Physics 451/551 Theoretical Mechanics. G. A. Krafft Old Dominion University Jefferson Lab Lecture 18
|
|
- Eugene Campbell
- 5 years ago
- Views:
Transcription
1 Physics 451/551 Theoretical Mechanics G. A. Krafft Old Dominion University Jefferson Lab Lecture 18 Theoretical Mechanics Fall 018
2 Properties of Sound Sound Waves Requires medium for propagation Mainly longitudinal (displacement along propagation direction) Wavelength much longer than interatomic spacing so can treat medium as continuous Fundamental functions Mass density Velocity field x, y, z, t v x, y, z, t Two fundamental equations Continuity equation (Conservation of mass) Velocity equation (Conservation of momentum) Newton s Law in disguise Theoretical Mechanics Fall 018
3 Fundamental Functions Density ρ(x,y,z), mass per unit volume M x, y, z, t lim V 0 V dm x, y, z, t dxdydz Velocity field v x, y, z, t o x, y, z v x, y, z, t Theoretical Mechanics Fall 018
4 Continuity Equation Consider mass entering differential volume element dy x, y, z dx dz Mass entering box in a short time Δt v x x, y, z, t vxx dx, y, z, tdydz t Take limit Δt 0,,,,,, vy x y z t vy x y dy z t dzdxt,,,,,, v z x y z t vz x y z dz t dxdy t x, y, z, t t x, y, z, tdxdydz t v dv Theoretical Mechanics Fall 018
5 dv dxdydz v dxdydz t t dv By Stoke s Theorem. Because true for all dv Mass current density (flux) (kg/(sec m )) Jm v Sometimes rendered in terms of the total time derivative (moving along with the flow) Incompressible flow v 0 and ρ constant dv t v 0 d v v 0 v t dt Theoretical Mechanics Fall 018
6 Pressure Scalar Displace material from a small volume dv with sides given by da. The pressure p is defined to the force acting on the area element df da Pressure is normal to the area element Doesn t depend on orientation of volume External forces (e.g., gravitational force) must be balanced by a pressure gradient to get a stationary fluid in equilibrium Pressure force (per unit volume) F pr p p x Theoretical Mechanics Fall 018
7 Fluid at rest Fluid in motion Hydrostatic Equilibrium f app p dv 0 f app p dv dv Fnet p fapp dv m dv dt dt As with density use total derivative (sometimes called material derivative or convective derivative) dv dt v v t v Theoretical Mechanics Fall 018
8 Fluid Dynamic Equations dv v p v v fapp dt t Manipulate with vector identity vv v v v v Final velocity equation v v v v v f p app t One more thing: equation of state relating p and ρ Theoretical Mechanics Fall 018
9 Energy Conservation For energy in a fixed volume 3 v Etot d x V ε internal energy per unit mass Work done (first law in co-moving frame) Mp Md pdv d p s, d Isentropic process (s constant, no heat transfer in) p t t Theoretical Mechanics Fall 018
10 1 1 t v v v v p v fapp p p v pv v t p p pv v p t 1 1 t 1 je v p v v p v fapp v Theoretical Mechanics Fall 018
11 Bernoulli s Theorem Exact first integral of velocity equation when Irrotational motion v 0 v External force conservative f U Flow incompressible with fixed ρ Bernouli s Theorem If flow compressible but isentropic app 0 p U t 0 p U t Theoretical Mechanics Fall 018
12 Kelvin s Theorem on Circulation Already discussed this in the Arnold material dv v v p v U dt t t To linear order t ds v C t C s, t t C s, t t v C s, t, t 1 v C C s, t t t t v C s, t t, t t ds s C s, t t v C s, t, t ds s Theoretical Mechanics Fall 018
13 d dv C,,,, C s t t ds v C s t t v C s, t, t ds dt dt s s p C v C U ds ds s s 0 (the integrand is exact!) The circulation is constant about any closed curve that moves with the fluid. If a fluid is stationary and acted on by a conservative force, the flow in a simply connected region necessarily remains irrotational. Theoretical Mechanics Fall 018
