Calculating Viscous Flow: Velocity Profiles in Rivers and Pipes
|
|
- Quentin Morton
- 7 years ago
- Views:
Transcription
1 previous inex next Calculating Viscous Flow: Velocity Profiles in Rivers an Pipes Michael Fowler, UVa 9/8/1 Introuction In this lecture, we ll erive the velocity istribution for two examples of laminar flow. First we ll consier a wie river, by which we mean wie compare with its epth (which we take to be uniform) an we ignore the more complicate flow pattern near the banks. Our secon example is smooth flow own a circular pipe. For the wie river, the water flow can be thought of as being in horizontal sheets, so all the water at the same epth is moving at the same velocity. As mentione in the last lecture, the flow can be picture as like a pile of printer paper left on a sloping esk: it all slies own, assume the bottom sheet stays stuck to the esk, each other sheet moves ownhill a little faster than the sheet immeiately beneath it. For flow own a circular pipe, the laminar sheets are hollow tubes centere on the line own the mile of the pipe. The fastest flowing flui is right at that central line. For both river an tube flow, the rag force between ajacent small elements of neighboring sheets is given by force per unit area F v z A where now the z-irection means perpenicular to the small element of sheet. A Flowing River: Fining the Velocity Profile For a river flowing steaily own a gentle incline uner gravity, we ll assume all the streamlines point in the same irection, the river is wie an of uniform epth, an the epth is much smaller than the with. This means almost all the flow is well away from the eges (the river banks), so we ll ignore the slowing own there, an just analyze the flow rate per meter of river with, taking it to be uniform across the river. The simplest basic question is: given the slope of the lan an the epth of the river, what is the total flow rate? To answer, we nee to fin the spee of flow v(z) as a function of epth (we know the water in contact with the river be isn t flowing at all), an then a the flow contributions from the ifferent epths (this will be an integral) to fin the total flow. The function v(z) is calle the velocity profile. We ll prove it looks something like this:
2 Stream velocity v(z) at ifferent epths (For a smoothly flowing river, the ownhill groun slope woul be imperceptible on this scale.) But how o we begin to calculate v(z)? Recall that (in an earlier lecture) to fin how hyrostatic pressure varie with epth, we mentally separate a cyliner of flui from its surrounings, an applie Newton s Laws: it wasn t moving, so we figure its weight ha to be balance by the sum of the pressure forces it experience from the rest of the flui surrouning it. In fact, its weight was balance by the ifference between the pressure unerneath an that on top. Taking a cue from that, here we isolate mentally a thin layer of the river, like one of those sheets of printer paper, lying between height z above the be an z z. This layer is moving, but at a steay spee, so the total force on it will still be zero. Like the whole river, this layer isn t quite horizontal, its weight has a small but nonzero component ragging it ownhill, an this weight component is balance by the ifference between the viscous force from the faster water above an that from slower water below. Bear in min that the iagram below is at a tiny angle to the horizontal: Forwar viscous rag from flui above mgsin ownhill Backwar viscous rag from flui below z + z z Obviously, for the forces to balance, the backwar rag on the thin layer from the slower moving water beneath has to be stronger than the forwar rag from the faster water above, so the rate of change of spee with height above the river be is ecreasing on going up from the be. Let us fin the total force (which must be zero) on one square meter of the thin layer of water between heights z an z z: First, gravity: if the river is flowing ownhill at some small angle, this square meter of the 3 layer (volume z m 1 m z m, ensity ) experiences a gravitational force mg sin gz tugging it ownstream (taking the small angle approximation, sin.)
