

### Class XI Physics Syllabus One Paper Three Hours Max Marks: 70

Class XI Physics Syllabus 2013 One Paper Three Hours Max Marks: 70 Class XI Weightage Unit I Physical World & Measurement 03 Unit II Kinematics 10 Unit III Laws of Motion 10 Unit IV Work, Energy & Power

### Fluid Animation. Christopher Batty November 17, 2011

Fluid Animation Christopher Batty November 17, 2011 What distinguishes fluids? What distinguishes fluids? No preferred shape Always flows when force is applied Deforms to fit its container Internal forces

### INDEX 363. Cartesian coordinates 19,20,42, 67, 83 Cartesian tensors 84, 87, 226

INDEX 363 A Absolute differentiation 120 Absolute scalar field 43 Absolute tensor 45,46,47,48 Acceleration 121, 190, 192 Action integral 198 Addition of systems 6, 51 Addition of tensors 6, 51 Adherence

### 3.8 The First Law of Thermodynamics and the Energy Equation

CEE 3310 Control Volume Analysis, Sep 30, 2011 65 Review Conservation of angular momentum 1-D form ( r F )ext = [ˆ ] ( r v)d + ( r v) out ṁ out ( r v) in ṁ in t CV 3.8 The First Law of Thermodynamics and

### The 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:

### Boundary-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

### Hydrodynamics. 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

### CH.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

### Unsteady Flow of a Newtonian Fluid in a Contracting and Expanding Pipe

Unsteady Flow of a Newtonian Fluid in a Contracting and Expanding Pipe T S L Radhika**, M B Srinivas, T Raja Rani*, A. Karthik BITS Pilani- Hyderabad campus, Hyderabad, Telangana, India. *MTC, Muscat,