Differential relations for fluid flow


 Griffin Lamb
 3 years ago
 Views:
Transcription
1 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 pattern as oppose to control volume technique that provide gross average g information about the flow. Integral Relations for CV M. Bahrami ENSC 283 Spring
2 Acceleration field of a fluid The Cartesian vector form of a velocity field in general is: The acceleration vector field can be calculated: The total acceleration vector: Integral Relations for CV M. Bahrami ENSC 283 Spring
3 Conservation of mass The conservation of mass for the element can be written as: Integral Relations for CV M. Bahrami ENSC 283 Spring
4 Cylindrical polar coordinates The continuity equation in cylindrical coordinates become: Integral Relations for CV M. Bahrami ENSC 283 Spring
5 Special cases For steady compressible flow, continuity equation simplifies to: For incompressible ibl flow, continuity it equation can be further simplified since density changes are negligible: Note: the continuity equation is always important and must always be satisfied for a rational analysis of a flow pattern. Integral Relations for CV M. Bahrami ENSC 283 Spring
6 Linear momentum equation In a Cartesian coordinates, the momentum equation can be written as: There are types of forces: body forces and surface forces. Body forces are due to external fields such as gravity and magnetism fields. We only consider gravity forces: Surface forces are due to the stresses on the sides of the control surface. These stresses are the sum of hydrostatic pressure plus viscous stresses which arise from the motion of the fluid. Integral Relations for CV M. Bahrami ENSC 283 Spring
7 Stress tensor Unlike velocity, stresses and strains are nine component tensors and require two subscripts to define each component. The net surface force due to stresses in the x direction can be found as: Integral Relations for CV M. Bahrami ENSC 283 Spring
8 Momentum equation cont d. Similarly we can find the net surface force in y and z direction. After summing them up and dividing through by the volume: Therefore the linear momentum equation for an infinitesimal element becomes: This is a vector equation, and can be written as: Integral Relations for CV M. Bahrami ENSC 283 Spring
9 Special cases of momentum eq. Euler s equation (inviscid flow), when the viscous terms are negligible: Navier Stoke equation (Newtonian fluid), For a Newtonian fluid, the viscous stresses are proportional to the element strain rates and the coefficient of viscosity. For a Newtonian fluid with constant density and viscosity, we get: Integral Relations for CV M. Bahrami ENSC 283 Spring
10 Navier Stokes equation cont d Incompressible flow Navier stokes equations with constant density. Navier Stokes equations have 4 unknowns: p, u, v, and w. They should be combined with the continuity equation to form four equations for theses unknowns. Navier Stokes equations have a limited number of analytical solutions; these equations typically are solved numerically using computational fluid dynamics (CFD) software and techniques. Integral Relations for CV M. Bahrami ENSC 283 Spring
11 Angular momentum equation Application of the integral theorem to a differential element gives that the shear stresses are symmetric: Therefore, there is no differential angular momentum equation. Integral Relations for CV M. Bahrami ENSC 283 Spring
12 Boundary conditions We have 3 equations to solve: i) continuity equation, ii) momentum, and iii) energy with 5 unknowns: ρ, V, p, u and T. We use data or algebraic expressions for state relations of thermodynamic properties such as ideal gas equation of state: Integral Relations for CV M. Bahrami ENSC 283 Spring
13 Important boundary conditions At solid wall: V fluid = V wall (no slip condition) T fluid = T wall (no temperature jump) At inlet or outlet section of the flow: V, p, T are known At a liquid gas interface: equality of vertical velocity across the interface (kinematic boundary condition) Mechanical equilibrium at liquid gas interface At a liquid gas interface: heat transfer must be the same Integral Relations for CV M. Bahrami ENSC 283 Spring
14 Incompressible flow const properties Flow with constant ρ, μ, and k is a basic simplification that is very common in engineering problem that leads to: Continuity equation: Momentum equation: For frictionless or inviscid flows in which μ=0. The momentum equation reduces to Euler s equation: Integral Relations for CV M. Bahrami ENSC 283 Spring
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
More informationChapter 9: Differential Analysis
91 Introduction 92 Conservation of Mass 93 The Stream Function 94 Conservation of Linear Momentum 95 Navier Stokes Equation 96 Differential Analysis Problems Recall 91 Introduction (1) Chap 5: Control
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 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 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 information Marine Hydrodynamics. Lecture 4. Knowns Equations # Unknowns # (conservation of mass) (conservation of momentum)
2.20  Marine Hydrodynamics, Spring 2005 Lecture 4 2.20  Marine Hydrodynamics Lecture 4 Introduction Governing Equations so far: Knowns Equations # Unknowns # density ρ( x, t) Continuity 1 velocities
More informationAngular momentum equation
Angular momentum equation For angular momentum equation, B =H O the angular momentum vector about point O which moments are desired. Where β is The Reynolds transport equation can be written as follows:
More information3. FORMS OF GOVERNING EQUATIONS IN CFD
3. FORMS OF GOVERNING EQUATIONS IN CFD 3.1. Governing and model equations in CFD Fluid flows are governed by the NavierStokes equations (NS), which simpler, inviscid, form is the Euler equations. For
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 informationContents. 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.........................