14 Lagrangian for Isentropic Flow Two independent field variables: ρ and Φ Lagrangian density t p U t L U t Canonical momenta L P / t P 0 L / t 0 0 Theoretical Mechanics Fall 018
15 Euler Lagrange Equations L L P 0 0 t t p Hamiltonian Density L L p P U t t H P U t P t L internal energy plus potential energy plus kinetic energy 0 Theoretical Mechanics Fall 018
16 Sound Waves Linearize about a uniform stationary state,, 0, x t v x t v p x t p p Continuity equation 1 0v 0 0 v 0 t 0 t Velocity equation v 1 p t Eisentropic equation of state p p0 p ps, 0 p0 p c s Theoretical Mechanics Fall 018
17 Flow Irrotational Take curl of velocity equation. Conclude flow irrotational v p v t t t 1 p c t 0 t 0 t Scalar wave equation 1 c t c t Boundary conditions nˆ nˆ V 0 for a fixed boundary 0 free surface t 0 Theoretical Mechanics Fall 018
18 3-D Plane Wave Solutions Ansatz, Re e i k xt k c v 0 0 ik 0 c i iv ik 0 c Energy flux 0 0 j E ik i k c k t 1 * 1 Re ˆ Theoretical Mechanics Fall 018
19 Helmholz Equation and Organ Pipes Theoretical Mechanics Fall 018
20 Theoretical Mechanics Fall 018
21 Green Function for Wave Equation Green Function in 3-D Apply Fourier Transforms u r u r f r 3 ipr f p d re f r 3 ipr f r d pe f p 3 1 Fourier transform equation to solve and integrate by parts twice p u p u p f p Theoretical Mechanics Fall 018
22 Green Function Solution The Fourier transform of the solution is The solution is u p 1 f p ipx u r 3 e d p p The Green function is p f 1 1 p p 3 ipr ipr 3 3 u r e e d pf r d r p ipr ipr 3 G r r e e d p 3 Theoretical Mechanics Fall 018
23 Alternate equation for Green function 1 ipr ipr 3 Gr r e e d p 3 r r Simplify iprcos iprsin 1 e 3 1 e sin 3 p 0 0 G R d p p dp d p R 1 psin pr 1 psin pr e dp 4 R dp 0 R p R p Yukawa potential (Green function) rr e Gr r 4 r r Theoretical Mechanics Fall 018
24 Helmholtz Equation Driven (Inhomogeneous) Wave Equation 1 c t Time Fourier Transform r, t f r, t Wave Equation Fourier Transformed r, f r, c 1 f r t 1 d e f r it r, t de r, it,, Theoretical Mechanics Fall 018
25 c Green function satisfies Green Function 3 r, t d r dtg r r, t t f r, t,, k k f k, ik r t 1 f k 3 r, t 4 d kd e k c ik r t ik rt r, t d r dt d k d e e f 4 r, t k c Theoretical Mechanics Fall 018
26 Green function is ik r t ik rt 1 3 e e G r r, t t d k d 4 k c Satisfies 1 c t G r r t t r r t t, Also, with causal boundary conditions is G r r, i r r / c e 4 r r Theoretical Mechanics Fall 018
27 Causal Boundary Conditions Can get causal B. C. by correct pole choice ω k plane kc i kc i i / c i / c Gives so-called retarded Green function Green function evaluated 3 ik R, d k e G R 3 k i / c 1 e e e kdk k i / c R ikr ikr ir/ c 8 ir 4 Theoretical Mechanics Fall 018
Physics 451/551 Theoretical Mechanics. G. A. Krafft Old Dominion University Jefferson Lab Lecture 18
Physics 451/551 Theoretical Mechanics G. A. Krafft Old Dominion University Jefferson Lab Lecture 18 Theoretical Mechanics Fall 18 Properties of Sound Sound Waves Requires medium for propagation Mainly
More informationContinuum 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
More informationThe 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
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 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 informationElectromagnetic. G. A. Krafft Jefferson Lab Jefferson Lab Professor of Physics Old Dominion University TAADI Electromagnetic Theory
TAAD1 Electromagnetic Theory G. A. Krafft Jefferson Lab Jefferson Lab Professor of Physics Old Dominion University 8-31-12 Classical Electrodynamics A main physics discovery of the last half of the 2 th
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 informationIn this section, mathematical description of the motion of fluid elements moving in a flow field is
