ME 3560 Fluid Mechanics

Size: px
Start display at page:

Download "ME 3560 Fluid Mechanics"

Transcription

1 ME 3560 Fluid Mechanics 1

2 4.1 The Velocity Field One of the most important parameters that need to be monitored when a fluid is flowing is the velocity. In general the flow parameters are described in terms of the motion of fluid particles rather than individual molecules. Thus, this motion can be described in terms of the velocity and acceleration of the fluid particles. In order to describe the flow parameters it is sought to provide at a given instant in time, a description of any fluid property (such as, p, V, and a) as a function of the fluid's location. This representation of fluid parameters as functions of the spatial coordinates is termed a field representation of the flow. The specific field representation may be different at different times, thus. Thus, to completely specify the velocity, V, in a process, the velocity field must be expressed as: V = V (x, y, z, t). V u( x, y, z, t)ˆ i v( x, y, z, t) ˆj w( x, y, z, t) kˆ 2

3 V u( x, y, z, y)ˆ i v( x, y, z, y) ˆj w( x, y, z, y) kˆ u, v, and w are the x, y, and z components of the velocity vector. The velocity of a particle is the time rate of change of the position vector for that particle. The position of particle A relative to the coordinate system is given by its position vector, r A, which (if the particle is moving) is a function of time. The velocity is a vector therefore it has both a direction and a magnitude, V= V =(u 2 + v 2 + w 2 ) 1/2 A change in velocity results in an acceleration. This acceleration may be due to a change in speed and/or direction. 3

4 4.1.1 Eulerian and Lagragian Flow Descriptions There are two general approaches in analyzing fluid mechanics problems: Eulerian method. Measures the flow parameters at fixed locations to then generate the parameters flowfield. Lagrangian method, involves following individual fluid particles as they move about and determining how the fluid properties associated with these particles change as a function of time One-, Two-, and Three-Dimensional Flows In general, a fluid flow is a rather complex three-dimensional, timedependent phenomenon. V u( x, y, z, t)ˆ i v( x, y, z, t)ˆj w( x, y, z, t) kˆ 4

5 However, sometimes it is possible to make simplifying assumptions that allow a much easier understanding of the problem without sacrificing needed accuracy: Assume the flow is two dimensional or one dimensional. Assume the flow is incompressible (even when dealing with gases) One of these simplifications involves approximating a real flow as a simpler one or two dimensional flow. 5

6 4.1.3 Steady and Unsteady Flows Steady flow. The velocity (or any other parameter: T, p,, etc.) ata given point in space does not vary with time V/t=0. Unsteady flows. Flow field parameters are time dependent V/t0. Among the various types of unsteady flows are nonperiodic flow, periodic flow, and truly random flow Streamlines, Streaklines, and Pathlines Streamline is a line that is everywhere tangent to the velocity field. If the flow is steady, nothing at a fixed point (including the velocity direction) changes with time, so the streamlines are fixed lines in space. For unsteady flows the streamlines may change shape with time. x y z u v w dy dx v u 6

7 Streakline consists of all particles in a flow that have previously passed through a common point. Streaklines can be generated by taking instantaneous photographs of marked particles that all passed through a given location in the flow field at some earlier time. Such a line can be produced by continuously injecting marked fluid (neutrally buoyant smoke in air, or dye in water) at a given location. 7

8 Pathline is the line traced out by a given particle as it flows from one point to another. The pathline is a Lagrangian concept that can be produced in the laboratory by marking a fluid particle (dying a small fluid element) and taking a time exposure photograph of its motion. Pathlines, Streamlines,andStreaklines are the same for steady flows. For unsteady flows none of these three types of lines need to be the same. 8

9 4.2 The Acceleration Field A flow can be studied by either (1) following individual particles (Lagrangian description) or (2) remaining fixed in space and observing different particles as they pass by (Eulerian description). In either case, to apply Newton's second law (F =ma) it is necessary to describe the particle acceleration in an appropriate fashion. For the infrequently used Lagrangian method, we describe the fluid acceleration just as is done in solid body dynamics a = a(t) for each particle. For the Eulerian description we describe the acceleration field as a function of position and time without actually following any particular particle, a = a(x, y, z, t). The acceleration of a particle is the time rate of change of its velocity. For unsteady flows the velocity at a given point in space varies with time. Also, a fluid particle may accelerate because its velocity changes as it flows from one point to another in space. 9

10 The Material Derivative z w w y w v x w u t w a z v w y v v x v u t v a z u w y u v x u u t u a z V w y V v x V u t V a z w y v x u t dt D dt DV a () () () () ()