More informationWhere does Bernoulli's Equation come from?
Where does Bernoulli's Equation come from? Introduction By now, you have seen the following equation many times, using it to solve simple fluid problems. P ρ + v + gz = constant (along a streamline) This
More informationFluid Dynamics Exercises and questions for the course
Fluid Dynamics Exercises and questions for the course January 15, 2014 A two dimensional flow field characterised by the following velocity components in polar coordinates is called a free vortex: u r
More informationChapter 2: Fluid Dynamics Review
7 Chapter 2: Fluid Dynamics Review This chapter serves as a short review of basic fluid mechanics. We derive the relevant transport equations (or conservation equations), state Newton s viscosity law leading
More informationvector 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
More information150A 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
More informationLesson 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
More informationDetailed 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
More informationREE Internal Fluid Flow Sheet 2  Solution Fundamentals of Fluid Mechanics
REE 307  Internal Fluid Flow Sheet 2  Solution Fundamentals of Fluid Mechanics 1. Is the following flows physically possible, that is, satisfy the continuity equation? Substitute the expressions for
More informationChapter 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
More informationPage 1. Neatly print your name: Signature: (Note that unsigned exams will be given a score of zero.)
Page 1 Neatly print your name: Signature: (Note that unsigned exams will be given a score of zero.) Circle your lecture section (1 point if not circled, or circled incorrectly): Prof. Vlachos Prof. Ardekani
More informationChapter 6: Momentum Analysis
61 Introduction 62Newton s Law and Conservation of Momentum 63 Choosing a Control Volume 64 Forces Acting on a Control Volume 65Linear Momentum Equation 66 Angular Momentum 67 The Second Law of
More information2. FLUIDFLOW EQUATIONS SPRING 2019
2. FLUIDFLOW EQUATIONS SPRING 2019 2.1 Introduction 2.2 Conservative differential equations 2.3 Nonconservative differential equations 2.4 Nondimensionalisation Summary Examples 2.1 Introduction Fluid
More informationHydrostatic. Pressure distribution in a static fluid and its effects on solid surfaces and on floating and submerged bodies.
Hydrostatic Pressure distribution in a static fluid and its effects on solid surfaces and on floating and submerged bodies. M. Bahrami ENSC 283 Spring 2009 1 Fluid at rest hydrostatic condition: when a
More informationUniversity of Hail Faculty of Engineering DEPARTMENT OF MECHANICAL ENGINEERING. ME Fluid Mechanics Lecture notes. Chapter 1
University of Hail Faculty of Engineering DEPARTMENT OF MECHANICAL ENGINEERING ME 311  Fluid Mechanics Lecture notes Chapter 1 Introduction and fluid properties Prepared by : Dr. N. Ait Messaoudene Based
More informationShell Balances in Fluid Mechanics
Shell Balances in Fluid Mechanics R. Shankar Subramanian Department of Chemical and Biomolecular Engineering Clarkson University When fluid flow occurs in a single direction everywhere in a system, shell
More informationIntroduction to Aerodynamics. Dr. Guven Aerospace Engineer (P.hD)
Introduction to Aerodynamics Dr. Guven Aerospace Engineer (P.hD) Aerodynamic Forces All aerodynamic forces are generated wither through pressure distribution or a shear stress distribution on a body. The
More informationFluid Mechanics. Spring 2009
Instructor: Dr. YangCheng Shih Department of Energy and Refrigerating AirConditioning Engineering National Taipei University of Technology Spring 2009 Chapter 1 Introduction 11 General Remarks 12 Scope
More informationAE/ME 339. Computational Fluid Dynamics (CFD) K. M. Isaac. Momentum equation. Computational Fluid Dynamics (AE/ME 339) MAEEM Dept.
