# CHAPTER 8 ENTROPY GENERATION AND TRANSPORT

Size: px
Start display at page:

Transcription

1 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? 2) How is the entropy of a fluid affected 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 and the energy conservation equation De P Dv. (8.1) ( e + k). (8.2) t U e P k U + Q G U 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 i) ( U. (8.3) x i U + P i i ) G 0 i t First let s look at the temporal and convective terms in this equation. bc 8.1 4/6/13

2 The kinetic energy equation t U ( i ) U ( i U ) - + U t i x. (8.4) U + + U t i U U x i - + U x i U x The sum of the underlined terms is zero by continuity. Thus U ( i ) U ( i U ) - + U t i k k U x t x (8.5) where, k 2. Use continuity again in the form, ku k U (8.6) x x k t Add the above two equations to get the temporal-convective part of the kinetic energy equation. U ( i ) U ( i U ) - + U t i x k ( ) t ( ku x ) (8.7) Now let s rearrange the pressure term. ( P i ) - x ( PU ) -- P U - x x (8.8) and the viscous stress term. i x ( ) i (8.9) x i x Finally, the kinetic energy equation is k ( ) t ( U x k + PU ) G P U + - x i x. (8.10) 4/6/ bc

3 Internal energy 8.3 INTERNAL ENERGY When the kinetic energy equation (8.10) is subtracted from the energy equation (8.2) the result is the conservation equation for internal energy. e t ( U x e + Q ) P U - + x i x. (8.11) Note that both of these equations are in the general conservation form ( ) -- + ( ) sources. (8.12) t 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 U - P U x (8.13) 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 Multiply by the pressure D U. (8.14) P U P D P Dv. (8.15) 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. bc 8.3 4/6/13

4 Viscous dissipation of kinetic energy 8.4 VISCOUS DISSIPATION OF KINETIC ENERGY Now let s look at the term i x. (8.16) 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 i --µ µ 3 v i S kk ( Si + W i ) (8.17) The sum over the antisymmetric part is zero. Thus 2µ ( S i S i ) We can write the first two terms as a square. 2 --µ ( S 3 ii S kk ) + µ v ( S ii S kk ). (8.18) 2µ 1 S i -- 3 i S 1 kk Si -- 3 i S kk + µ ( S S ) v ii kk (8.19) The scalar quantity,, is the viscous dissipation of kinetic energy and it is clear from the fact that (8.19) 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 mean ( 1 3) ii P + µ v S kk. (8.20) In terms of flow forces, Stokes hypothesis implies that, in a viscous 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 4/6/ bc

5 Entropy trace( S i 1 3 ) 0. (8.21) ( ) i S kk Under Stokes hypothesis the second term in the dissipation expression (8.19) is zero and the only contributor to kinetic energy dissipation is the anisotropic (off diagonal) part of the rate of strain tensor. 8.5 ENTROPY We are now in a position to say something about the entropy of the system. Recall the Gibbs equation De P -- D T Ds (8.22) Write the internal energy equation, e t ( U x e + Q ) P U - + x (8.23) in terms of the substantial derivative De P U + - x Q - + x. (8.24) We have already written down the continuity equation in the form required to replace the second term on the left hand side. P -- D P U - x (8.25) The energy equation now becomes De P -- D Q - + x. (8.26) 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 Q - + x. (8.27) bc 8.5 4/6/13

6 Entropy Let s put this in the form of a conservation equation for the entropy. The approach is to use the continuity equation yet again! s s U t x Q T x T s su t s U x x (8.28) Add the two to get s t ( U x s) Q - T x T. (8.29) The heat flux term can be rearranged into a divergence term and a squared term. Q T Q Q T x T T x x For a linear conducting material, the heat flux is (8.30) Q k T. (8.31) x Let k --- T T T x x. (8.32) Finally, the conservation equation for the entropy becomes s t + x U s k T T x T. (8.33) The integral form of the entropy transport equation on a general control volume is D k sdv + ( U U A )s T n T x da V() t At () V() t dv T (8.34) 4/6/ bc