11 D( )/dt is termed the material derivative or substantial derivative. A shorthand notation for the material derivative operator is D() dt () t ( V )() Unsteady Effects The material derivative formula contains two types of terms: Terms involving the time derivative ()/t: Local Derivative Terms involving spatial derivatives ()/x, ()/y,and()/z. ()/t represents the effects of the unsteadiness of the flow. V/t is termed the local acceleration.forsteadyflow:()/t=0 Physically, there is no change in flow parameters at a fixed point in space if the flow is steady. There may be a change of those parameters for a fluid particle as it moves about. If a flow is unsteady, its parameter values (V, T,, etc.) at any location may change with time 11

12 4.2.3 Convective Effects The portion of the material derivative represented by the spatial derivatives is termed the convective derivative. The convective derivative represents the fact that a flow property associated with a fluid particle may vary because of the motion of the particle from one point in space to another point in space. This contribution to the time rate of change of the parameter for the particle can occur whether the flow is steady or unsteady. It is due to the convection, or motion, of the particle through space in which there is a gradient [( )=()/xî+()/x ĵ + ()/z k] The portion of a due to the term (V )V is the convective acceleration 12

13 4.3 Control Volume and System Representations A system is a collection of matter of fixed identity (always the same atoms or fluid particles), which may move, flow, and interact with its surroundings. A system is a specific, identifiable quantity of matter. It may consist of a relatively large amount of mass or it may be an infinitesimal size. A system may interact with its surroundings by various means (by the transfer of heat or the exertion of a pressure force, for example). A system may continually change size and shape, but it always contains the same mass. 13

14 A control volume, is a volume in space (a geometric entity, independent of mass) through which fluid may flow. In fluid mechanics, it is difficult to identify and keep track of a specific quantity of matter. In several cases, the main interest is in determining the forces put on a device rather than in the information obtained by following a given portion of the air (a system) as it flows along. For these situations it is more adequate to use the control volume approach. Identify a specific volume in space (a volume associated with the device of interest) and analyze the fluid flow within, through, or around that volume. 14

15 In general, the control volume can be a moving volume, although for most situations we will use only fixed, nondeformable control volumes. The matter within a control volume may change with time as the fluid flows through it. The amount of mass within the volume may change with time. The control volume itself is a specific geometric entity, independent of the flowing fluid. 15

16 All of the laws governing the motion of a fluid are stated in their basic form in terms of a system approach. For example, the mass of a system remains constant, or the time rate of change of momentum of a system is equal to the sum of all the forces acting on the system. 16

17 4.4 The Reynolds Transport Theorem The Reynolds transport theorem provides a mathematical way to relate the properties of a flow between a control volume and a system. Extensive Property, is a property whose value is directly proportional to the amount of the mass being considered. -Mass -Energy -Momentum Intensive Property, is a property whose value is independent of the amount of mass. -Density -Temperature -Velocity -Pressure 17

18 B = m b=1 B = mv b=v B = E b=e An extensive property B is related to its corresponding intensive property b by B = mb In general, for a system, an extensive property B can be determined as: B b dv sys The rate of change of an extensive property in a system is expressed as db dt sys sys d b dv sys dt 18

19 In a similar way, the rate of change of an extensive property in a control volume is expressed as d b dv db cv cv dt dt Derivation of the Reynolds Transport Theorem Consider two instants in time: t and t+t. Assume that at t the control volume and the system coincide, thus B sys ( t) B ( t) cv 19

20 Then, at a time t+t B sys ( t t) B ( t t) B ( t t) B ( t t) cv I II The change in the amount of B in the system in the time interval δt divided by this time interval is given by Bsys Bsys ( t t) Bsys ( t) t t Bsys Bcv ( t t) BI ( t t) BII ( t t) t t B sys ( t) 20

21 Since at the initial time t we have B sys (t)=b cv (t), B t sys B cv ( t t) Bcv ( t) BI ( t t) BII ( t t) t t t For δt 0, the left-hand side this equation is DB sys /Dt. DB sys /Dt represents the time rate of change of property B associated with a system (a given portion of fluid) as it moves along. For δt 0,thefirsttermontheright-handsideofthisequationisthe time rate of change of the amount of B within the control volume lim t0 B cv ( t t) t B cv ( t) B t cv cv bdv t 21

22 The term B II ( t t) ( 2 b2 )( VII ) 2b2 A2V 2t Represents the amount of the extensive parameter B flowing out of the control volume, across the control surface. Thus, the rate at which this property flows from the control volume is: B out lim t0 B II ( t t) t A V b 2 22