Jun. 05, 015 Chapter 6. Differential Analysis of Fluid Flow 6.1 Fluid Element Kinematics In this section, mathematical description of the motion of fluid elements moving in a flow field is given. A small
More information3.5 Vorticity Equation
.0 - Marine Hydrodynamics, Spring 005 Lecture 9.0 - Marine Hydrodynamics Lecture 9 Lecture 9 is structured as follows: In paragraph 3.5 we return to the full Navier-Stokes equations (unsteady, viscous
More informationSection 2: Lecture 1 Integral Form of the Conservation Equations for Compressible Flow
Section 2: Lecture 1 Integral Form of the Conservation Equations for Compressible Flow Anderson: Chapter 2 pp. 41-54 1 Equation of State: Section 1 Review p = R g T " > R g = R u M w - R u = 8314.4126
More informationFundamentals 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/
More informationFluid Dynamics. Massimo Ricotti. University of Maryland. Fluid Dynamics p.1/14
Fluid Dynamics p.1/14 Fluid Dynamics Massimo Ricotti ricotti@astro.umd.edu University of Maryland Fluid Dynamics p.2/14 The equations of fluid dynamics are coupled PDEs that form an IVP (hyperbolic). Use
More informationMechanics Physics 151
Mechanics Physics 151 Lecture 4 Continuous Systems and Fields (Chapter 13) What We Did Last Time Built Lagrangian formalism for continuous system Lagrangian L Lagrange s equation = L dxdydz Derived simple
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 informationDivergence Theorem and Its Application in Characterizing
Divergence Theorem and Its Application in Characterizing Fluid Flow Let v be the velocity of flow of a fluid element and ρ(x, y, z, t) be the mass density of fluid at a point (x, y, z) at time t. Thus,
More informationModels of ocean circulation are all based on the equations of motion.
Equations of motion Models of ocean circulation are all based on the equations of motion. Only in simple cases the equations of motion can be solved analytically, usually they must be solved numerically.
More informationAE/ME 339. K. M. Isaac. 9/22/2005 Topic 6 FluidFlowEquations_Introduction. Computational Fluid Dynamics (AE/ME 339) MAEEM Dept.
AE/ME 339 Computational Fluid Dynamics (CFD) 1...in the phrase computational fluid dynamics the word computational is simply an adjective to fluid dynamics.... -John D. Anderson 2 1 Equations of Fluid
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 information0.2. CONSERVATION LAW FOR FLUID 9
0.2. CONSERVATION LAW FOR FLUID 9 Consider x-component of Eq. (26), we have D(ρu) + ρu( v) dv t = ρg x dv t S pi ds, (27) where ρg x is the x-component of the bodily force, and the surface integral is
More informationConcepts in Engineering Mathematics: Lecture 39
Concepts in Engineering Mathematics: Lecture 39 Part IV: Vector Calculus Lecture 39 Version: 0.94 Dec7.15 Jont B. Allen; UIUC Urbana IL, USA December 9, 2015 Jont B. Allen; UIUC Urbana IL, USA Concepts
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 informationPressure in stationary and moving fluid Lab- Lab On- On Chip: Lecture 2
Pressure in stationary and moving fluid Lab-On-Chip: Lecture Lecture plan what is pressure e and how it s distributed in static fluid water pressure in engineering problems buoyancy y and archimedes law;
More informationOPAC102. The Acoustic Wave Equation
OPAC102 The Acoustic Wave Equation Acoustic waves in fluid Acoustic waves constitute one kind of pressure fluctuation that can exist in a compressible fluid. The restoring forces responsible for propagating
More informationMAC2313 Final A. (5 pts) 1. How many of the following are necessarily true? i. The vector field F = 2x + 3y, 3x 5y is conservative.
