Module 1 : The equation of continuity. Lecture 1: Equation of Continuity

Size: px
Start display at page:

Download "Module 1 : The equation of continuity. Lecture 1: Equation of Continuity"

Transcription

1 1 Module 1 : The equaton of contnuty Lecture 1: Equaton of Contnuty

2 2 Advanced Heat and Mass Transfer: Modules 1. THE EQUATION OF CONTINUITY : Lectures 1-6 () () () (v) (v) Overall Mass Balance Momentum Balance Energy Balance Specal Mass Balance Equaton for the fluxes 2. DIFFUSIE HEAT AND MASS TRANSFER: Lectures 7-20 () () () (v) (v) Steady and Unsteady/One and Multple Dmensons Mass Transfer wth Chemcal Reacton Perturbaton Technques Movng Boundary Problems Smultaneous Heat and Mass Transfer 3. CONECTIE HEAT AND MASS TRANSFER: Lectures () () () (v) Flow Insde Ducts Dsperson Lamnar Boundary Layers Mass Transfer wth Chemcal Reactons

3 3 (v) (v) Asymptotc Methods Smultaneous Momentum, Heat and Mass Transfer (v) Natural Convecton 4. MULTICOMPONENT TRANSPORT: Lectures () () () (v) Bnary Systems Mut-component Flux Equatons Thermal Dffuson Dmensonal Analyss 5. MASS TRANSFER IN TURBULANT FLOWS: Lectures () () () Tme Averagng and Eddy scosty Unversal elocty Mass Transfer n Turbulent Ppe Flow Reference Books 1. Brd, R.B., Stewart, W.E. and Lghtfoot, E.N., Transport Phenomenon, Wley (1960). 2. Carslaw, H.S. and Jaeger, J.C., Conducton of heat n Solds, (2 nd ed) Oxford (1975). 3. Slattery, J., Momentum, Energy and Mass Transfer n Contnua, (2 nd ed) Krueger (1981). Transport Processes

4 4 Goals of the Course To relate mathematcal symbols to physcal realty To revew several classc problems To show examples of how to approach the unknown THE EQUATION OF CONTINUITY The Contnuum Approxmaton Feld varables (e.g. velocty) at a pont are spatal averages over a small volume around that pont, where has to be such that l << 1/3 << D, (1.1a) where, l s a characterstc mcroscopc length scale, whch can be of molecular dmensons or the dstance between molecules n a gas or the partcle sze n a sold/flud two-phase system, and D s a characterstc macroscopc length scale. For example, for flow n a ppe, D can be the ppe dameter. The contnuum approxmaton consders the fluds to be contnuous. Thus, the flud propertes such as temperature, pressure, densty and velocty of the flud are taken to be well defned at nfntely small ponts (.e. at mcroscopc level), defnng a reference element of volume, let s call ths volume RE, at the geometrc order of the dstance between the two adjacent molecules of flud. Propertes are assumed to vary contnuously from one pont to another, and are averaged over the volume RE. The fact that the flud s made up of dscrete molecules s gnored.

5 5 Euleran and Lagrangan coordnates Euleran Coordnate: n ths system the ndependent varables are x, y, z and t or x (=1, 2, 3) and t. Ths s a fxed coordnate system. The basc conservaton equaton are n the Euleran frame, R = R (x, t). In the Lagrangan frame, attenton s fxed on a partcular mass of flud as t flows, R = R (x o, t), where the coordnate x o specfes whch flud element s beng consdered. Materal Dervatve Consder a varable α such that α = α ( x, t) (1.1b) Then the total dfferental of α can be expressed as δα = δt + δx (1.1c) Dvson by a tme dfferental δt leads to the followng expresson: δα δt δx = + (1.1d) δt After takng the lmt δt 0, we obtan for the materal dervatve Dα Dt = + ν (1.1e) where, ν = s the flud velocty n drecton, n the lmt of δt 0 Dα s called the Materal Dervatve or Lagrangan Dervatve n tme and Dt Dervatve n tme. s the Euleran

6 6 The Materal dervatve or Lagrangan tme dervatve represents n total change n α as seen by an observer who s movng wth a partcular flud element. In the Lagrangan frame, we observe the partcle for a tme δt as t flows. The poston of a partcle changes by δx whle α changes as δα. Tme dervatves The tme dervatve s a dervatve of a functon wth respect to tme. It mples the rate of change of value of a functon wth respect to tme t. Partal tme dervatve Total tme dervatve (at a pont) By dvdng by dt, the total dfferental can be wrtten as total tme dervatve dc dt dx dy dc dt y dt z dt = (1.1f) Ths expresson represents the change n the tme of the functon c.e., (/) as we move about wth arbtrary veloctes n the coordnate drectons.e., (dx/dt, dy/dt and dz/dt). Substantal Dervatve or Materal Dervatve If we constran the moton to follow the moton of the ndvdual flud partcles, we obtan the Substantal Dervatve or Materal Dervatve (also known as Convectve Dervatve) gven by Dc Substantal tme Dervatveor Convectve Dervatve: = + U dt x + U y + U y z z (1.1g)

7 7 where U x, U y and U z are the components of the local flud velocty U n x, y and z drectons, respectvely. The Mass Contnuty Equaton The contnuty equaton s an overall mass balance about a control volume. Consder a volume element of volume fxed n space as shown n fgure below. Here the volume s bounded by a surface S wth outward unt normal vector n Fg. 1.1 Control olume for Mass Contnuty equaton ( Accumulat on of mass nsde ) = ( The Net nflux of mass through surface S) v ρ d = ρu n ds s (1.2) Here, ρ s the mass densty. The mnus sgn (-) n front of the ntegral s because of the choce of n pontng outwards. The Dvergence Theorem (Gauss) for a vector feld A gves