23 The term B I ( t t) ( 1 b1 )( VI ) 1b1 AV 1 1t Represents the amount of the extensive parameter B flowing into the control volume, across the control surface Thus, the rate at which this property flows from the control volume is: B in lim t0 B I ( t t) t AV b 1 23

24 By combining the previous equations, a relation between the time rate of change of B for the system and that for the control volume is given by DB Dt DB Dt sys sys B t B t cv cv B out B in 2 A2V 2b2 1AV 1 1b1 A generalization of the previous equation is given by the Reynolds Transport Theorem DB Dt sys t cv b dv cs bv nda ˆ 24

25 Selection of a Control Volume CV is typically fixed and non deforming. CV includes all relevant inlets and outlets where information is available or required. If possible CS should be perpendicular to the velocity vector to simplify V Contributions to the surface integrals must be simple and relevant. In CS other than inlets and outlets, select solid boundaries where V =0. Sign of V nˆ nˆ n n V in V nˆ in V in V V out nˆ out V out 25

ROAD MAP... D-0: Reynolds Transport Theorem D-1: Conservation of Mass D-2: Conservation of Momentum D-3: Conservation of Energy

ROAD MAP... D-0: Reynolds Transport Theorem D-1: Conservation of Mass D-2: Conservation of Momentum D-3: Conservation of Energy ES06 Fluid Mechani UNIT D: Flow Field Analysis ROAD MAP... D-0: Reynolds Transport Theorem D-1: Conservation of Mass D-: Conservation of Momentum D-3: Conservation of Energy ES06 Fluid Mechani Unit D-0:

More information

Chapter 4: Fluid Kinematics

Chapter 4: Fluid Kinematics Overview Fluid kinematics deals with the motion of fluids without considering the forces and moments which create the motion. Items discussed in this Chapter. Material derivative and its relationship to

More information

M E 320 Professor John M. Cimbala Lecture 10. The Reynolds Transport Theorem (RTT) (Section 4-6)

M E 320 Professor John M. Cimbala Lecture 10. The Reynolds Transport Theorem (RTT) (Section 4-6) 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 information

Correlation between System (Lagrangian) concept Control-volume (Eulerian) concept for comprehensive understanding of fluid motion?

Correlation between System (Lagrangian) concept Control-volume (Eulerian) concept for comprehensive understanding of fluid motion? The Reynolds Tanspot Theoem Coelation between System (Lagangian) concept Contol-volume (Euleian) concept fo compehensive undestanding of fluid motion? Reynolds Tanspot Theoem Let s set a fundamental equation

More information

AE/ME 339. K. M. Isaac. 9/22/2005 Topic 6 FluidFlowEquations_Introduction. Computational Fluid Dynamics (AE/ME 339) MAEEM Dept.

AE/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 information

2. Conservation of Mass

2. 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 information

Where does Bernoulli's Equation come from?

Where 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 information

V (r,t) = i ˆ u( x, y,z,t) + ˆ j v( x, y,z,t) + k ˆ w( x, y, z,t)

V (r,t) = i ˆ u( x, y,z,t) + ˆ j v( x, y,z,t) + k ˆ w( x, y, z,t) IV. DIFFERENTIAL RELATIONS FOR A FLUID PARTICLE This chapter presents the development and application of the basic differential equations of fluid motion. Simplifications in the general equations and common

More information

M E 320 Professor John M. Cimbala Lecture 10

M 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 information

Chapter 4 Continuity Equation and Reynolds Transport Theorem

Chapter 4 Continuity Equation and Reynolds Transport Theorem Chapter 4 Continuity Equation and Reynolds Transport Theorem 4.1 Control Volume 4. The Continuity Equation for One-Dimensional Steady Flow 4.3 The Continuity Equation for Two-Dimensional Steady Flow 4.4

More information

6. Basic basic equations I ( )

6. Basic basic equations I ( ) 6. Basic basic equations I (4.2-4.4) Steady and uniform flows, streamline, streamtube One-, two-, and three-dimensional flow Laminar and turbulent flow Reynolds number System and control volume Continuity

More information

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

MAE 3130: Fluid Mechanics Lecture 7: Differential Analysis/Part 1 Spring 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

Chapter 5 Control Volume Approach and Continuity Equation

Chapter 5 Control Volume Approach and Continuity Equation Chapter 5 Control Volume Approach and Continuity Equation Lagrangian and Eulerian Approach To evaluate the pressure and velocities at arbitrary locations in a flow field. The flow into a sudden contraction,

More information

In this section, mathematical description of the motion of fluid elements moving in a flow field is