3 3 Next, the viscous rag forces: the square meter of layer experiences two viscous forces, one from the slower water below, equal to v z /, tening to slow it own, one from the faster water above it, v z z /, tening to spee it up. Gravity must balance out the ifference between the two viscous forces: g z v z z v z 0 We can alreay see from this equation that, unlike the flui between the plates, v(z) can t possibly be linear in z the equation woul not balance if v / were the same at z an z z! Diviing throughout by an by z, v z z v z z g. Taking now the limit z 0, an recalling the efinition of the ifferential f x f x x f x lim x x 0 x we fin the ifferential equation v z g. The solution of this equation is easy: with C, D constants of integration. g vz z Cz D Remember that the velocity v(z) is zero at the bottom of the river, z = 0, so the constant D must be zero, an can be roppe immeiately. But we re not through we haven t foun C. To o that, we nee to go to the top. Velocity Profile Near the River Surface What happens to the thin layer of river water at the very top the layer in contact with the air? Assuming there is negligible win, there is essentially zero parallel-to-the-surface force from above.
4 4 So the balance of forces equation for the top layer is just gz vz 0. We can take this top layer to be as thin as we like, so let s look what happens in the limit of extreme thinness, z 0. The term gzthen goes to zero, so the other term must as well. Since is constant, this means v z 0 at the surface z h. So the velocity profile function v z has zero slope at the river surface. With this new information, we can finally fix the arbitrary integration constant C. Now the velocity profile so v z gives an h 0 g h C. g vz z Cz, v z g z C, Putting this value for C into v(z) we have the final result: g vz z h z. This velocity profile v(z) is half the top part of a parabola:
5 5 Total River Flow Knowing the velocity profile v(z) enables us to compute the total flow of water in the river. As explaine earlier, we re assuming a wie river having uniform epth, ignoring the slowown near the eges of the river, taking the same v(z) all the way across. We ll calculate the flow across one meter of with of the river, so the total flow is our result multiplie by the river s with. The flow contribution from a single layer of thickness z at height z is vz z cubic meters per secon across one meter of with. The total flow is the sum over all layers. In the limit of many infinitely thin layers, that is, z 0, the sum becomes an integral, an the total flow rate h h I v( z) ( g / ) z( h z) ( g / 3 ) h in cubic meters per secon per meter of with of the river. It is worth thinking about what this result means physically. The interesting part is that the flow is proportional to h 3, where h is the epth of the river. So, if there s a storm an the river is twice as eep as normal, an flowing steaily, the flow rate will be eight times normal. Exercise: plot on a graph the velocity profiles for two rivers, one of epth h an one h, having the same values of, g, an. What is the ratio of the surface velocities of the two rivers? Suppose that one meter below the surface of one of the rivers, the water is flowing 0.5 m.sec -1 slower than it is flowing at the surface. Woul that also be true of the other river? Flow own a Circular Tube (Poiseuille Flow) The flow rate for smooth flow through a pipe of circular cross-section can be foun by essentially the same metho. (This was the flow pattern analyze by Poiseuille an use by him to confirm Newton s postulate of flui flow behavior being governe by a coefficient of viscosity.) In the pipe, the flow is fastest in the mile, an the water in contact with the pipe wall (like that at the river be) oesn t flow at all. The river s flow pattern was most naturally analyze by thinking of flat layers of water, all the water in one layer having the same spee. What woul be the corresponing picture for flow own a pipe? Here all the flui at the same istance from the center moves own the pipe at the same spee instea of flat layers of flui, we have concentric hollow cyliners of flui, one insie the next, with a tiny ro of the fastest flui right at the center. This is again laminar flow, even though this time the sheets are rolle into tubes. The blue circular area on the cross-section of the pipe shown below represents one of these cyliners of flui all the flui between r an rrfrom the central line. Each of these hollow cyliners of water is pushe along the pipe by the pressure ifference between the ens of the pipe. Each feels viscous forces from its two neighboring cyliners: the next bigger one, which surrouns it, tening to slow it own, but the next smaller one (insie it) tening to spee it up. Writing own the ifferential equation is a little more tricky that for the