MAC2313 Final A (5 pts) 1. How many of the following are necessarily true? i. The vector field F = 2x + 3y, 3x 5y is conservative. ii. The vector field F = 5(x 2 + y 2 ) 3/2 x, y is radial. iii. All constant
More informationFluid mechanics and living organisms
Physics of the Human Body 37 Chapter 4: In this chapter we discuss the basic laws of fluid flow as they apply to life processes at various size scales For example, fluid dynamics at low Reynolds number
More informationModule 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
More informationKINEMATICS OF CONTINUA
KINEMATICS OF CONTINUA Introduction Deformation of a continuum Configurations of a continuum Deformation mapping Descriptions of motion Material time derivative Velocity and acceleration Transformation
More informationAdvection, Conservation, Conserved Physical Quantities, Wave Equations
EP711 Supplementary Material Thursday, September 4, 2014 Advection, Conservation, Conserved Physical Quantities, Wave Equations Jonathan B. Snively!Embry-Riddle Aeronautical University Contents EP711 Supplementary
More informationChapter 3 - Vector Calculus
Chapter 3 - Vector Calculus Gradient in Cartesian coordinate system f ( x, y, z,...) dr ( dx, dy, dz,...) Then, f f f f,,,... x y z f f f df dx dy dz... f dr x y z df 0 (constant f contour) f dr 0 or f
More informationLecture 1: Introduction to Linear and Non-Linear Waves
Lecture 1: Introduction to Linear and Non-Linear Waves Lecturer: Harvey Segur. Write-up: Michael Bates June 15, 2009 1 Introduction to Water Waves 1.1 Motivation and Basic Properties There are many types
More informationChapter 5. Sound Waves and Vortices. 5.1 Sound waves
Chapter 5 Sound Waves and Vortices In this chapter we explore a set of characteristic solutions to the uid equations with the goal of familiarizing the reader with typical behaviors in uid dynamics. Sound
More informationFundamentals of Fluid Dynamics: Elementary Viscous Flow
Fundamentals of Fluid Dynamics: Elementary Viscous Flow Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pan.pl/ tzielins/ Institute of Fundamental Technological Research
More informationFundamentals of Acoustics
Fundamentals of Acoustics Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pan.pl/ tzielins/ Institute of Fundamental Technological Research of the Polish Academy of Sciences
More informationIntroduction to Magnetohydrodynamics (MHD)
Introduction to Magnetohydrodynamics (MHD) Tony Arber University of Warwick 4th SOLARNET Summer School on Solar MHD and Reconnection Aim Derivation of MHD equations from conservation laws Quasi-neutrality
More informationAST242 LECTURE NOTES PART 5
AST242 LECTURE NOTES PART 5 Contents 1. Waves and instabilities 1 1.1. Sound waves compressive waves in 1D 1 2. Jeans Instability 5 3. Stratified Fluid Flows Waves or Instabilities on a Fluid Boundary
More informationModule 2: Governing Equations and Hypersonic Relations
Module 2: Governing Equations and Hypersonic Relations Lecture -2: Mass Conservation Equation 2.1 The Differential Equation for mass conservation: Let consider an infinitely small elemental control volume
More informationPhysics 5153 Classical Mechanics. Canonical Transformations-1
1 Introduction Physics 5153 Classical Mechanics Canonical Transformations The choice of generalized coordinates used to describe a physical system is completely arbitrary, but the Lagrangian is invariant
More informationTopics in Fluid Dynamics: Classical physics and recent mathematics
Topics in Fluid Dynamics: Classical physics and recent mathematics Toan T. Nguyen 1,2 Penn State University Graduate Student Seminar @ PSU Jan 18th, 2018 1 Homepage: http://math.psu.edu/nguyen 2 Math blog:
More informationAstrophysical Fluid Dynamics
Astrophysical Fluid Dynamics What is a Fluid? I. What is a fluid? I.1 The Fluid approximation: The fluid is an idealized concept in which the matter is described as a continuous medium with certain macroscopic
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 information- Marine Hydrodynamics. Lecture 14. F, M = [linear function of m ij ] [function of instantaneous U, U, Ω] not of motion history.