In 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 information

MOMENTUM PRINCIPLE. Review: Last time, we derived the Reynolds Transport Theorem: Chapter 6. where B is any extensive property (proportional to mass),

MOMENTUM 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 information

Getting started: CFD notation

Getting started: CFD notation PDE of p-th order Getting started: CFD notation f ( u,x, t, u x 1,..., u x n, u, 2 u x 1 x 2,..., p u p ) = 0 scalar unknowns u = u(x, t), x R n, t R, n = 1,2,3 vector unknowns v = v(x, t), v R m, m =

More information

COURSE 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 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 information

LECTURE NOTES - III. Prof. Dr. Atıl BULU

LECTURE NOTES - III. Prof. Dr. Atıl BULU LECTURE NOTES - III «FLUID MECHANICS» Istanbul Technical University College of Civil Engineering Civil Engineering Department Hydraulics Division CHAPTER KINEMATICS OF FLUIDS.. FLUID IN MOTION Fluid motion

More information

Chapter 6: Momentum Analysis

Chapter 6: Momentum Analysis 6-1 Introduction 6-2Newton s Law and Conservation of Momentum 6-3 Choosing a Control Volume 6-4 Forces Acting on a Control Volume 6-5Linear Momentum Equation 6-6 Angular Momentum 6-7 The Second Law of

More information

The most common methods to identify velocity of flow are pathlines, streaklines and streamlines.

The most common methods to identify velocity of flow are pathlines, streaklines and streamlines. 4 FLUID FLOW 4.1 Introduction Many civil engineering problems in fluid mechanics are concerned with fluids in motion. The distribution of potable water, the collection of domestic sewage and storm water,

More information

Unit C-1: List of Subjects

Unit C-1: List of Subjects Unit C-: List of Subjects The elocity Field The Acceleration Field The Material or Substantial Derivative Steady Flow and Streamlines Fluid Particle in a Flow Field F=ma along a Streamline Bernoulli s

More information

CHAPTER 4. Basics of Fluid Dynamics

CHAPTER 4. Basics of Fluid Dynamics CHAPTER 4 Basics of Fluid Dynamics What is a fluid? A fluid is a substance that can flow, has no fixed shape, and offers little resistance to an external stress In a fluid the constituent particles (atoms,

More information

KINEMATICS OF CONTINUA

KINEMATICS 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 information

Math Review Night: Work and the Dot Product

Math Review Night: Work and the Dot Product Math Review Night: Work and the Dot Product Dot Product A scalar quantity Magnitude: A B = A B cosθ The dot product can be positive, zero, or negative Two types of projections: the dot product is the parallel

More information

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

Fluid Mechanics Prof. S. K. Som Department of Mechanical Engineering Indian Institute of Technology, Kharagpur Fluid Mechanics Prof. S. K. Som Department of Mechanical Engineering Indian Institute of Technology, Kharagpur Lecture - 11 Kinematics of Fluid Part - II Good afternoon, I welcome you all to this session

More information

Fundamentals of Fluid Dynamics: Ideal Flow Theory & Basic Aerodynamics

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

More information

Objectives. 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 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 information

AA210A Fundamentals of Compressible Flow. Chapter 1 - Introduction to fluid flow

AA210A Fundamentals of Compressible Flow. Chapter 1 - Introduction to fluid flow AA210A Fundamentals of Compressible Flow Chapter 1 - Introduction to fluid flow 1 1.2 Conservation of mass Mass flux in the x-direction [ ρu ] = M L 3 L T = M L 2 T Momentum per unit volume Mass per unit

More information

3.8 The First Law of Thermodynamics and the Energy Equation

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

More information

Principles of Convection

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

More information

SYSTEMS VS. CONTROL VOLUMES. Control volume CV (open system): Arbitrary geometric space, surrounded by control surfaces (CS)

SYSTEMS VS. CONTROL VOLUMES. Control volume CV (open system): Arbitrary geometric space, surrounded by control surfaces (CS) SYSTEMS VS. CONTROL VOLUMES System (closed system): Predefined mass m, surrounded by a system boundary Control volume CV (open system): Arbitrary geometric space, surrounded by control surfaces (CS) Many

More information

Lecture 4. Differential Analysis of Fluid Flow Navier-Stockes equation

Lecture 4. Differential Analysis of Fluid Flow Navier-Stockes equation Lecture 4 Differential Analysis of Fluid Flow Navier-Stockes equation Newton second law and conservation of momentum & momentum-of-momentum A jet of fluid deflected by an object puts a force on the object.

More information

Last name: First name: Student ID: Discussion: You solution procedure should be legible and complete for full credit (use scratch paper as needed).