6 6 river, because we must take into account that the two surfaces of the hollow cyliner (insie an outsie) have ifferent areas, rl r r L. It turns out that the velocity profile is an again parabolic: the etails are given below. a Velocity profile for laminar flow own a circular pipe Circular Pipe Flow: Mathematical Details Suppose the pipe has raius a, length L an pressure rop P, pressure rop per meter P/ L. Let us focus on the flui in the cyliner between r an rrfrom the line own the mile, an we ll take the cyliner to have unit length, for convenience. The pressure force maintaining the flui motion is the ifference between pressure x area for the two ens of this one meter long hollow cyliner: P net pressure force r r. L (We re assuming r r, since we ll be taking the r 0 limit, so the en area r r. The equality becomes exact in the limit.) This force exactly balances the ifference between the outer surface viscous rag force from the slower surrouning flui an the inner viscous force from the central faster-moving flui, very similar to the situation in the previous analysis of river flow. Using F / A v z /, an remembering that the inner an outer surfaces of the cyliner have slightly ifferent areas, that P is positive, but v / r is negative, the force equation is: Rearranging, P r r r r v r r r v r. L r r
7 7 v r r v r r r r P r r r L r v r r r in the limit r 0, remembering the efinition of the ifferential (see the similar analysis above for the river). This can now be integrate to give v P r r C r L where C is a constant of integration. Diviing both sies by r an integrating again P r vr C ln r D. L 4 The constant C must be zero, since physically the flui velocity is finite at r = 0. The constant D is etermine by the requirement that the flui spee is zero where the flui is in contact with the tube, at r = a. The flui velocity is therefore v r P L a r To fin the total flow rate I own the pipe, we integrate over the flow in each hollow cyliner of water: in cubic meters per secon. a a 3 P a r r P I rv rr r a L 4 8 L 0 0 Notice the flow rate goes as the fourth power of the raius, so oubling the raius results in a sixteen-fol increase in flow. That is why narrowing of arteries is so serious previous inex next Thanks to Lina Fahlberg-Stojanovska for spotting a sign error in the earlier version.
Answers to the Practice Problems for Test 2
Answers to the Practice Problems for Test 2 Davi Murphy. Fin f (x) if it is known that x [f(2x)] = x2. By the chain rule, x [f(2x)] = f (2x) 2, so 2f (2x) = x 2. Hence f (2x) = x 2 /2, but the lefthan
More informationExample Optimization Problems selected from Section 4.7
Example Optimization Problems selecte from Section 4.7 19) We are aske to fin the points ( X, Y ) on the ellipse 4x 2 + y 2 = 4 that are farthest away from the point ( 1, 0 ) ; as it happens, this point
More informationAs customary, choice (a) is the correct answer in all the following problems.
PHY2049 Summer 2012 Instructor: Francisco Rojas Exam 1 As customary, choice (a) is the correct answer in all the following problems. Problem 1 A uniformly charge (thin) non-conucting ro is locate on the
More informationLagrangian and Hamiltonian Mechanics
Lagrangian an Hamiltonian Mechanics D.G. Simpson, Ph.D. Department of Physical Sciences an Engineering Prince George s Community College December 5, 007 Introuction In this course we have been stuying
More informationFluid Pressure and Fluid Force
0_0707.q //0 : PM Page 07 SECTION 7.7 Section 7.7 Flui Pressure an Flui Force 07 Flui Pressure an Flui Force Fin flui pressure an flui force. Flui Pressure an Flui Force Swimmers know that the eeper an
More informationReading: Ryden chs. 3 & 4, Shu chs. 15 & 16. For the enthusiasts, Shu chs. 13 & 14.