8 8 ( ) d = ( A n)ds S Fg.1.2. Gauss Dvergence theorem A (1.3a) Where s gradent (a vector) and can be expressed as = + j + k (1.3b) y z In equaton (1.3a), the left hand sde s the volume ntegral over the volume, the rght hand sde s the surface ntegral over the boundary of the volume. The closed manfold d s qute generally the boundary of orented by outward-pontng normals and n s the outward pontng unt normal feld of the boundary d. For A = ρu, equaton (1.2) becomes ( U ) d = ( ρu n)ds ρ (1.3c) S Substtute equaton (1.3c) on the rght hand sde of equaton (1.2) we get, ρ d = ( ρu )d (1.3d) or

9 9 ρ + U (1.3e) ( ρ ) d = 0 Equaton (1.3e) s known as Mass Contnuty equaton. Snce ths equaton must hold for arbtrary, Mass Contnuty Equaton becomes ρ + or ( ρ ) = 0 U (1.4a) Dρ + U Dt ρ( ) = 0 (1.4b) From equaton (1.1g), we know that Dρ Dt ρ = + U ( ρ) (1.4c) Incompressble fluds: Incompressble fluds are those fluds that do not exhbt any varaton n ρ densty ether n space or tme. Therefore for ncompressble fluds ρ = 0 and = 0. If ρ s constant (for ncompressble fluds) n space and tme, then the equaton of contnuty for ncompressble fluds becomes U =0 (1.4d) Equaton (1.4b) can also be wrtten as: 1 D Dt = ( U ) (1.4e) where, 1 = and ρ 1 D Dt s the rate of dlaton of the flud.

10 10 Applcaton: We can apply the prncple of contnuty to ppes wth cross sectons whch changes along ther length. See Fg 1.3 below. Fg.1.3. flud flowng through convergent-dvergent secton of the ppe A lqud s flowng from left to rght and the ppe s narrowng n the same drecton. By the contnuty prncple, the mass flow rate must be the same at each secton.

χ x B E (c) Figure 2.1.1: (a) a material particle in a body, (b) a place in space, (c) a configuration of the body

χ x B E (c) Figure 2.1.1: (a) a material particle in a body, (b) a place in space, (c) a configuration of the body Secton.. Moton.. The Materal Body and Moton hyscal materals n the real world are modeled usng an abstract mathematcal entty called a body. Ths body conssts of an nfnte number of materal partcles. Shown

More information

Macroscopic Momentum Balances

Macroscopic Momentum Balances Lecture 13 F. Morrson CM3110 2013 10/22/2013 CM3110 Transport I Part I: Flud Mechancs Macroscopc Momentum Balances Professor Fath Morrson Department of Chemcal Engneerng Mchgan Technologcal Unersty 1 Macroscopc

More information

ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM

ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look

More information

Kinematics of Fluids. Lecture 16. (Refer the text book CONTINUUM MECHANICS by GEORGE E. MASE, Schaum s Outlines) 17/02/2017

Kinematics of Fluids. Lecture 16. (Refer the text book CONTINUUM MECHANICS by GEORGE E. MASE, Schaum s Outlines) 17/02/2017 17/0/017 Lecture 16 (Refer the text boo CONTINUUM MECHANICS by GEORGE E. MASE, Schaum s Outlnes) Knematcs of Fluds Last class, we started dscussng about the nematcs of fluds. Recall the Lagrangan and Euleran

More information

Week3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity

Week3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1-s tme nterval. The velocty of the partcle

More information

PHYS 705: Classical Mechanics. Calculus of Variations II

PHYS 705: Classical Mechanics. Calculus of Variations II 1 PHYS 705: Classcal Mechancs Calculus of Varatons II 2 Calculus of Varatons: Generalzaton (no constrant yet) Suppose now that F depends on several dependent varables : We need to fnd such that has a statonary

More information

Lagrangian Field Theory

Lagrangian Field Theory Lagrangan Feld Theory Adam Lott PHY 391 Aprl 6, 017 1 Introducton Ths paper s a summary of Chapter of Mandl and Shaw s Quantum Feld Theory [1]. The frst thng to do s to fx the notaton. For the most part,

More information

PHYS 705: Classical Mechanics. Newtonian Mechanics

PHYS 705: Classical Mechanics. Newtonian Mechanics 1 PHYS 705: Classcal Mechancs Newtonan Mechancs Quck Revew of Newtonan Mechancs Basc Descrpton: -An dealzed pont partcle or a system of pont partcles n an nertal reference frame [Rgd bodes (ch. 5 later)]

More information

Physics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian-1

Physics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian-1 P. Guterrez Physcs 5153 Classcal Mechancs D Alembert s Prncple and The Lagrangan 1 Introducton The prncple of vrtual work provdes a method of solvng problems of statc equlbrum wthout havng to consder the

More information

Mathematical Preparations

Mathematical Preparations 1 Introducton Mathematcal Preparatons The theory of relatvty was developed to explan experments whch studed the propagaton of electromagnetc radaton n movng coordnate systems. Wthn expermental error the

More information

Physics 181. Particle Systems

Physics 181. Particle Systems Physcs 181 Partcle Systems Overvew In these notes we dscuss the varables approprate to the descrpton of systems of partcles, ther defntons, ther relatons, and ther conservatons laws. We consder a system