7 Entropy The right hand side of (8.33) 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 subect to the no slip condition on the walls. The fluid is stirred by a fan that is then turned off and the flow is allowed to settle. V A Figure 8.1 Fluid stirred by a fan in an adiabatic box. The entropy of the fluid in the box behaves according to d dt ---- s V U s k d T n T x da + V A dv T. (8.35) 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. S V sdv (8.36) Thus as the closed, adiabatic, system comes to rest, the entropy continues to increase until all the gradients in velocity and temperature have vanished. ds dt V T d V > 0 V (8.37) bc 8.7 4/6/13

8 Problems 8.6 PROBLEMS Problem 1 - Show that (8.18) can be expressed in the form (8.19). Problem 2 - Imagine a spherically symmetric flow in the absence of shear. 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, µ. Suppose the fluid is a monatomic gas for which µ 0. Does there exist a function f for which the dissipation is zero? Refer to Appendix 2 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 300K 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 area of the propeller is 1m 2.and the volume of the box is 1m 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. 4/6/ bc

9 Problems g State 1 State 2 m h 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? bc 8.9 4/6/13

10 Problems 4/6/ bc

### Entropy 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?

### 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

### 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,

### 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

### 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

### 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.

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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,

### 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

### 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.

### 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

### 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

### 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 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

### 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

### 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

### 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

### 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

### 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

### 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.........................

### 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

### 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

### 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 =

### 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

### 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

### 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)

### 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

### 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

### 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

### 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

### 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

### ( ) 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

### 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

### 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

### 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

### Conservation Laws in Ideal MHD

Conservation Laws in Ideal MHD Nick Murphy Harvard-Smithsonian Center for Astrophysics Astronomy 253: Plasma Astrophysics February 3, 2016 These lecture notes are largely based on Plasma Physics for Astrophysics

### 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

### 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

### 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

### 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

### 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

### 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/

### 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

### 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.

### 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

### vector H. If O is the point about which moments are desired, the angular moment about O is given:

The angular momentum A control volume analysis can be applied to the angular momentum, by letting B equal to angularmomentum vector H. If O is the point about which moments are desired, the angular moment

### 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

### 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

### ESCI 485 Air/Sea Interaction Lesson 1 Stresses and Fluxes Dr. DeCaria

ESCI 485 Air/Sea Interaction Lesson 1 Stresses and Fluxes Dr DeCaria References: An Introduction to Dynamic Meteorology, Holton MOMENTUM EQUATIONS The momentum equations governing the ocean or atmosphere

### 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

### 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

### 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,

### Continuum Mechanics Lecture 5 Ideal fluids

Continuum Mechanics Lecture 5 Ideal fluids Prof. http://www.itp.uzh.ch/~teyssier Outline - Helmholtz decomposition - Divergence and curl theorem - Kelvin s circulation theorem - The vorticity equation

### 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.,

### Introduction to Heat and Mass Transfer. Week 10

Introduction to Heat and Mass Transfer Week 10 Concentration Boundary Layer No concentration jump condition requires species adjacent to surface to have same concentration as at the surface Owing to concentration

### 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

### 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

### 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

### - 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

### 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

### Lesson 6 Review of fundamentals: Fluid flow

Lesson 6 Review of fundamentals: Fluid flow The specific objective of this lesson is to conduct a brief review of the fundamentals of fluid flow and present: A general equation for conservation of mass

### Linear Transport Relations (LTR)

Linear Transport Relations (LTR) Much of Transport Phenomena deals with the exchange of momentum, mass, or heat between two (or many) objects. Often, the most mathematically simple way to consider how

### 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

### Chapter 1 Fluid Characteristics

Chapter 1 Fluid Characteristics 1.1 Introduction 1.1.1 Phases Solid increasing increasing spacing and intermolecular liquid latitude of cohesive Fluid gas (vapor) molecular force plasma motion 1.1.2 Fluidity