2.20 - Marine Hydrodynamics, Spring 2005 ecture 14 2.20 - Marine Hydrodynamics ecture 14 3.20 Some Properties of Added-Mass Coefficients 1. m ij = ρ [function of geometry only] F, M = [linear function
More informationFORMULA SHEET. General formulas:
FORMULA SHEET You may use this formula sheet during the Advanced Transport Phenomena course and it should contain all formulas you need during this course. Note that the weeks are numbered from 1.1 to
More informationSolutions of M3-4A16 Assessed Problems # 3 [#1] Exercises in exterior calculus operations
D. D. Holm Solutions to M3-4A16 Assessed Problems # 3 15 Dec 2010 1 Solutions of M3-4A16 Assessed Problems # 3 [#1] Exercises in exterior calculus operations Vector notation for differential basis elements:
More informationA Brief Revision of Vector Calculus and Maxwell s Equations
A Brief Revision of Vector Calculus and Maxwell s Equations Debapratim Ghosh Electronic Systems Group Department of Electrical Engineering Indian Institute of Technology Bombay e-mail: dghosh@ee.iitb.ac.in
More informationPressure in stationary and moving fluid. Lab-On-Chip: Lecture 2
Pressure in stationary and moving fluid Lab-On-Chip: Lecture Fluid Statics No shearing stress.no relative movement between adjacent fluid particles, i.e. static or moving as a single block Pressure at
More informationProf. dr. A. Achterberg, Astronomical Dept., IMAPP, Radboud Universiteit
Prof. dr. A. Achterberg, Astronomical Dept., IMAPP, Radboud Universiteit Central concepts: Phase velocity: velocity with which surfaces of constant phase move Group velocity: velocity with which slow
More informationMATH 228: Calculus III (FALL 2016) Sample Problems for FINAL EXAM SOLUTIONS
MATH 228: Calculus III (FALL 216) Sample Problems for FINAL EXAM SOLUTIONS MATH 228 Page 2 Problem 1. (2pts) Evaluate the line integral C xy dx + (x + y) dy along the parabola y x2 from ( 1, 1) to (2,
More informationMAE 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
More informationSolution Set Two. 1 Problem #1: Projectile Motion Cartesian Coordinates Polar Coordinates... 3
: Solution Set Two Northwestern University, Classical Mechanics Classical Mechanics, Third Ed.- Goldstein October 7, 2015 Contents 1 Problem #1: Projectile Motion. 2 1.1 Cartesian Coordinates....................................
More informationPhysics 607 Final Exam
Physics 607 Final Exam Please be well-organized, and show all significant steps clearly in all problems. You are graded on your work, so please do not just write down answers with no explanation! Do all
More informationLecture 41: Highlights
Lecture 41: Highlights The goal of this lecture is to remind you of some of the key points that we ve covered this semester Note that this is not the complete set of topics that may appear on the final
More information2.20 Fall 2018 Math Review
2.20 Fall 2018 Math Review September 10, 2018 These notes are to help you through the math used in this class. This is just a refresher, so if you never learned one of these topics you should look more
More informationDivergence Theorem December 2013
Divergence Theorem 17.3 11 December 2013 Fundamental Theorem, Four Ways. b F (x) dx = F (b) F (a) a [a, b] F (x) on boundary of If C path from P to Q, ( φ) ds = φ(q) φ(p) C φ on boundary of C Green s Theorem:
More informationThe Divergence Theorem Stokes Theorem Applications of Vector Calculus. Calculus. Vector Calculus (III)
Calculus Vector Calculus (III) Outline 1 The Divergence Theorem 2 Stokes Theorem 3 Applications of Vector Calculus The Divergence Theorem (I) Recall that at the end of section 12.5, we had rewritten Green
More informationPRACTICE PROBLEMS. Please let me know if you find any mistakes in the text so that i can fix them. 1. Mixed partial derivatives.