Last name: First name: Student ID: Discussion: You solution procedure should be legible and complete for full credit (use scratch paper as needed). University of California, Berkeley Mechanical Engineering ME 106, Fluid Mechanics ODK/Midterm 2, Fall 2015 Last name: First name: Student ID: Discussion: Notes: You solution procedure should be legible

More information

Kinematics (2) - Motion in Three Dimensions

Kinematics (2) - Motion in Three Dimensions Kinematics (2) - Motion in Three Dimensions 1. Introduction Kinematics is a branch of mechanics which describes the motion of objects without consideration of the circumstances leading to the motion. 2.

More information

ENV5056 Numerical Modeling of Flow and Contaminant Transport in Rivers. Asst. Prof. Dr. Orhan GÜNDÜZ

ENV5056 Numerical Modeling of Flow and Contaminant Transport in Rivers. Asst. Prof. Dr. Orhan GÜNDÜZ ENV5056 Numerical Modeling of Flow and Contaminant Transport in Rivers Reynolds Transport Theorem Asst. Prof. Dr. Orhan GÜNDÜZ We are sometimes interested in what happens to a particular part of the fluid

More information

Convection Heat Transfer

Convection 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 information

Chapter 6: Momentum Analysis of Flow Systems

Chapter 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 information

Lecture 2 Flow classifications and continuity

Lecture 2 Flow classifications and continuity Lecture 2 Flow classifications and continuity Dr Tim Gough: t.gough@bradford.ac.uk General information 1 No tutorial week 3 3 rd October 2013 this Thursday. Attempt tutorial based on examples from today

More information

Basic equations of motion in fluid mechanics

Basic 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 information

3.5 Vorticity Equation

3.5 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 information

PHYS 172: Modern Mechanics. Summer Lecture 2 Velocity and Momentum Read:

PHYS 172: Modern Mechanics. Summer Lecture 2 Velocity and Momentum Read: PHYS 172: Modern Mechanics Summer 2010 p sys F net t E W Q sys surr surr L sys net t Lecture 2 Velocity and Momentum Read: 1.6-1.9 Math Experience A) Currently taking Calculus B) Currently taking Calculus

More information

Shell/Integral Balances (SIB)

Shell/Integral Balances (SIB) Shell/Integral Balances (SIB) Shell/Integral Balances Shell or integral (macroscopic) balances are often relatively simple to solve, both conceptually and mechanically, as only limited data is necessary.

More information

Energy CEEN 598D: Fluid Mechanics for Hydro Systems. Lindsay Bearup Berthoud Hall 121

Energy CEEN 598D: Fluid Mechanics for Hydro Systems. Lindsay Bearup Berthoud Hall 121 Energy CEEN 598D: Fluid Mechanics for Hydro Systems Lindsay Bearup lbearup@mines.edu Berthoud Hall 11 GEGN 498A Fall 013 For the record: Material DerivaIves db sys dt Rate of change of property B of system

More information

Chapter 3 Bernoulli Equation

Chapter 3 Bernoulli Equation 1 Bernoulli Equation 3.1 Flow Patterns: Streamlines, Pathlines, Streaklines 1) A streamline, is a line that is everywhere tangent to the velocity vector at a given instant. Examples of streamlines around

More information

CE 204 FLUID MECHANICS

CE 204 FLUID MECHANICS CE 204 FLUID MECHANICS Onur AKAY Assistant Professor Okan University Department of Civil Engineering Akfırat Campus 34959 Tuzla-Istanbul/TURKEY Phone: +90-216-677-1630 ext.1974 Fax: +90-216-677-1486 E-mail:

More information

Kinetic Energy and Work

Kinetic Energy and Work Kinetic Energy and Work 8.01 W06D1 Today s Readings: Chapter 13 The Concept of Energy and Conservation of Energy, Sections 13.1-13.8 Announcements Problem Set 4 due Week 6 Tuesday at 9 pm in box outside

More information

4.1 LAWS OF MECHANICS - Review

4.1 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 information

18.02 Multivariable Calculus Fall 2007

18.02 Multivariable Calculus Fall 2007 MIT OpenourseWare http://ocw.mit.edu 18.02 Multivariable alculus Fall 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.02 Lecture 21. Test for

More information

Chapter 2: Basic Governing Equations

Chapter 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 information

Fundamentals of Fluid Mechanics

Fundamentals 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 information

Conservation of Angular Momentum

Conservation of Angular Momentum 10 March 2017 Conservation of ngular Momentum Lecture 23 In the last class, we discussed about the conservation of angular momentum principle. Using RTT, the angular momentum principle was given as DHo

More information

Module 2 : Lecture 1 GOVERNING EQUATIONS OF FLUID MOTION (Fundamental Aspects)

Module 2 : Lecture 1 GOVERNING EQUATIONS OF FLUID MOTION (Fundamental Aspects) Module : Lecture 1 GOVERNING EQUATIONS OF FLUID MOTION (Fundamental Aspects) Descriptions of Fluid Motion A fluid is composed of different particles for which the properties may change with respect to

More information

MULTIPLE-CHOICE PROBLEMS:(Two marks per answer) (Circle the Letter Beside the Most Correct Answer in the Questions Below.)