7 Shocks Reaing: Ryen chs 3 & 4, Shu chs 5 & 6 For the enthusiasts, Shu chs 3 & 4 A goo article for further reaing: Shull & Draine, The physics of interstellar shock waves, in Interstellar processes; Proceeings
More information10.2 Systems of Linear Equations: Matrices
SECTION 0.2 Systems of Linear Equations: Matrices 7 0.2 Systems of Linear Equations: Matrices OBJECTIVES Write the Augmente Matrix of a System of Linear Equations 2 Write the System from the Augmente Matrix
More informationf(x) = a x, h(5) = ( 1) 5 1 = 2 2 1
Exponential Functions an their Derivatives Exponential functions are functions of the form f(x) = a x, where a is a positive constant referre to as the base. The functions f(x) = x, g(x) = e x, an h(x)
More informationThe Quick Calculus Tutorial
The Quick Calculus Tutorial This text is a quick introuction into Calculus ieas an techniques. It is esigne to help you if you take the Calculus base course Physics 211 at the same time with Calculus I,
More informationMathematics. Circles. hsn.uk.net. Higher. Contents. Circles 119 HSN22400
hsn.uk.net Higher Mathematics UNIT OUTCOME 4 Circles Contents Circles 119 1 Representing a Circle 119 Testing a Point 10 3 The General Equation of a Circle 10 4 Intersection of a Line an a Circle 1 5 Tangents
More information11 CHAPTER 11: FOOTINGS
CHAPTER ELEVEN FOOTINGS 1 11 CHAPTER 11: FOOTINGS 11.1 Introuction Footings are structural elements that transmit column or wall loas to the unerlying soil below the structure. Footings are esigne to transmit
More informationLecture L25-3D Rigid Body Kinematics
J. Peraire, S. Winall 16.07 Dynamics Fall 2008 Version 2.0 Lecture L25-3D Rigi Boy Kinematics In this lecture, we consier the motion of a 3D rigi boy. We shall see that in the general three-imensional
More informationDIFFRACTION AND INTERFERENCE
DIFFRACTION AND INTERFERENCE In this experiment you will emonstrate the wave nature of light by investigating how it bens aroun eges an how it interferes constructively an estructively. You will observe
More informationHomework 8. problems: 10.40, 10.73, 11.55, 12.43
Hoework 8 probles: 0.0, 0.7,.55,. Proble 0.0 A block of ass kg an a block of ass 6 kg are connecte by a assless strint over a pulley in the shape of a soli isk having raius R0.5 an ass M0 kg. These blocks
More informationNotes on tangents to parabolas
Notes on tangents to parabolas (These are notes for a talk I gave on 2007 March 30.) The point of this talk is not to publicize new results. The most recent material in it is the concept of Bézier curves,
More informationf(x + h) f(x) h as representing the slope of a secant line. As h goes to 0, the slope of the secant line approaches the slope of the tangent line.
Derivative of f(z) Dr. E. Jacobs Te erivative of a function is efine as a limit: f (x) 0 f(x + ) f(x) We can visualize te expression f(x+) f(x) as representing te slope of a secant line. As goes to 0,
More informationMath 230.01, Fall 2012: HW 1 Solutions
Math 3., Fall : HW Solutions Problem (p.9 #). Suppose a wor is picke at ranom from this sentence. Fin: a) the chance the wor has at least letters; SOLUTION: All wors are equally likely to be chosen. The
More informationMechanics 1: Conservation of Energy and Momentum
Mechanics : Conservation of Energy and Momentum If a certain quantity associated with a system does not change in time. We say that it is conserved, and the system possesses a conservation law. Conservation
More informationDifferential Relations for Fluid Flow. Acceleration field of a fluid. The differential equation of mass conservation
Differential Relations for Fluid Flow In this approach, we apply our four basic conservation laws to an infinitesimally small control volume. The differential approach provides point by point details of
More informationRules for Finding Derivatives
3 Rules for Fining Derivatives It is teious to compute a limit every time we nee to know the erivative of a function. Fortunately, we can evelop a small collection of examples an rules that allow us to
More informationHeat Transfer Prof. Dr. Aloke Kumar Ghosal Department of Chemical Engineering Indian Institute of Technology, Guwahati
Heat Transfer Prof. Dr. Aloke Kumar Ghosal Department of Chemical Engineering Indian Institute of Technology, Guwahati Module No. # 02 One Dimensional Steady State Heat Transfer Lecture No. # 05 Extended
More informationIntroduction to Integration Part 1: Anti-Differentiation