More information

Physics 5153 Classical Mechanics. Principle of Virtual Work-1

Physics 5153 Classical Mechanics. Principle of Virtual Work-1 P. Guterrez 1 Introducton Physcs 5153 Classcal Mechancs Prncple of Vrtual Work The frst varatonal prncple we encounter n mechancs s the prncple of vrtual work. It establshes the equlbrum condton of a mechancal

More information

2.29 Numerical Fluid Mechanics Fall 2011 Lecture 6

2.29 Numerical Fluid Mechanics Fall 2011 Lecture 6 REVIEW of Lecture 5 2.29 Numercal Flud Mechancs Fall 2011 Lecture 6 Contnuum Hypothess and conservaton laws Macroscopc Propertes Materal covered n class: Dfferental forms of conservaton laws Materal Dervatve

More information

CHAPTER 14 GENERAL PERTURBATION THEORY

CHAPTER 14 GENERAL PERTURBATION THEORY CHAPTER 4 GENERAL PERTURBATION THEORY 4 Introducton A partcle n orbt around a pont mass or a sphercally symmetrc mass dstrbuton s movng n a gravtatonal potental of the form GM / r In ths potental t moves

More information

CHAPTER 6. LAGRANGE S EQUATIONS (Analytical Mechanics)

CHAPTER 6. LAGRANGE S EQUATIONS (Analytical Mechanics) CHAPTER 6 LAGRANGE S EQUATIONS (Analytcal Mechancs) 1 Ex. 1: Consder a partcle movng on a fxed horzontal surface. r P Let, be the poston and F be the total force on the partcle. The FBD s: -mgk F 1 x O

More information

More metrics on cartesian products

More metrics on cartesian products More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of

More information

Transfer Functions. Convenient representation of a linear, dynamic model. A transfer function (TF) relates one input and one output: ( ) system

Transfer Functions. Convenient representation of a linear, dynamic model. A transfer function (TF) relates one input and one output: ( ) system Transfer Functons Convenent representaton of a lnear, dynamc model. A transfer functon (TF) relates one nput and one output: x t X s y t system Y s The followng termnology s used: x y nput output forcng

More information

Open Systems: Chemical Potential and Partial Molar Quantities Chemical Potential

Open Systems: Chemical Potential and Partial Molar Quantities Chemical Potential Open Systems: Chemcal Potental and Partal Molar Quanttes Chemcal Potental For closed systems, we have derved the followng relatonshps: du = TdS pdv dh = TdS + Vdp da = SdT pdv dg = VdP SdT For open systems,

More information

Module 3: Element Properties Lecture 1: Natural Coordinates

Module 3: Element Properties Lecture 1: Natural Coordinates Module 3: Element Propertes Lecture : Natural Coordnates Natural coordnate system s bascally a local coordnate system whch allows the specfcaton of a pont wthn the element by a set of dmensonless numbers

More information

Physics 207: Lecture 20. Today s Agenda Homework for Monday

Physics 207: Lecture 20. Today s Agenda Homework for Monday Physcs 207: Lecture 20 Today s Agenda Homework for Monday Recap: Systems of Partcles Center of mass Velocty and acceleraton of the center of mass Dynamcs of the center of mass Lnear Momentum Example problems

More information

MATH 5630: Discrete Time-Space Model Hung Phan, UMass Lowell March 1, 2018

MATH 5630: Discrete Time-Space Model Hung Phan, UMass Lowell March 1, 2018 MATH 5630: Dscrete Tme-Space Model Hung Phan, UMass Lowell March, 08 Newton s Law of Coolng Consder the coolng of a well strred coffee so that the temperature does not depend on space Newton s law of collng

More information

Week 9 Chapter 10 Section 1-5

Week 9 Chapter 10 Section 1-5 Week 9 Chapter 10 Secton 1-5 Rotaton Rgd Object A rgd object s one that s nondeformable The relatve locatons of all partcles makng up the object reman constant All real objects are deformable to some extent,

More information

(Online First)A Lattice Boltzmann Scheme for Diffusion Equation in Spherical Coordinate

(Online First)A Lattice Boltzmann Scheme for Diffusion Equation in Spherical Coordinate Internatonal Journal of Mathematcs and Systems Scence (018) Volume 1 do:10.494/jmss.v1.815 (Onlne Frst)A Lattce Boltzmann Scheme for Dffuson Equaton n Sphercal Coordnate Debabrata Datta 1 *, T K Pal 1

More information

Thermodynamics General

Thermodynamics General Thermodynamcs General Lecture 1 Lecture 1 s devoted to establshng buldng blocks for dscussng thermodynamcs. In addton, the equaton of state wll be establshed. I. Buldng blocks for thermodynamcs A. Dmensons,

More information

The Feynman path integral

The Feynman path integral The Feynman path ntegral Aprl 3, 205 Hesenberg and Schrödnger pctures The Schrödnger wave functon places the tme dependence of a physcal system n the state, ψ, t, where the state s a vector n Hlbert space

More information

Georgia Tech PHYS 6124 Mathematical Methods of Physics I

Georgia Tech PHYS 6124 Mathematical Methods of Physics I Georga Tech PHYS 624 Mathematcal Methods of Physcs I Instructor: Predrag Cvtanovć Fall semester 202 Homework Set #7 due October 30 202 == show all your work for maxmum credt == put labels ttle legends

More information

Publication 2006/01. Transport Equations in Incompressible. Lars Davidson

Publication 2006/01. Transport Equations in Incompressible. Lars Davidson Publcaton 2006/01 Transport Equatons n Incompressble URANS and LES Lars Davdson Dvson of Flud Dynamcs Department of Appled Mechancs Chalmers Unversty of Technology Göteborg, Sweden, May 2006 Transport