AE/ME 339 Computational Fluid Dynamics (CFD) 9//005 Topic7_NS_ F0 1 Momentum equation 9//005 Topic7_NS_ F0 1 Consider the moving fluid element model shown in Figure.b Basis is Newton s nd Law which says
More informationAE/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 Nonconservation form D Dt. V 0.(2.29) Conservation form ( V ) 0...(2.33) t 12/21/01
More informationComputational 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
More informationChapter 6: Incompressible Inviscid Flow
Chapter 6: Incompressible Inviscid Flow 61 Introduction 62 Nondimensionalization of the NSE 63 Creeping Flow 64 Inviscid Regions of Flow 65 Irrotational Flow Approximation 66 Elementary Planar Irrotational
More informationfluid mechanics as a prominent discipline of application for numerical
1. fluid mechanics as a prominent discipline of application for numerical simulations: experimental fluid mechanics: wind tunnel studies, laser Doppler anemometry, hot wire techniques,... theoretical fluid
More informationCOURSE NUMBER: ME 321 Fluid Mechanics I 3 credit hour. Basic Equations in fluid Dynamics
COURSE NUMBER: ME 321 Fluid Mechanics I 3 credit hour Basic Equations in fluid Dynamics Course teacher Dr. M. Mahbubur Razzaque Professor Department of Mechanical Engineering BUET 1 Description of Fluid
More informationIntroduction to Marine Hydrodynamics
1896 1920 1987 2006 Introduction to Marine Hydrodynamics (NA235) Department of Naval Architecture and Ocean Engineering School of Naval Architecture, Ocean & Civil Engineering First Assignment The first
More informationThe Shape of a Rain Drop as determined from the NavierStokes equation John Caleb Speirs Classical Mechanics PHGN 505 December 12th, 2011
The Shape of a Rain Drop as determined from the NavierStokes equation John Caleb Speirs Classical Mechanics PHGN 505 December 12th, 2011 Derivation of NavierStokes Equation 1 The total stress tensor
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 informationExercise: concepts from chapter 10
Reading:, Ch 10 1) The flow of magma with a viscosity as great as 10 10 Pa s, let alone that of rock with a viscosity of 10 20 Pa s, is difficult to comprehend because our common eperience is with s like
More informationM E 320 Professor John M. Cimbala Lecture 10. The Reynolds Transport Theorem (RTT) (Section 46)
M E 320 Professor John M. Cimbala Lecture 10 Today, we will: Discuss the Reynolds Transport Theorem (RTT) Show how the RTT applies to the conservation laws Begin Chapter 5 Conservation Laws D. The Reynolds
More informationME3560 Tentative Schedule Spring 2019
ME3560 Tentative Schedule Spring 2019 Week Number Date Lecture Topics Covered Prior to Lecture Read Section Assignment Prep Problems for Prep Probs. Must be Solved by 1 Monday 1/7/2019 1 Introduction to
More informationChapter 6: Momentum Analysis of Flow Systems
Chapter 6: Momentum Analysis of Flow Systems Introduction Fluid flow problems can be analyzed using one of three basic approaches: differential, experimental, and integral (or control volume). In Chap.