Mathematics Learning Centre Introuction to Integration Part : Anti-Differentiation Mary Barnes c 999 University of Syney Contents For Reference. Table of erivatives......2 New notation.... 2 Introuction
More informationGiven three vectors A, B, andc. We list three products with formula (A B) C = B(A C) A(B C); A (B C) =B(A C) C(A B);
1.1.4. Prouct of three vectors. Given three vectors A, B, anc. We list three proucts with formula (A B) C = B(A C) A(B C); A (B C) =B(A C) C(A B); a 1 a 2 a 3 (A B) C = b 1 b 2 b 3 c 1 c 2 c 3 where the
More informationWork and Energy. Physics 1425 Lecture 12. Michael Fowler, UVa
Work and Energy Physics 1425 Lecture 12 Michael Fowler, UVa What is Work and What Isn t? In physics, work has a very restricted meaning! Doing homework isn t work. Carrying somebody a mile on a level road
More informationChapter 6 Work and Energy
Chapter 6 WORK AND ENERGY PREVIEW Work is the scalar product of the force acting on an object and the displacement through which it acts. When work is done on or by a system, the energy of that system
More informationLines. We have learned that the graph of a linear equation. y = mx +b
Section 0. Lines We have learne that the graph of a linear equation = m +b is a nonvertical line with slope m an -intercept (0, b). We can also look at the angle that such a line makes with the -ais. This
More informationPhysics: Principles and Applications, 6e Giancoli Chapter 2 Describing Motion: Kinematics in One Dimension
Physics: Principles and Applications, 6e Giancoli Chapter 2 Describing Motion: Kinematics in One Dimension Conceptual Questions 1) Suppose that an object travels from one point in space to another. Make
More informationSections 3.1/3.2: Introducing the Derivative/Rules of Differentiation
Sections 3.1/3.2: Introucing te Derivative/Rules of Differentiation 1 Tangent Line Before looking at te erivative, refer back to Section 2.1, looking at average velocity an instantaneous velocity. Here
More information1. Mass, Force and Gravity
STE Physics Intro Name 1. Mass, Force and Gravity Before attempting to understand force, we need to look at mass and acceleration. a) What does mass measure? The quantity of matter(atoms) b) What is the
More informationXI / PHYSICS FLUIDS IN MOTION 11/PA
Viscosity It is the property of a liquid due to which it flows in the form of layers and each layer opposes the motion of its adjacent layer. Cause of viscosity Consider two neighboring liquid layers A
More informationHere the units used are radians and sin x = sin(x radians). Recall that sin x and cos x are defined and continuous everywhere and
Lecture 9 : Derivatives of Trigonometric Functions (Please review Trigonometry uner Algebra/Precalculus Review on the class webpage.) In this section we will look at the erivatives of the trigonometric
More informationCURRENCY OPTION PRICING II
Jones Grauate School Rice University Masa Watanabe INTERNATIONAL FINANCE MGMT 657 Calibrating the Binomial Tree to Volatility Black-Scholes Moel for Currency Options Properties of the BS Moel Option Sensitivity
More informationFRICTION, WORK, AND THE INCLINED PLANE
FRICTION, WORK, AND THE INCLINED PLANE Objective: To measure the coefficient of static and inetic friction between a bloc and an inclined plane and to examine the relationship between the plane s angle
More informationp atmospheric Statics : Pressure Hydrostatic Pressure: linear change in pressure with depth Measure depth, h, from free surface Pressure Head p gh
IVE1400: n Introduction to Fluid Mechanics Statics : Pressure : Statics r P Sleigh: P..Sleigh@leeds.ac.uk r J Noakes:.J.Noakes@leeds.ac.uk January 008 Module web site: www.efm.leeds.ac.uk/ive/fluidslevel1
More informationi( t) L i( t) 56mH 1.1A t = τ ln 1 = ln 1 ln 1 6.67ms
Exam III PHY 49 Summer C July 16, 8 1. In the circuit shown, L = 56 mh, R = 4.6 Ω an V = 1. V. The switch S has been open for a long time then is suenly close at t =. At what value of t (in msec) will
More informationPhysics Midterm Review Packet January 2010
Physics Midterm Review Packet January 2010 This Packet is a Study Guide, not a replacement for studying from your notes, tests, quizzes, and textbook. Midterm Date: Thursday, January 28 th 8:15-10:15 Room:
More informationFluids and Solids: Fundamentals
Fluids and Solids: Fundamentals We normally recognize three states of matter: solid; liquid and gas. However, liquid and gas are both fluids: in contrast to solids they lack the ability to resist deformation.