PRACTICE PROBLEMS Please let me know if you find any mistakes in the text so that i can fix them. 1.1. Let Show that f is C 1 and yet How is that possible? 1. Mixed partial derivatives f(x, y) = {xy x
More information18.325: Vortex Dynamics
8.35: Vortex Dynamics Problem Sheet. Fluid is barotropic which means p = p(. The Euler equation, in presence of a conservative body force, is Du Dt = p χ. This can be written, on use of a vector identity,
More informationDivergence Theorem Fundamental Theorem, Four Ways. 3D Fundamental Theorem. Divergence Theorem
Divergence Theorem 17.3 11 December 213 Fundamental Theorem, Four Ways. b F (x) dx = F (b) F (a) a [a, b] F (x) on boundary of If C path from P to Q, ( φ) ds = φ(q) φ(p) C φ on boundary of C Green s Theorem:
More informationLecture 4 Notes: 06 / 30. Energy carried by a wave
Lecture 4 Notes: 06 / 30 Energy carried by a wave We want to find the total energy (kinetic and potential) in a sine wave on a string. A small segment of a string at a fixed point x 0 behaves as a harmonic
More informationSummary of various integrals
ummary of various integrals Here s an arbitrary compilation of information about integrals Moisés made on a cold ecember night. 1 General things o not mix scalars and vectors! In particular ome integrals
More informationChapter 2: Basic Governing Equations
-1 Reynolds Transport Theorem (RTT) - Continuity Equation -3 The Linear Momentum Equation -4 The First Law of Thermodynamics -5 General Equation in Conservative Form -6 General Equation in Non-Conservative
More information2 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)
More informationLecture II: Vector and Multivariate Calculus
Lecture II: Vector and Multivariate Calculus Dot Product a, b R ' ', a ( b = +,- a + ( b + R. a ( b = a b cos θ. θ convex angle between the vectors. Squared norm of vector: a 3 = a ( a. Alternative notation:
More informationTransport processes. 7. Semester Chemical Engineering Civil Engineering
Transport processes 7. Semester Chemical Engineering Civil Engineering 1 Course plan 1. Elementary Fluid Dynamics 2. Fluid Kinematics 3. Finite Control Volume nalysis 4. Differential nalysis of Fluid Flow
More informationThermodynamics: A Brief Introduction. Thermodynamics: A Brief Introduction
Brief review or introduction to Classical Thermodynamics Hopefully you remember this equation from chemistry. The Gibbs Free Energy (G) as related to enthalpy (H) and entropy (S) and temperature (T). Δ
More informationJ. Szantyr Lecture No. 3 Fluid in equilibrium
J. Szantyr Lecture No. 3 Fluid in equilibrium Internal forces mutual interactions of the selected mass elements of the analysed region of fluid, forces having a surface character, forming pairs acting
More informationREVIEW. Hamilton s principle. based on FW-18. Variational statement of mechanics: (for conservative forces) action Equivalent to Newton s laws!