More information

The equation of motion of a dynamical system is given by a set of differential equations. That is (1)

The equation of motion of a dynamical system is given by a set of differential equations. That is (1) Dynamcal Systems Many engneerng and natural systems are dynamcal systems. For example a pendulum s a dynamcal system. State l The state of the dynamcal system specfes t condtons. For a pendulum n the absence

More information

Spin-rotation coupling of the angularly accelerated rigid body

Spin-rotation coupling of the angularly accelerated rigid body Spn-rotaton couplng of the angularly accelerated rgd body Loua Hassan Elzen Basher Khartoum, Sudan. Postal code:11123 E-mal: louaelzen@gmal.com November 1, 2017 All Rghts Reserved. Abstract Ths paper s

More information

Lecture 20: Noether s Theorem

Lecture 20: Noether s Theorem Lecture 20: Noether s Theorem In our revew of Newtonan Mechancs, we were remnded that some quanttes (energy, lnear momentum, and angular momentum) are conserved That s, they are constant f no external

More information

Canonical transformations

Canonical transformations Canoncal transformatons November 23, 2014 Recall that we have defned a symplectc transformaton to be any lnear transformaton M A B leavng the symplectc form nvarant, Ω AB M A CM B DΩ CD Coordnate transformatons,

More information

The Geometry of Logit and Probit

The Geometry of Logit and Probit The Geometry of Logt and Probt Ths short note s meant as a supplement to Chapters and 3 of Spatal Models of Parlamentary Votng and the notaton and reference to fgures n the text below s to those two chapters.

More information

Conservation of Angular Momentum = "Spin"

Conservation of Angular Momentum = Spin Page 1 of 6 Conservaton of Angular Momentum = "Spn" We can assgn a drecton to the angular velocty: drecton of = drecton of axs + rght hand rule (wth rght hand, curl fngers n drecton of rotaton, thumb ponts

More information

In this section is given an overview of the common elasticity models.

In this section is given an overview of the common elasticity models. Secton 4.1 4.1 Elastc Solds In ths secton s gven an overvew of the common elastcty models. 4.1.1 The Lnear Elastc Sold The classcal Lnear Elastc model, or Hooean model, has the followng lnear relatonshp

More information

Tensor Smooth Length for SPH Modelling of High Speed Impact

Tensor Smooth Length for SPH Modelling of High Speed Impact Tensor Smooth Length for SPH Modellng of Hgh Speed Impact Roman Cherepanov and Alexander Gerasmov Insttute of Appled mathematcs and mechancs, Tomsk State Unversty 634050, Lenna av. 36, Tomsk, Russa RCherepanov82@gmal.com,Ger@npmm.tsu.ru

More information

Classical Mechanics Virtual Work & d Alembert s Principle

Classical Mechanics Virtual Work & d Alembert s Principle Classcal Mechancs Vrtual Work & d Alembert s Prncple Dpan Kumar Ghosh UM-DAE Centre for Excellence n Basc Scences Kalna, Mumba 400098 August 15, 2016 1 Constrants Moton of a system of partcles s often

More information

EPR Paradox and the Physical Meaning of an Experiment in Quantum Mechanics. Vesselin C. Noninski

EPR Paradox and the Physical Meaning of an Experiment in Quantum Mechanics. Vesselin C. Noninski EPR Paradox and the Physcal Meanng of an Experment n Quantum Mechancs Vesseln C Nonnsk vesselnnonnsk@verzonnet Abstract It s shown that there s one purely determnstc outcome when measurement s made on

More information

2 Finite difference basics

2 Finite difference basics Numersche Methoden 1, WS 11/12 B.J.P. Kaus 2 Fnte dfference bascs Consder the one- The bascs of the fnte dfference method are best understood wth an example. dmensonal transent heat conducton equaton T

More information

STUDY ON TWO PHASE FLOW IN MICRO CHANNEL BASED ON EXPERI- MENTS AND NUMERICAL EXAMINATIONS

STUDY ON TWO PHASE FLOW IN MICRO CHANNEL BASED ON EXPERI- MENTS AND NUMERICAL EXAMINATIONS Blucher Mechancal Engneerng Proceedngs May 0, vol., num. www.proceedngs.blucher.com.br/evento/0wccm STUDY ON TWO PHASE FLOW IN MICRO CHANNEL BASED ON EXPERI- MENTS AND NUMERICAL EXAMINATIONS Takahko Kurahash,

More information

Solutions to Exercises in Astrophysical Gas Dynamics

Solutions to Exercises in Astrophysical Gas Dynamics 1 Solutons to Exercses n Astrophyscal Gas Dynamcs 1. (a). Snce u 1, v are vectors then, under an orthogonal transformaton, u = a j u j v = a k u k Therefore, u v = a j a k u j v k = δ jk u j v k = u j

More information

Inductance Calculation for Conductors of Arbitrary Shape

Inductance Calculation for Conductors of Arbitrary Shape CRYO/02/028 Aprl 5, 2002 Inductance Calculaton for Conductors of Arbtrary Shape L. Bottura Dstrbuton: Internal Summary In ths note we descrbe a method for the numercal calculaton of nductances among conductors

More information

1. Why turbulence occur? Hydrodynamic Instability. Hydrodynamic Instability. Centrifugal Instability: Rayleigh-Benard Instability:

1. Why turbulence occur? Hydrodynamic Instability. Hydrodynamic Instability. Centrifugal Instability: Rayleigh-Benard Instability: . Why turbulence occur? Hydrodynamc Instablty Hydrodynamc Instablty T Centrfugal Instablty: Ω Raylegh-Benard Instablty: Drvng force: centrfugal force Drvng force: buoyancy flud Dampng force: vscous dsspaton