More informationC B A T 3 T 2 T 1. 1. What is the magnitude of the force T 1? A) 37.5 N B) 75.0 N C) 113 N D) 157 N E) 192 N
Three boxes are connected by massless strings and are resting on a frictionless table. Each box has a mass of 15 kg, and the tension T 1 in the right string is accelerating the boxes to the right at a
More informationFreely Falling Objects
Freely Falling Objects Physics 1425 Lecture 3 Michael Fowler, UVa. Today s Topics In the previous lecture, we analyzed onedimensional motion, defining displacement, velocity, and acceleration and finding
More informationx x y y Then, my slope is =. Notice, if we use the slope formula, we ll get the same thing: m =
Slope and Lines The slope of a line is a ratio that measures the incline of the line. As a result, the smaller the incline, the closer the slope is to zero and the steeper the incline, the farther the
More informationPractice Problems on Boundary Layers. Answer(s): D = 107 N D = 152 N. C. Wassgren, Purdue University Page 1 of 17 Last Updated: 2010 Nov 22
BL_01 A thin flat plate 55 by 110 cm is immersed in a 6 m/s stream of SAE 10 oil at 20 C. Compute the total skin friction drag if the stream is parallel to (a) the long side and (b) the short side. D =
More information1. the acceleration of the body decreases by. 2. the acceleration of the body increases by. 3. the body falls 9.8 m during each second.
Answer, Key Homework 3 Davi McIntyre 45123 Mar 25, 2004 1 This print-out shoul have 21 questions. Multiple-choice questions may continue on the next column or pae fin all choices before makin your selection.
More informationLAB 6 - GRAVITATIONAL AND PASSIVE FORCES
L06-1 Name Date Partners LAB 6 - GRAVITATIONAL AND PASSIVE FORCES OBJECTIVES And thus Nature will be very conformable to herself and very simple, performing all the great Motions of the heavenly Bodies
More informationHow To Understand The Theory Of Gravity
Newton s Law of Gravity and Kepler s Laws Michael Fowler Phys 142E Lec 9 2/6/09. These notes are partly adapted from my Physics 152 lectures, where more mathematical details can be found. The Universal
More informationAcceleration of Gravity Lab Basic Version
Acceleration of Gravity Lab Basic Version In this lab you will explore the motion of falling objects. As an object begins to fall, it moves faster and faster (its velocity increases) due to the acceleration
More informationPhysics Kinematics Model
Physics Kinematics Model I. Overview Active Physics introduces the concept of average velocity and average acceleration. This unit supplements Active Physics by addressing the concept of instantaneous
More informationIf you put the same book on a tilted surface the normal force will be less. The magnitude of the normal force will equal: N = W cos θ
Experiment 4 ormal and Frictional Forces Preparation Prepare for this week's quiz by reviewing last week's experiment Read this week's experiment and the section in your textbook dealing with normal forces
More information15.2. First-Order Linear Differential Equations. First-Order Linear Differential Equations Bernoulli Equations Applications
00 CHAPTER 5 Differential Equations SECTION 5. First-Orer Linear Differential Equations First-Orer Linear Differential Equations Bernoulli Equations Applications First-Orer Linear Differential Equations
More informationPLOTTING DATA AND INTERPRETING GRAPHS
PLOTTING DATA AND INTERPRETING GRAPHS Fundamentals of Graphing One of the most important sets of skills in science and mathematics is the ability to construct graphs and to interpret the information they