MULTIPLE-CHOICE PROBLEMS:(Two marks per answer) (Circle the Letter Beside the Most Correct Answer in the Questions Below.) MULTIPLE-CHOICE PROLEMS:(Two marks per answer) (Circle the Letter eside the Most Correct Answer in the Questions elow.) 1. The absolute viscosity µ of a fluid is primarily a function of: a. Density. b.

More information

CHAPTER 6 VECTOR CALCULUS. We ve spent a lot of time so far just looking at all the different ways you can graph

CHAPTER 6 VECTOR CALCULUS. We ve spent a lot of time so far just looking at all the different ways you can graph CHAPTER 6 VECTOR CALCULUS We ve spent a lot of time so far just looking at all the different ways you can graph things and describe things in three dimensions, and it certainly seems like there is a lot

More information

Fluid Dynamics Exercises and questions for the course

Fluid Dynamics Exercises and questions for the course Fluid Dynamics Exercises and questions for the course January 15, 2014 A two dimensional flow field characterised by the following velocity components in polar coordinates is called a free vortex: u r

More information

Module 12: Work and the Scalar Product

Module 12: Work and the Scalar Product Module 1: Work and the Scalar Product 1.1 Scalar Product (Dot Product) We shall introduce a vector operation, called the dot product or scalar product that takes any two vectors and generates a scalar

More information

Fluid Mechanics. du dy

Fluid Mechanics. du dy FLUID MECHANICS Technical English - I 1 th week Fluid Mechanics FLUID STATICS FLUID DYNAMICS Fluid Statics or Hydrostatics is the study of fluids at rest. The main equation required for this is Newton's

More information

MECH 5810 Module 3: Conservation of Linear Momentum

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

More information

i.e. the conservation of mass, the conservation of linear momentum, the conservation of energy.

i.e. the conservation of mass, the conservation of linear momentum, the conservation of energy. 04/04/2017 LECTURE 33 Geometric Interpretation of Stream Function: In the last class, you came to know about the different types of boundary conditions that needs to be applied to solve the governing equations

More information

Module 2: Governing Equations and Hypersonic Relations

Module 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 information

The conservation equations

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

More information

Fluids Lecture 1 Notes

Fluids Lecture 1 Notes Fluids Lecture Notes. Introductory Concepts and Definitions. Properties of Fluids Reading: Anderson. (optional),.,.3,.4 Introductory Concepts and Definitions Fluid Mechanics and Fluid Dynamics encompass

More information

Computational Fluid Dynamics Prof. Dr. Suman Chakraborty Department of Mechanical Engineering Indian Institute of Technology, Kharagpur

Computational Fluid Dynamics Prof. Dr. Suman Chakraborty Department of Mechanical Engineering Indian Institute of Technology, Kharagpur Computational Fluid Dynamics Prof. Dr. Suman Chakraborty Department of Mechanical Engineering Indian Institute of Technology, Kharagpur Lecture No. # 02 Conservation of Mass and Momentum: Continuity and

More information

CHAPTER 3 BASIC EQUATIONS IN FLUID MECHANICS NOOR ALIZA AHMAD

CHAPTER 3 BASIC EQUATIONS IN FLUID MECHANICS NOOR ALIZA AHMAD CHAPTER 3 BASIC EQUATIONS IN FLUID MECHANICS 1 INTRODUCTION Flow often referred as an ideal fluid. We presume that such a fluid has no viscosity. However, this is an idealized situation that does not exist.