More information

Mechanics Physics 151

Mechanics Physics 151 Mechancs Physcs 151 Lecture 3 Lagrange s Equatons (Goldsten Chapter 1) Hamlton s Prncple (Chapter 2) What We Dd Last Tme! Dscussed mult-partcle systems! Internal and external forces! Laws of acton and

More information

Thermal-Fluids I. Chapter 18 Transient heat conduction. Dr. Primal Fernando Ph: (850)

Thermal-Fluids I. Chapter 18 Transient heat conduction. Dr. Primal Fernando Ph: (850) hermal-fluds I Chapter 18 ransent heat conducton Dr. Prmal Fernando prmal@eng.fsu.edu Ph: (850) 410-6323 1 ransent heat conducton In general, he temperature of a body vares wth tme as well as poston. In

More information

n α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0

n α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0 MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector

More information

Celestial Mechanics. Basic Orbits. Why circles? Tycho Brahe. PHY celestial-mechanics - J. Hedberg

Celestial Mechanics. Basic Orbits. Why circles? Tycho Brahe. PHY celestial-mechanics - J. Hedberg PHY 454 - celestal-mechancs - J. Hedberg - 207 Celestal Mechancs. Basc Orbts. Why crcles? 2. Tycho Brahe 3. Kepler 4. 3 laws of orbtng bodes 2. Newtonan Mechancs 3. Newton's Laws. Law of Gravtaton 2. The

More information

Calculus of Variations Basics

Calculus of Variations Basics Chapter 1 Calculus of Varatons Bascs 1.1 Varaton of a General Functonal In ths chapter, we derve the general formula for the varaton of a functonal of the form J [y 1,y 2,,y n ] F x,y 1,y 2,,y n,y 1,y

More information

Appendix B. The Finite Difference Scheme

Appendix B. The Finite Difference Scheme 140 APPENDIXES Appendx B. The Fnte Dfference Scheme In ths appendx we present numercal technques whch are used to approxmate solutons of system 3.1 3.3. A comprehensve treatment of theoretcal and mplementaton

More information

Computational Fluid Dynamics. Smoothed Particle Hydrodynamics. Simulations. Smoothing Kernels and Basis of SPH

Computational Fluid Dynamics. Smoothed Particle Hydrodynamics. Simulations. Smoothing Kernels and Basis of SPH Computatonal Flud Dynamcs If you want to learn a bt more of the math behnd flud dynamcs, read my prevous post about the Naver- Stokes equatons and Newtonan fluds. The equatons derved n the post are the

More information

Physics 607 Exam 1. ( ) = 1, Γ( z +1) = zγ( z) x n e x2 dx = 1. e x2

Physics 607 Exam 1. ( ) = 1, Γ( z +1) = zγ( z) x n e x2 dx = 1. e x2 Physcs 607 Exam 1 Please be well-organzed, and show all sgnfcant steps clearly n all problems. You are graded on your wor, so please do not just wrte down answers wth no explanaton! Do all your wor on

More information

coordinates. Then, the position vectors are described by

coordinates. Then, the position vectors are described by Revewng, what we have dscussed so far: Generalzed coordnates Any number of varables (say, n) suffcent to specfy the confguraton of the system at each nstant to tme (need not be the mnmum number). In general,

More information

Turbulent Flow. Turbulent Flow

Turbulent Flow. Turbulent Flow http://www.youtube.com/watch?v=xoll2kedog&feature=related http://br.youtube.com/watch?v=7kkftgx2any http://br.youtube.com/watch?v=vqhxihpvcvu 1. Caothc fluctuatons wth a wde range of frequences and

More information

Perfect Fluid Cosmological Model in the Frame Work Lyra s Manifold

Perfect Fluid Cosmological Model in the Frame Work Lyra s Manifold Prespacetme Journal December 06 Volume 7 Issue 6 pp. 095-099 Pund, A. M. & Avachar, G.., Perfect Flud Cosmologcal Model n the Frame Work Lyra s Manfold Perfect Flud Cosmologcal Model n the Frame Work Lyra

More information

Astrophysical Fluid Dynamics What is a Fluid?

Astrophysical Fluid Dynamics What is a Fluid? Astrophyscal Flud Dynamcs What s a Flud? 1 I. What s a flud? I.1 The Flud approxmaton: The flud s an dealzed concept n whch the matter s descrbed as a contnuous medum wth certan macroscopc propertes that

More information

Normally, in one phase reservoir simulation we would deal with one of the following fluid systems:

Normally, in one phase reservoir simulation we would deal with one of the following fluid systems: TPG4160 Reservor Smulaton 2017 page 1 of 9 ONE-DIMENSIONAL, ONE-PHASE RESERVOIR SIMULATION Flud systems The term sngle phase apples to any system wth only one phase present n the reservor In some cases

More information

Inner Product. Euclidean Space. Orthonormal Basis. Orthogonal

Inner Product. Euclidean Space. Orthonormal Basis. Orthogonal Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,

More information

PES 1120 Spring 2014, Spendier Lecture 6/Page 1

PES 1120 Spring 2014, Spendier Lecture 6/Page 1 PES 110 Sprng 014, Spender Lecture 6/Page 1 Lecture today: Chapter 1) Electrc feld due to charge dstrbutons -> charged rod -> charged rng We ntroduced the electrc feld, E. I defned t as an nvsble aura