Hamilton s principle Variational statement of mechanics: (for conservative forces) action Equivalent to Newton s laws! based on FW-18 REVIEW the particle takes the path that minimizes the integrated difference
More informationMaxwell's Equations and Conservation Laws
Maxwell's Equations and Conservation Laws 1 Reading: Jackson 6.1 through 6.4, 6.7 Ampère's Law, since identically. Although for magnetostatics, generally Maxwell suggested: Use Gauss's Law to rewrite continuity
More informationAccelerator Physics NMI and Synchrotron Radiation. G. A. Krafft Old Dominion University Jefferson Lab Lecture 16
Accelerator Physics NMI and Synchrotron Radiation G. A. Krafft Old Dominion University Jefferson Lab Lecture 16 Graduate Accelerator Physics Fall 17 Oscillation Frequency nq I n i Z c E Re Z 1 mode has
More informationThe Euler equations in fluid mechanics
The Euler equations in fluid mechanics Jordan Bell jordan.bell@gmail.com Department of Mathematics, niversity of Toronto April 14, 014 1 Continuity equation Let Ω be a domain in R n and let C (Ω R); perhaps
More informationElectric and Magnetic Forces in Lagrangian and Hamiltonian Formalism
Electric and Magnetic Forces in Lagrangian and Hamiltonian Formalism Benjamin Hornberger 1/26/1 Phy 55, Classical Electrodynamics, Prof. Goldhaber Lecture notes from Oct. 26, 21 Lecture held by Prof. Weisberger
More informationCHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION
CHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION 7.1 THE NAVIER-STOKES EQUATIONS Under the assumption of a Newtonian stress-rate-of-strain constitutive equation and a linear, thermally conductive medium,
More informationOffshore Hydromechanics
Offshore Hydromechanics Module 1 : Hydrostatics Constant Flows Surface Waves OE4620 Offshore Hydromechanics Ir. W.E. de Vries Offshore Engineering Today First hour: Schedule for remainder of hydromechanics
More informationEKC314: TRANSPORT PHENOMENA Core Course for B.Eng.(Chemical Engineering) Semester II (2008/2009)
EKC314: TRANSPORT PHENOMENA Core Course for B.Eng.(Chemical Engineering) Semester II (2008/2009) Dr. Mohamad Hekarl Uzir-chhekarl@eng.usm.my School of Chemical Engineering Engineering Campus, Universiti
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 information1 Equations of motion
Part A Fluid Dynamics & Waves Draft date: 21 January 2014 1 1 1 Equations of motion 1.1 Introduction In this section we will derive the equations of motion for an inviscid fluid, that is a fluid with zero
More informationAE301 Aerodynamics I UNIT B: Theory of Aerodynamics
AE301 Aerodynamics I UNIT B: Theory of Aerodynamics ROAD MAP... B-1: Mathematics for Aerodynamics B-: Flow Field Representations B-3: Potential Flow Analysis B-4: Applications of Potential Flow Analysis
More informationBasic equations of motion in fluid mechanics
1 Annex 1 Basic equations of motion in fluid mechanics 1.1 Introduction It is assumed that the reader of this book is familiar with the basic laws of fluid mechanics. Nevertheless some of these laws will
More informationG : Statistical Mechanics
G5.651: Statistical Mechanics Notes for Lecture 1 I. DERIVATION OF THE DISCRETIZED PATH INTEGRAL We begin our discussion of the Feynman path integral with the canonical ensemble. The epressions for the
More informationIntroduction to Fluid Dynamics
Introduction to Fluid Dynamics Roger K. Smith Skript - auf englisch! Umsonst im Internet http://www.meteo.physik.uni-muenchen.de Wählen: Lehre Manuskripte Download User Name: meteo Password: download Aim
More informationIdeas from Vector Calculus Kurt Bryan
Ideas from Vector Calculus Kurt Bryan Most of the facts I state below are for functions of two or three variables, but with noted exceptions all are true for functions of n variables..1 Tangent Line Approximation
More informationLagrangian Formulation of Elastic Wave Equation on Riemannian Manifolds
RWE-C3-EAFIT Lagrangian Formulation of Elastic Wave Equation on Riemannian Manifolds Hector Roman Quiceno E. Advisors Ph.D Jairo Alberto Villegas G. Ph.D Diego Alberto Gutierrez I. Centro de Ciencias de
More information2. Conservation of Mass