More informationFluid Mechanics II Viscosity and shear stresses
Fluid Mechanics II Viscosity and shear stresses Shear stresses in a Newtonian fluid A fluid at rest can not resist shearing forces. Under the action of such forces it deforms continuously, however small
More informationHomework #4 Solution. μ 1. μ 2
Homework #4 Solution 4.20 in Middleman We have two viscous liquids that are immiscible (e.g. water and oil), layered between two solid surfaces, where the top boundary is translating: y = B y = kb y =
More informationIntroduction to Fluid Mechanics
Introduction to Fluid Mechanics TienTsan Shieh April 16, 2009 What is a Fluid? The key distinction between a fluid and a solid lies in the mode of resistance to change of shape. The fluid, unlike the
More informationM E 320 Professor John M. Cimbala Lecture 10
M E 320 Professor John M. Cimbala Lecture 10 Today, we will: Finish our example problem rates of motion and deformation of fluid particles Discuss the Reynolds Transport Theorem (RTT) Show how the RTT
More informationFundamentals of Fluid Mechanics
Sixth Edition Fundamentals of Fluid Mechanics International Student Version BRUCE R. MUNSON DONALD F. YOUNG Department of Aerospace Engineering and Engineering Mechanics THEODORE H. OKIISHI Department
More informationCONVECTIVE HEAT TRANSFER
CONVECTIVE HEAT TRANSFER Mohammad Goharkhah Department of Mechanical Engineering, Sahand Unversity of Technology, Tabriz, Iran CHAPTER 3 LAMINAR BOUNDARY LAYER FLOW LAMINAR BOUNDARY LAYER FLOW Boundary
More informationObjectives. Conservation of mass principle: Mass Equation The Bernoulli equation Conservation of energy principle: Energy equation
Objectives Conservation of mass principle: Mass Equation The Bernoulli equation Conservation of energy principle: Energy equation Conservation of Mass Conservation of Mass Mass, like energy, is a conserved
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 informationNumerical Simulation of Newtonian and NonNewtonian Flows in Bypass
Numerical Simulation of Newtonian and NonNewtonian Flows in Bypass Vladimír Prokop, Karel Kozel Czech Technical University Faculty of Mechanical Engineering Department of Technical Mathematics Vladimír
More informationChemical Engineering 374
Chemical Engineering 374 Fluid Mechanics Exam 3 Review 1 Spiritual Thought 2 Ether 12:27 6 And now, I, Moroni, would speak somewhat concerning these things; I would show unto the world that faith is things
More informationSimplified Model of WWER440 Fuel Assembly for ThermoHydraulic Analysis
1 Portál pre odborné publikovanie ISSN 13380087 Simplified Model of WWER440 Fuel Assembly for ThermoHydraulic Analysis Jakubec Jakub Elektrotechnika 13.02.2013 This work deals with thermohydraulic processes
More informationINDEX 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
More informationFluid Dynamics: Theory, Computation, and Numerical Simulation Second Edition
Fluid Dynamics: Theory, Computation, and Numerical Simulation Second Edition C. Pozrikidis m Springer Contents Preface v 1 Introduction to Kinematics 1 1.1 Fluids and solids 1 1.2 Fluid parcels and flow
More informationIntroduction 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
More informationMicroscopic Momentum Balance Equation (NavierStokes)
CM3110 Transport I Part I: Fluid Mechanics Microscopic Momentum Balance Equation (NavierStokes) Professor Faith Morrison Department of Chemical Engineering Michigan Technological University 1 Microscopic
More informationMOMENTUM PRINCIPLE. Review: Last time, we derived the Reynolds Transport Theorem: Chapter 6. where B is any extensive property (proportional to mass),
Chapter 6 MOMENTUM PRINCIPLE Review: Last time, we derived the Reynolds Transport Theorem: where B is any extensive property (proportional to mass), and b is the corresponding intensive property (B / m
More informationChapter 1. Continuum mechanics review. 1.1 Definitions and nomenclature
Chapter 1 Continuum mechanics review We will assume some familiarity with continuum mechanics as discussed in the context of an introductory geodynamics course; a good reference for such problems is Turcotte
More informationModule 3: "Thin Film Hydrodynamics" Lecture 11: "" The Lecture Contains: Micro and Nano Scale Hydrodynamics with and without Free Surfaces
The Lecture Contains: Micro and Nano Scale Hydrodynamics with and without Free Surfaces Order of Magnitude Analysis file:///e /courses/colloid_interface_science/lecture11/11_1.htm[6/16/2012 1:39:56 PM]
More informationCFD SIMULATIONS OF FLOW, HEAT AND MASS TRANSFER IN THINFILM EVAPORATOR
Distillation Absorption 2010 A.B. de Haan, H. Kooijman and A. Górak (Editors) All rights reserved by authors as per DA2010 copyright notice CFD SIMULATIONS OF FLOW, HEAT AND MASS TRANSFER IN THINFILM
More informationChapter 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
More informationAA210A 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
More information7 The NavierStokes Equations
18.354/12.27 Spring 214 7 The NavierStokes Equations In the previous section, we have seen how one can deduce the general structure of hydrodynamic equations from purely macroscopic considerations and