More information

Math Review 1: Vectors

Math Review 1: Vectors Math Review 1: Vectors Coordinate System Coordinate system: used to describe the position of a point in space and consists of 1. An origin as the reference point 2. A set of coordinate axes with scales

More information

High Speed Aerodynamics. Copyright 2009 Narayanan Komerath

High Speed Aerodynamics. Copyright 2009 Narayanan Komerath Welcome to High Speed Aerodynamics 1 Lift, drag and pitching moment? Linearized Potential Flow Transformations Compressible Boundary Layer WHAT IS HIGH SPEED AERODYNAMICS? Airfoil section? Thin airfoil

More information

CH.1. DESCRIPTION OF MOTION. Continuum Mechanics Course (MMC)

CH.1. DESCRIPTION OF MOTION. Continuum Mechanics Course (MMC) CH.1. DESCRIPTION OF MOTION Continuum Mechanics Course (MMC) Overview 1.1. Definition of the Continuous Medium 1.1.1. Concept of Continuum 1.1.. Continuous Medium or Continuum 1.. Equations of Motion 1..1

More information

Astrophysical Fluid Dynamics

Astrophysical 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 information

CE 204 FLUID MECHANICS

CE 204 FLUID MECHANICS CE 204 FLUID MECHANICS Onur AKAY Assistant Professor Okan University Department of Civil Engineering Akfırat Campus 34959 Tuzla-Istanbul/TURKEY Phone: +90-216-677-1630 ext.1974 Fax: +90-216-677-1486 E-mail:

More information

ME3560 Tentative Schedule Spring 2019

ME3560 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 information

Phys101 First Major-111 Zero Version Monday, October 17, 2011 Page: 1

Phys101 First Major-111 Zero Version Monday, October 17, 2011 Page: 1 Monday, October 17, 011 Page: 1 Q1. 1 b The speed-time relation of a moving particle is given by: v = at +, where v is the speed, t t + c is the time and a, b, c are constants. The dimensional formulae

More information

Notes: DERIVATIVES. Velocity and Other Rates of Change

Notes: DERIVATIVES. Velocity and Other Rates of Change Notes: DERIVATIVES Velocity and Oter Rates of Cange I. Average Rate of Cange A.) Def.- Te average rate of cange of f(x) on te interval [a, b] is f( b) f( a) b a secant ( ) ( ) m troug a, f ( a ) and b,

More information

FLUID MECHANICS. Gaza. Chapter CHAPTER 44. Motion of Fluid Particles and Streams. Dr. Khalil Mahmoud ALASTAL

FLUID MECHANICS. Gaza. Chapter CHAPTER 44. Motion of Fluid Particles and Streams. Dr. Khalil Mahmoud ALASTAL FLUID MECHANICS Gaza Chapter CHAPTER 44 Motion of Fluid Particles and Streams Dr. Khalil Mahmoud ALASTAL Objectives of this Chapter: Introduce concepts necessary to analyze fluids in motion. Identify differences

More information

ME3560 Tentative Schedule Fall 2018

ME3560 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 information

Page 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.) 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 information

Chapter 9: Differential Analysis

Chapter 9: Differential Analysis 9-1 Introduction 9-2 Conservation of Mass 9-3 The Stream Function 9-4 Conservation of Linear Momentum 9-5 Navier Stokes Equation 9-6 Differential Analysis Problems Recall 9-1 Introduction (1) Chap 5: Control

More information

!! +! 2!! +!"!! =!! +! 2!! +!"!! +!!"!"!"

!! +! 2!! +!!! =!! +! 2!! +!!! +!!!! Homework 4 Solutions 1. (15 points) Bernoulli s equation can be adapted for use in evaluating unsteady flow conditions, such as those encountered during start- up processes. For example, consider the large

More information

1 Equations of motion

1 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 information

FLUID MECHANICS FLUID KINEMATICS VELOCITY FIELD FLOW PATTERNS

FLUID MECHANICS FLUID KINEMATICS VELOCITY FIELD FLOW PATTERNS www.getmyuni.com FLUID MECHANICS FLUID KINEMATICS Fluid Kinematics gives the geometry of fluid motion. It is a branch of fluid mechanics, which describes the fluid motion, and it s consequences without

More information

2.29 Numerical Fluid Mechanics Fall 2011 Lecture 5

2.29 Numerical Fluid Mechanics Fall 2011 Lecture 5 .9 Numerical Fluid Mechanics Fall 011 Lecture 5 REVIEW Lecture 4 Roots of nonlinear equations: Open Methods Fixed-point Iteration (General method or Picard Iteration), with examples Iteration rule: x g(

More information

Basic Fluid Mechanics

Basic Fluid Mechanics Basic Fluid Mechanics Chapter 3B: Conservation of Mass C3B: Conservation of Mass 1 3.2 Governing Equations There are two basic types of governing equations that we will encounter in this course Differential

More information

Introduction to Fluid Dynamics

Introduction 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 information

EKC314: 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) 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 information