More information

CONTROLLED FLOW SIMULATION USING SPH METHOD

CONTROLLED FLOW SIMULATION USING SPH METHOD HERI COADA AIR FORCE ACADEMY ROMAIA ITERATIOAL COFERECE of SCIETIFIC PAPER AFASES 01 Brasov, 4-6 May 01 GEERAL M.R. STEFAIK ARMED FORCES ACADEMY SLOVAK REPUBLIC COTROLLED FLOW SIMULATIO USIG SPH METHOD

More information

Formal solvers of the RT equation

Formal solvers of the RT equation Formal solvers of the RT equaton Formal RT solvers Runge- Kutta (reference solver) Pskunov N.: 979, Master Thess Long characterstcs (Feautrer scheme) Cannon C.J.: 970, ApJ 6, 55 Short characterstcs (Hermtan

More information

Basic concept of reactive flows. Basic concept of reactive flows Combustion Mixing and reaction in high viscous fluid Application of Chaos

Basic concept of reactive flows. Basic concept of reactive flows Combustion Mixing and reaction in high viscous fluid Application of Chaos Introducton to Toshhsa Ueda School of Scence for Open and Envronmental Systems Keo Unversty, Japan Combuston Mxng and reacton n hgh vscous flud Applcaton of Chaos Keo Unversty 1 Keo Unversty 2 What s reactve

More information

NUMERICAL DIFFERENTIATION

NUMERICAL DIFFERENTIATION NUMERICAL DIFFERENTIATION 1 Introducton Dfferentaton s a method to compute the rate at whch a dependent output y changes wth respect to the change n the ndependent nput x. Ths rate of change s called the

More information

Physics 207 Lecture 6

Physics 207 Lecture 6 Physcs 207 Lecture 6 Agenda: Physcs 207, Lecture 6, Sept. 25 Chapter 4 Frames of reference Chapter 5 ewton s Law Mass Inerta s (contact and non-contact) Frcton (a external force that opposes moton) Free

More information

First Law: A body at rest remains at rest, a body in motion continues to move at constant velocity, unless acted upon by an external force.

First Law: A body at rest remains at rest, a body in motion continues to move at constant velocity, unless acted upon by an external force. Secton 1. Dynamcs (Newton s Laws of Moton) Two approaches: 1) Gven all the forces actng on a body, predct the subsequent (changes n) moton. 2) Gven the (changes n) moton of a body, nfer what forces act

More information

CHEMICAL ENGINEERING

CHEMICAL ENGINEERING Postal Correspondence GATE & PSUs -MT To Buy Postal Correspondence Packages call at 0-9990657855 1 TABLE OF CONTENT S. No. Ttle Page no. 1. Introducton 3 2. Dffuson 10 3. Dryng and Humdfcaton 24 4. Absorpton

More information

Lecture 5.8 Flux Vector Splitting

Lecture 5.8 Flux Vector Splitting Lecture 5.8 Flux Vector Splttng 1 Flux Vector Splttng The vector E n (5.7.) can be rewrtten as E = AU (5.8.1) (wth A as gven n (5.7.4) or (5.7.6) ) whenever, the equaton of state s of the separable form

More information

π e ax2 dx = x 2 e ax2 dx or x 3 e ax2 dx = 1 x 4 e ax2 dx = 3 π 8a 5/2 (a) We are considering the Maxwell velocity distribution function: 2πτ/m

π e ax2 dx = x 2 e ax2 dx or x 3 e ax2 dx = 1 x 4 e ax2 dx = 3 π 8a 5/2 (a) We are considering the Maxwell velocity distribution function: 2πτ/m Homework Solutons Problem In solvng ths problem, we wll need to calculate some moments of the Gaussan dstrbuton. The brute-force method s to ntegrate by parts but there s a nce trck. The followng ntegrals

More information

Module 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur

Module 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur Module 3 LOSSY IMAGE COMPRESSION SYSTEMS Verson ECE IIT, Kharagpur Lesson 6 Theory of Quantzaton Verson ECE IIT, Kharagpur Instructonal Objectves At the end of ths lesson, the students should be able to:

More information

CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE

CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE Analytcal soluton s usually not possble when exctaton vares arbtrarly wth tme or f the system s nonlnear. Such problems can be solved by numercal tmesteppng

More information

A particle in a state of uniform motion remain in that state of motion unless acted upon by external force.

A particle in a state of uniform motion remain in that state of motion unless acted upon by external force. The fundamental prncples of classcal mechancs were lad down by Galleo and Newton n the 16th and 17th centures. In 1686, Newton wrote the Prncpa where he gave us three laws of moton, one law of gravty,

More information

Chapter 8. Potential Energy and Conservation of Energy

Chapter 8. Potential Energy and Conservation of Energy Chapter 8 Potental Energy and Conservaton of Energy In ths chapter we wll ntroduce the followng concepts: Potental Energy Conservatve and non-conservatve forces Mechancal Energy Conservaton of Mechancal

More information

Quantum Mechanics I Problem set No.1

Quantum Mechanics I Problem set No.1 Quantum Mechancs I Problem set No.1 Septembe0, 2017 1 The Least Acton Prncple The acton reads S = d t L(q, q) (1) accordng to the least (extremal) acton prncple, the varaton of acton s zero 0 = δs = t

More information

AERODYNAMICS I LECTURE 6 AERODYNAMICS OF A WING FUNDAMENTALS OF THE LIFTING-LINE THEORY

AERODYNAMICS I LECTURE 6 AERODYNAMICS OF A WING FUNDAMENTALS OF THE LIFTING-LINE THEORY LECTURE 6 AERODYNAMICS OF A WING FUNDAMENTALS OF THE LIFTING-LINE THEORY The Bot-Savart Law The velocty nduced by the sngular vortex lne wth the crculaton can be determned by means of the Bot- Savart formula