More informationCourse 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
More informationME3560 Tentative Schedule Fall 2018
ME3560 Tentative Schedule Fall 2018 Week Number 1 Wednesday 8/29/2018 1 Date Lecture Topics Covered Introduction to course, syllabus and class policies. Math Review. Differentiation. Prior to Lecture Read
More informationQuick Recapitulation of Fluid Mechanics
Quick Recapitulation of Fluid Mechanics Amey Joshi 07Feb018 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 informationBasic hydrodynamics. David Gurarie. 1 Newtonian fluids: Euler and NavierStokes equations
Basic hydrodynamics David Gurarie 1 Newtonian fluids: Euler and NavierStokes equations The basic hydrodynamic equations in the Eulerian form consist of conservation of mass, momentum and energy. We denote
More informationFluid Mechanics Theory I
Fluid Mechanics Theory I Last Class: 1. Introduction 2. MicroTAS or Lab on a Chip 3. Microfluidics Length Scale 4. Fundamentals 5. Different Aspects of Microfluidcs Today s Contents: 1. Introduction to
More informationComputer Fluid Dynamics E181107
Computer Fluid Dynamics E181107 2181106 Transport equations, Navier Stokes equations Remark: foils with black background could be skipped, they are aimed to the more advanced courses Rudolf Žitný, Ústav
More informationOE4625 Dredge Pumps and Slurry Transport. Vaclav Matousek October 13, 2004
OE465 Vaclav Matousek October 13, 004 1 Dredge Vermelding Pumps onderdeel and Slurry organisatie Transport OE465 Vaclav Matousek October 13, 004 Dredge Vermelding Pumps onderdeel and Slurry organisatie
More informationFluid 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.
More informationNavierStokes Equation: Principle of Conservation of Momentum
Naviertokes Equation: Principle of Conservation of Momentum R. hankar ubramanian Department of Chemical and Biomolecular Engineering Clarkson University Newton formulated the principle of conservation
More informationGetting started: CFD notation
PDE of pth 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 informationInterpreting 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
More information1. 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
More informationCHAPTER 7 NUMERICAL MODELLING OF A SPIRAL HEAT EXCHANGER USING CFD TECHNIQUE
CHAPTER 7 NUMERICAL MODELLING OF A SPIRAL HEAT EXCHANGER USING CFD TECHNIQUE In this chapter, the governing equations for the proposed numerical model with discretisation methods are presented. Spiral
More information3.8 The First Law of Thermodynamics and the Energy Equation
CEE 3310 Control Volume Analysis, Sep 30, 2011 65 Review Conservation of angular momentum 1D form ( r F )ext = [ˆ ] ( r v)d + ( r v) out ṁ out ( r v) in ṁ in t CV 3.8 The First Law of Thermodynamics and
More informationME 309 Fluid Mechanics Fall 2010 Exam 2 1A. 1B.
Fall 010 Exam 1A. 1B. Fall 010 Exam 1C. Water is flowing through a 180º bend. The inner and outer radii of the bend are 0.75 and 1.5 m, respectively. The velocity profile is approximated as C/r where C
More informationExercise 5: Exact Solutions to the NavierStokes Equations I
Fluid Mechanics, SG4, HT009 September 5, 009 Exercise 5: Exact Solutions to the NavierStokes Equations I Example : Plane Couette Flow Consider the flow of a viscous Newtonian fluid between two parallel
More informationFluid Mechanics Qualifying Examination Sample Exam 2
Fluid Mechanics Qualifying Examination Sample Exam 2 Allotted Time: 3 Hours The exam is closed book and closed notes. Students are allowed one (doublesided) formula sheet. There are five questions on
More informationTURBULENT FLOW ACROSS A ROTATING CYLINDER WITH SURFACE ROUGHNESS
HEFAT2014 10 th International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics 14 16 July 2014 Orlando, Florida TURBULENT FLOW ACROSS A ROTATING CYLINDER WITH SURFACE ROUGHNESS Everts, M.,
More informationSimulation of Tjunction using LBM and VOF ENERGY 224 Final Project Yifan Wang,
Simulation of Tjunction using LBM and VOF ENERGY 224 Final Project Yifan Wang, yfwang09@stanford.edu 1. Problem setting In this project, we present a benchmark simulation for segmented flows, which contain
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 NonConservative
More informationLecture 3. Properties of Fluids 11/01/2017. There are thermodynamic properties of fluids like:
11/01/2017 Lecture 3 Properties of Fluids There are thermodynamic properties of fluids like: Pressure, p (N/m 2 ) or [ML 1 T 2 ], Density, ρ (kg/m 3 ) or [ML 3 ], Specific weight, γ = ρg (N/m 3 ) or
More informationDetailed 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. Secondorder tensors 3. Volume vs. velocity 4. Del operator B. Chapter 1: Review of Basic
More informationME3250 Fluid Dynamics I
ME3250 Fluid Dynamics I Section I, Fall 2012 Instructor: Prof. Zhuyin Ren Department of Mechanical Engineering University of Connecticut Course Information Website: http://www.engr.uconn.edu/~rzr11001/me3250_f12/
More informationP = 1 3 (σ xx + σ yy + σ zz ) = F A. It is created by the bombardment of the surface by molecules of fluid.