### 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

### 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

### SW103: Lecture 2. Magnetohydrodynamics and MHD models

SW103: Lecture 2 Magnetohydrodynamics and MHD models Scale sizes in the Solar Terrestrial System: or why we use MagnetoHydroDynamics Sun-Earth distance = 1 Astronomical Unit (AU) 200 R Sun 20,000 R E 1

### Principles of Convection

Principles of Convection Point Conduction & convection are similar both require the presence of a material medium. But convection requires the presence of fluid motion. Heat transfer through the: Solid

### Chapter (3) TURBULENCE KINETIC ENERGY

Chapter (3) TURBULENCE KINETIC ENERGY 3.1 The TKE budget Derivation : The definition of TKE presented is TKE/m= e = 0.5 ( u 2 + v 2 + w 2 ). we recognize immediately that TKE/m is nothing more than the

### n 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

### Miami-Dade Community College PHY 2053 College Physics I

Miami-Dade Community College PHY 2053 College Physics I PHY 2053 3 credits Course Description PHY 2053, College physics I, is the first semester of a two semester physics-without-calculus sequence. This

### Detailed Outline, M E 320 Fluid Flow, Spring Semester 2015

Detailed Outline, M E 320 Fluid Flow, Spring Semester 2015 I. Introduction (Chapters 1 and 2) A. What is Fluid Mechanics? 1. What is a fluid? 2. What is mechanics? B. Classification of Fluid Flows 1. Viscous

### 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

### Publication 97/2. An Introduction to Turbulence Models. Lars Davidson, lada

ublication 97/ An ntroduction to Turbulence Models Lars Davidson http://www.tfd.chalmers.se/ lada Department of Thermo and Fluid Dynamics CHALMERS UNVERSTY OF TECHNOLOGY Göteborg Sweden November 3 Nomenclature

### 4. The Green Kubo Relations

4. The Green Kubo Relations 4.1 The Langevin Equation In 1828 the botanist Robert Brown observed the motion of pollen grains suspended in a fluid. Although the system was allowed to come to equilibrium,

### 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

### 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

### 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 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(

### MECH 5810 Module 3: Conservation of Linear Momentum

MECH 5810 Module 3: Conservation of Linear Momentum D.J. Willis Department of Mechanical Engineering University of Massachusetts, Lowell MECH 5810 Advanced Fluid Dynamics Fall 2017 Outline 1 Announcements

### MAE 3130: Fluid Mechanics Lecture 7: Differential Analysis/Part 1 Spring Dr. Jason Roney Mechanical and Aerospace Engineering

MAE 3130: Fluid Mechanics Lecture 7: Differential Analysis/Part 1 Spring 2003 Dr. Jason Roney Mechanical and Aerospace Engineering Outline Introduction Kinematics Review Conservation of Mass Stream Function

### 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

### 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

### Detailed Outline, M E 521: Foundations of Fluid Mechanics I

Detailed Outline, M E 521: Foundations of Fluid Mechanics I I. Introduction and Review A. Notation 1. Vectors 2. Second-order tensors 3. Volume vs. velocity 4. Del operator B. Chapter 1: Review of Basic

### MECHANICAL PROPERTIES OF FLUIDS:

Important Definitions: MECHANICAL PROPERTIES OF FLUIDS: Fluid: A substance that can flow is called Fluid Both liquids and gases are fluids Pressure: The normal force acting per unit area of a surface is

### 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

### Month. March APRIL. The Orchid School Baner Weekly Syllabus Overview Std : XI Subject : Physics. Activities/ FAs Planned.

The Orchid School Baner Weekly Syllabus Overview 2015-2016 Std : XI Subject : Physics Month Lesson / Topic Expected Learning Objective Activities/ FAs Planned Remark March Physical World and Measurement

### Review 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