More information

Lecture Note 3. Eshelby s Inclusion II

Lecture Note 3. Eshelby s Inclusion II ME340B Elastcty of Mcroscopc Structures Stanford Unversty Wnter 004 Lecture Note 3. Eshelby s Incluson II Chrs Wenberger and We Ca c All rghts reserved January 6, 004 Contents 1 Incluson energy n an nfnte

More information

GENERAL EQUATIONS OF PHYSICO-CHEMICAL

GENERAL EQUATIONS OF PHYSICO-CHEMICAL GENERAL EQUATIONS OF PHYSICO-CHEMICAL PROCESSES Causes and conons for the evoluton of a system... 1 Integral formulaton of balance equatons... 2 Dfferental formulaton of balance equatons... 3 Boundary

More information

Mechanics Physics 151

Mechanics Physics 151 Mechancs Physcs 5 Lecture 7 Specal Relatvty (Chapter 7) What We Dd Last Tme Worked on relatvstc knematcs Essental tool for epermental physcs Basc technques are easy: Defne all 4 vectors Calculate c-o-m

More information

Analytical classical dynamics

Analytical classical dynamics Analytcal classcal ynamcs by Youun Hu Insttute of plasma physcs, Chnese Acaemy of Scences Emal: yhu@pp.cas.cn Abstract These notes were ntally wrtten when I rea tzpatrck s book[] an were later revse to

More information

3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X

3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number

More information

SIMULATION OF WAVE PROPAGATION IN AN HETEROGENEOUS ELASTIC ROD

SIMULATION OF WAVE PROPAGATION IN AN HETEROGENEOUS ELASTIC ROD SIMUATION OF WAVE POPAGATION IN AN HETEOGENEOUS EASTIC OD ogéro M Saldanha da Gama Unversdade do Estado do o de Janero ua Sào Francsco Xaver 54, sala 5 A 559-9, o de Janero, Brasl e-mal: rsgama@domancombr

More information

Introduction to Turbulence Modeling

Introduction to Turbulence Modeling Introducton to Turbulence Modelng Professor Ismal B. Celk West Vrgna nversty Ismal.Celk@mal.wvu.edu CFD Lab. - West Vrgna nversty I-1 Introducton to Turbulence CFD Lab. - West Vrgna nversty I-2 Introducton

More information

MAGNUM - A Fortran Library for the Calculation of Magnetic Configurations

MAGNUM - A Fortran Library for the Calculation of Magnetic Configurations CRYO/6/34 September, 3, 6 MAGNUM - A Fortran Lbrary for the Calculaton of Magnetc Confguratons L. Bottura Dstrbuton: Keywords: P. Bruzzone, M. Calv, J. Lster, C. Marnucc (EPFL/CRPP), A. Portone (EFDA-

More information

Moments of Inertia. and reminds us of the analogous equation for linear momentum p= mv, which is of the form. The kinetic energy of the body is.

Moments of Inertia. and reminds us of the analogous equation for linear momentum p= mv, which is of the form. The kinetic energy of the body is. Moments of Inerta Suppose a body s movng on a crcular path wth constant speed Let s consder two quanttes: the body s angular momentum L about the center of the crcle, and ts knetc energy T How are these

More information

Multicomponent Flows

Multicomponent Flows Mole Fracton emperature (K) ransport School of Aerospace Engneerng Equatons for Multcomponent Flows Jerry Setzman.2 25.15 2.1.5 CH4 H2O HCO x 1 emperature Methane Flame.1.2.3 Dstance (cm) 15 1 5 ransport

More information

A PROCEDURE FOR SIMULATING THE NONLINEAR CONDUCTION HEAT TRANSFER IN A BODY WITH TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY.

A PROCEDURE FOR SIMULATING THE NONLINEAR CONDUCTION HEAT TRANSFER IN A BODY WITH TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY. Proceedngs of the th Brazlan Congress of Thermal Scences and Engneerng -- ENCIT 006 Braz. Soc. of Mechancal Scences and Engneerng -- ABCM, Curtba, Brazl,- Dec. 5-8, 006 A PROCEDURE FOR SIMULATING THE NONLINEAR

More information

G. Bergeles Department of Mechanical Engineering, Nat. Technical University of Athens, Greece

G. Bergeles Department of Mechanical Engineering, Nat. Technical University of Athens, Greece ELEMENTARY FLUID DYNAMICS G. Bergeles Department of Mechancal Engneerng, Nat. Techncal Unversty of Athens, Greece Keywords: Flud Mechancs, Statcs, Knematcs, Naver-Stokes, Mass Conservaton, Momentum balance,

More information

Elshaboury SM et al.; Sch. J. Phys. Math. Stat., 2015; Vol-2; Issue-2B (Mar-May); pp

Elshaboury SM et al.; Sch. J. Phys. Math. Stat., 2015; Vol-2; Issue-2B (Mar-May); pp Elshabour SM et al.; Sch. J. Phs. Math. Stat. 5; Vol-; Issue-B (Mar-Ma); pp-69-75 Scholars Journal of Phscs Mathematcs Statstcs Sch. J. Phs. Math. Stat. 5; (B):69-75 Scholars Academc Scentfc Publshers

More information

arxiv: v1 [physics.flu-dyn] 16 Sep 2013

arxiv: v1 [physics.flu-dyn] 16 Sep 2013 Three-Dmensonal Smoothed Partcle Hydrodynamcs Method for Smulatng Free Surface Flows Rzal Dw Prayogo a,b, Chrstan Fredy Naa a a Faculty of Mathematcs and Natural Scences, Insttut Teknolog Bandung, Jl.