CEE 3310 Thermodynamic Properties, Aug. 27, 2010 11 1.4 Review A fluid is a substance that can not support a shear stress. Liquids differ from gasses in that liquids that do not completely fill a container
More informationLecture 2: Hydrodynamics at milli micrometer scale
1 at milli micrometer scale Introduction Flows at milli and micro meter scales are found in various fields, used for several processes and open up possibilities for new applications: Injection Engineering
More informationFLUID MECHANICS PROF. DR. METİN GÜNER COMPILER
FLUID MECHANICS PROF. DR. METİN GÜNER COMPILER ANKARA UNIVERSITY FACULTY OF AGRICULTURE DEPARTMENT OF AGRICULTURAL MACHINERY AND TECHNOLOGIES ENGINEERING 1 5. FLOW IN PIPES 5.1.3. Pressure and Shear Stress
More informationBoundary Conditions in Fluid Mechanics
Boundary Conditions in Fluid Mechanics R. Shankar Subramanian Department of Chemical and Biomolecular Engineering Clarkson University The governing equations for the velocity and pressure fields are partial
More informationIII.3 Momentum balance: Euler and Navier Stokes equations
32 Fundamental equations of nonrelativistic 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
More informationPIPE FLOWS: LECTURE /04/2017. Yesterday, for the example problem Δp = f(v, ρ, μ, L, D) We came up with the non dimensional relation
/04/07 ECTURE 4 PIPE FOWS: Yesterday, for the example problem Δp = f(v, ρ, μ,, ) We came up with the non dimensional relation f (, ) 3 V or, p f(, ) You can plot π versus π with π 3 as a parameter. Or,
More information( ) Notes. Fluid mechanics. Inviscid Euler model. Lagrangian viewpoint. " = " x,t,#, #
Notes Assignment 4 due today (when I check email tomorrow morning) Don t be afraid to make assumptions, approximate quantities, In particular, method for computing time step bound (look at max eigenvalue
More 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 informationCHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION
CHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION 7.1 THE NAVIERSTOKES EQUATIONS Under the assumption of a Newtonian stressrateofstrain constitutive equation and a linear, thermally conductive medium,
More information6.1 Momentum Equation for Frictionless Flow: Euler s Equation The equations of motion for frictionless flow, called Euler s
Chapter 6 INCOMPRESSIBLE INVISCID FLOW All real fluids possess viscosity. However in many flow cases it is reasonable to neglect the effects of viscosity. It is useful to investigate the dynamics of an
More informationNotes 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.
More informationDIMENSIONAL ANALYSIS IN MOMENTUM TRANSFER
DIMENSIONAL ANALYSIS IN MOMENTUM TRANSFER FT I Alda Simões Techniques for Dimensional Analysis Fluid Dynamics: Microscopic analysis, theory Physical modelling Differential balances Limited to simple geometries
More informationIntroduction to Aerospace Engineering
Introduction to Aerospace Engineering Lecture slides Challenge the future 300 Introduction to Aerospace Engineering Aerodynamics 5 & 6 Prof. H. Bijl ir. N. Timmer Delft University of Technology 5. Compressibility
More informationSummary PHY101 ( 2 ) T / Hanadi Al Harbi
الكمية Physical Quantity القانون Low التعريف Definition الوحدة SI Unit Linear Momentum P = mθ be equal to the mass of an object times its velocity. Kg. m/s vector quantity Stress F \ A the external force
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 information