2 Conservation of Mass The equation of mass conservation expresses a budget for the addition and removal of mass from a defined region of fluid Consider a fixed, non-deforming volume of fluid, V, called
More informationCONSERVATION OF MASS AND BALANCE OF LINEAR MOMENTUM
CONSERVATION OF MASS AND BALANCE OF LINEAR MOMENTUM Summary of integral theorems Material time derivative Reynolds transport theorem Principle of conservation of mass Principle of balance of linear momentum
More informationMathematical Models of Fluids
SOUND WAVES Mathematical Models of Fluids Fluids molecules roam and collide no springs Collisions cause pressure in fluid (Units: Pascal Pa = N/m 2 ) 2 mathematical models for fluid motion: 1) Bulk properties
More informationWebster s horn model on Bernoulli flow
Webster s horn model on Bernoulli flow Aalto University, Dept. Mathematics and Systems Analysis January 5th, 2018 Incompressible, steady Bernoulli principle Consider a straight tube Ω R 3 havin circular
More information[#1] R 3 bracket for the spherical pendulum
.. Holm Tuesday 11 January 2011 Solutions to MSc Enhanced Coursework for MA16 1 M3/4A16 MSc Enhanced Coursework arryl Holm Solutions Tuesday 11 January 2011 [#1] R 3 bracket for the spherical pendulum
More informationDisclaimer: This Final Exam Study Guide is meant to help you start studying. It is not necessarily a complete list of everything you need to know.
Disclaimer: This is meant to help you start studying. It is not necessarily a complete list of everything you need to know. The MTH 234 final exam mainly consists of standard response questions where students
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 informationConvection Heat Transfer
Convection Heat Transfer Department of Chemical Eng., Isfahan University of Technology, Isfahan, Iran Seyed Gholamreza Etemad Winter 2013 Heat convection: Introduction Difference between the temperature
More informationMicroscopic 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
More informationVector Calculus. A primer
Vector Calculus A primer Functions of Several Variables A single function of several variables: f: R $ R, f x (, x ),, x $ = y. Partial derivative vector, or gradient, is a vector: f = y,, y x ( x $ Multi-Valued
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 informationVarious lecture notes for
Various lecture notes for 18311. R. R. Rosales (MIT, Math. Dept., 2-337) April 12, 2013 Abstract Notes, both complete and/or incomplete, for MIT s 18.311 (Principles of Applied Mathematics). These notes
More information1. 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
More informationMultivariable Calculus
Multivariable Calculus In thermodynamics, we will frequently deal with functions of more than one variable e.g., P PT, V, n, U UT, V, n, U UT, P, n U = energy n = # moles etensive variable: depends on
More information2, where dp is the constant, R is the radius of
Dynamics of Viscous Flows (Lectures 8 to ) Q. Choose the correct answer (i) The average velocity of a one-dimensional incompressible fully developed viscous flow between two fixed parallel plates is m/s.
More informationVorticity Equation Marine Hydrodynamics Lecture 9. Return to viscous incompressible flow. N-S equation: v. Now: v = v + = 0 incompressible
13.01 Marine Hydrodynamics, Fall 004 Lecture 9 Copyright c 004 MIT - Department of Ocean Engineering, All rights reserved. Vorticity Equation 13.01 - Marine Hydrodynamics Lecture 9 Return to viscous incompressible
More information1 Particles in a room
Massachusetts Institute of Technology MITES 208 Physics III Lecture 07: Statistical Physics of the Ideal Gas In these notes we derive the partition function for a gas of non-interacting particles in a
More information4.1 LAWS OF MECHANICS - Review
4.1 LAWS OF MECHANICS - Review Ch4 9 SYSTEM System: Moving Fluid Definitions: System is defined as an arbitrary quantity of mass of fixed identity. Surrounding is everything external to this system. Boundary
More informationNumerical Methods for Partial Differential Equations
Numerical Methods for Partial Differential Equations Eric de Sturler University of Illinois at Urbana-Champaign The calculus of variations deals with maxima, minima, and stationary values of (definite)
More information