More information

Chapter 11 Angular Momentum

Chapter 11 Angular Momentum Chapter 11 Angular Momentum Analyss Model: Nonsolated System (Angular Momentum) Angular Momentum of a Rotatng Rgd Object Analyss Model: Isolated System (Angular Momentum) Angular Momentum of a Partcle

More information

Lecture 12: Discrete Laplacian

Lecture 12: Discrete Laplacian Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly

More information

Marginal Effects in Probit Models: Interpretation and Testing. 1. Interpreting Probit Coefficients

Marginal Effects in Probit Models: Interpretation and Testing. 1. Interpreting Probit Coefficients ECON 5 -- NOE 15 Margnal Effects n Probt Models: Interpretaton and estng hs note ntroduces you to the two types of margnal effects n probt models: margnal ndex effects, and margnal probablty effects. It

More information

Introduction to circuit analysis. Classification of Materials

Introduction to circuit analysis. Classification of Materials Introducton to crcut analyss OUTLINE Electrcal quanttes Charge Current Voltage Power The deal basc crcut element Sgn conventons Current versus voltage (I-V) graph Readng: 1.2, 1.3,1.6 Lecture 2, Slde 1

More information

Three views of mechanics

Three views of mechanics Three vews of mechancs John Hubbard, n L. Gross s course February 1, 211 1 Introducton A mechancal system s manfold wth a Remannan metrc K : T M R called knetc energy and a functon V : M R called potental

More information

Lecture 6/7 (February 10/12, 2014) DIRAC EQUATION. The non-relativistic Schrödinger equation was obtained by noting that the Hamiltonian 2

Lecture 6/7 (February 10/12, 2014) DIRAC EQUATION. The non-relativistic Schrödinger equation was obtained by noting that the Hamiltonian 2 P470 Lecture 6/7 (February 10/1, 014) DIRAC EQUATION The non-relatvstc Schrödnger equaton was obtaned by notng that the Hamltonan H = P (1) m can be transformed nto an operator form wth the substtutons

More information

Linear Momentum. Center of Mass.

Linear Momentum. Center of Mass. Lecture 6 Chapter 9 Physcs I 03.3.04 Lnear omentum. Center of ass. Course webste: http://faculty.uml.edu/ndry_danylov/teachng/physcsi Lecture Capture: http://echo360.uml.edu/danylov03/physcssprng.html

More information

Numerical Transient Heat Conduction Experiment

Numerical Transient Heat Conduction Experiment Numercal ransent Heat Conducton Experment OBJECIVE 1. o demonstrate the basc prncples of conducton heat transfer.. o show how the thermal conductvty of a sold can be measured. 3. o demonstrate the use

More information

Chapter 3. r r. Position, Velocity, and Acceleration Revisited

Chapter 3. r r. Position, Velocity, and Acceleration Revisited Chapter 3 Poston, Velocty, and Acceleraton Revsted The poston vector of a partcle s a vector drawn from the orgn to the locaton of the partcle. In two dmensons: r = x ˆ+ yj ˆ (1) The dsplacement vector

More information

Buckingham s pi-theorem

Buckingham s pi-theorem TMA495 Mathematcal modellng 2004 Buckngham s p-theorem Harald Hanche-Olsen hanche@math.ntnu.no Theory Ths note s about physcal quanttes R,...,R n. We lke to measure them n a consstent system of unts, such

More information

Week 8: Chapter 9. Linear Momentum. Newton Law and Momentum. Linear Momentum, cont. Conservation of Linear Momentum. Conservation of Momentum, 2

Week 8: Chapter 9. Linear Momentum. Newton Law and Momentum. Linear Momentum, cont. Conservation of Linear Momentum. Conservation of Momentum, 2 Lnear omentum Week 8: Chapter 9 Lnear omentum and Collsons The lnear momentum of a partcle, or an object that can be modeled as a partcle, of mass m movng wth a velocty v s defned to be the product of

More information

Physics 106a, Caltech 11 October, Lecture 4: Constraints, Virtual Work, etc. Constraints

Physics 106a, Caltech 11 October, Lecture 4: Constraints, Virtual Work, etc. Constraints Physcs 106a, Caltech 11 October, 2018 Lecture 4: Constrants, Vrtual Work, etc. Many, f not all, dynamcal problems we want to solve are constraned: not all of the possble 3 coordnates for M partcles (or

More information

CALCU THIRD EDITION. Monfy J. Strauss. Texas Tech University. Gerald L Bradley. Claremont McKenna College. Karl J. Smith. Santa Rosa Junior College

CALCU THIRD EDITION. Monfy J. Strauss. Texas Tech University. Gerald L Bradley. Claremont McKenna College. Karl J. Smith. Santa Rosa Junior College CALCU THIRD EDITION Monfy J. Strauss Texas Tech Unversty Gerald L Bradley Claremont McKenna College Karl J. Smth Santa Rosa Junor College Prentce Hall, Upper Saddle Rver, New Jersey 07458 Preface x 1 Functons

More information

Supplementary Notes for Chapter 9 Mixture Thermodynamics

Supplementary Notes for Chapter 9 Mixture Thermodynamics Supplementary Notes for Chapter 9 Mxture Thermodynamcs Key ponts Nne major topcs of Chapter 9 are revewed below: 1. Notaton and operatonal equatons for mxtures 2. PVTN EOSs for mxtures 3. General effects

More information