Chapter 9: Differential Analysis of Fluid Flow

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 information

CHAPTER 2 INVISCID FLOW

CHAPTER 2 INVISCID FLOW CHAPTER 2 INVISCID FLOW Changes due to motion through a field; Newton s second law (f = ma) applied to a fluid: Euler s equation; Euler s equation integrated along a streamline: Bernoulli s equation; Bernoulli

More information

t 0. Show the necessary work and make sure that your conclusion is clear. (10 points) MTH 254 Test 1 No Calculator Portion Given: October 14, 2015

t 0. Show the necessary work and make sure that your conclusion is clear. (10 points) MTH 254 Test 1 No Calculator Portion Given: October 14, 2015 MTH 254 Test 1 No Calculator Portion Given: October 14, 2015 Name 1. Figures A F on page 2 of the supplement show portions of six different vector valued functions along with one surface upon which the

More information

Dynamic Meteorology 1

Dynamic Meteorology 1 Dynamic Meteorology 1 Lecture 14 Sahraei Department of Physics, Razi University http://www.razi.ac.ir/sahraei Buys-Ballot rule (Northern Hemisphere) If the wind blows into your back, the Low will be to

More information

Exam #2: Fluid Kinematics and Conservation Laws April 13, 2016, 7:00 p.m. 8:40 p.m. in CE 118

Exam #2: Fluid Kinematics and Conservation Laws April 13, 2016, 7:00 p.m. 8:40 p.m. in CE 118 CVEN 311-501 (Socolofsky) Fluid Dynamics Exam #2: Fluid Kinematics and Conservation Laws April 13, 2016, 7:00 p.m. 8:40 p.m. in CE 118 Name: : UIN: : Instructions: Fill in your name and UIN in the space

More information

NPTEL Quiz Hydraulics

NPTEL Quiz Hydraulics Introduction NPTEL Quiz Hydraulics 1. An ideal fluid is a. One which obeys Newton s law of viscosity b. Frictionless and incompressible c. Very viscous d. Frictionless and compressible 2. The unit of kinematic

More information

13.7 Power Applied by a Constant Force

13.7 Power Applied by a Constant Force 13.7 Power Applied by a Constant Force Suppose that an applied force F a acts on a body during a time interval Δt, and the displacement of the point of application of the force is in the x -direction by

More information

Therefore, the control volume in this case can be treated as a solid body, with a net force or thrust of. bm # V

Therefore, the control volume in this case can be treated as a solid body, with a net force or thrust of. bm # V When the mass m of the control volume remains nearly constant, the first term of the Eq. 6 8 simply becomes mass times acceleration since 39 CHAPTER 6 d(mv ) CV m dv CV CV (ma ) CV Therefore, the control

More information

2.2 Average vs. Instantaneous Description

2.2 Average vs. Instantaneous Description 2 KINEMATICS 2.2 Average vs. Instantaneous Description Name: 2.2 Average vs. Instantaneous Description 2.2.1 Average vs. Instantaneous Velocity In the previous activity, you figured out that you can calculate

More information

Review of fluid dynamics

Review of fluid dynamics Chapter 2 Review of fluid dynamics 2.1 Preliminaries ome basic concepts: A fluid is a substance that deforms continuously under stress. A Material olume is a tagged region that moves with the fluid. Hence

More information

2.25 Advanced Fluid Mechanics

2.25 Advanced Fluid Mechanics MIT Department of Mechanical Engineering.5 Advanced Fluid Mechanics Problem 4.05 This problem is from Advanced Fluid Mechanics Problems by A.H. Shapiro and A.A. Sonin Consider the frictionless, steady

More information

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

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

More information

CONSERVATION OF MASS AND BALANCE OF LINEAR MOMENTUM

CONSERVATION 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 information

AE301 Aerodynamics I UNIT B: Theory of Aerodynamics

AE301 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 information

ME 321: FLUID MECHANICS-I

ME 321: FLUID MECHANICS-I 6/07/08 ME 3: LUID MECHANI-I Dr. A.B.M. Toufique Hasan Professor Department of Mechanical Engineering Bangladesh Universit of Engineering & Technolog (BUET), Dhaka Lecture- 4/07/08 Momentum Principle teacher.buet.ac.bd/toufiquehasan/

More information

CLASS Fourth Units (First part)

CLASS Fourth Units (First part) CLASS Fourth Units (First part) Overview Fluid Kinematics deals with the motion of fluids without considering the forces and moments which create the motion. Items discussed in this Chapter. Material derivative

More information

6. Laminar and turbulent boundary layers

6. Laminar and turbulent boundary layers 6. Laminar and turbulent boundary layers John Richard Thome 8 avril 2008 John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril 2008 1 / 34 6.1 Some introductory ideas Figure 6.1 A boundary

More information