More information10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED
CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations
More informationELEMENTS OF METRIC GEAR TECHNOLOGY
ELEMENS OF MEC GE ECHNOLOGY SECON SPU GE CLCULONS PHONE:..00 FX:.. WWW.SDP-S.COM. Stanar Spur Gear 0 0 Figure - shows the meshing of stanar spur gears. he meshing of stanar spur gears means pitch circles
More informationChapter 3 Falling Objects and Projectile Motion
Chapter 3 Falling Objects and Projectile Motion Gravity influences motion in a particular way. How does a dropped object behave?!does the object accelerate, or is the speed constant?!do two objects behave
More informationMath 1B, lecture 5: area and volume
Math B, lecture 5: area and volume Nathan Pflueger 6 September 2 Introduction This lecture and the next will be concerned with the computation of areas of regions in the plane, and volumes of regions in
More informationIntroduction to COMSOL. The Navier-Stokes Equations
Flow Between Parallel Plates Modified from the COMSOL ChE Library module rev 10/13/08 Modified by Robert P. Hesketh, Chemical Engineering, Rowan University Fall 2008 Introduction to COMSOL The following
More information6. Vectors. 1 2009-2016 Scott Surgent (surgent@asu.edu)
6. Vectors For purposes of applications in calculus and physics, a vector has both a direction and a magnitude (length), and is usually represented as an arrow. The start of the arrow is the vector s foot,
More informationUnsteady Flow Visualization by Animating Evenly-Spaced Streamlines
EUROGRAPHICS 2000 / M. Gross an F.R.A. Hopgoo Volume 19, (2000), Number 3 (Guest Eitors) Unsteay Flow Visualization by Animating Evenly-Space Bruno Jobar an Wilfri Lefer Université u Littoral Côte Opale,
More informationVISUAL PHYSICS School of Physics University of Sydney Australia. Why do cars need different oils in hot and cold countries?
VISUAL PHYSICS School of Physics University of Sydney Australia FLUID FLOW VISCOSITY POISEUILLE'S LAW? Why do cars need different oils in hot and cold countries? Why does the engine runs more freely as
More information20. Product rule, Quotient rule
20. Prouct rule, 20.1. Prouct rule Prouct rule, Prouct rule We have seen that the erivative of a sum is the sum of the erivatives: [f(x) + g(x)] = x x [f(x)] + x [(g(x)]. One might expect from this that
More informationOpen channel flow Basic principle
Open channel flow Basic principle INTRODUCTION Flow in rivers, irrigation canals, drainage ditches and aqueducts are some examples for open channel flow. These flows occur with a free surface and the pressure
More informationName Partners Date. Energy Diagrams I
Name Partners Date Visual Quantum Mechanics The Next Generation Energy Diagrams I Goal Changes in energy are a good way to describe an object s motion. Here you will construct energy diagrams for a toy
More informationLAB 6: GRAVITATIONAL AND PASSIVE FORCES
55 Name Date Partners LAB 6: GRAVITATIONAL AND PASSIVE FORCES And thus Nature will be very conformable to herself and very simple, performing all the great Motions of the heavenly Bodies by the attraction
More informationLab 7: Rotational Motion
Lab 7: Rotational Motion Equipment: DataStudio, rotary motion sensor mounted on 80 cm rod and heavy duty bench clamp (PASCO ME-9472), string with loop at one end and small white bead at the other end (125
More informationPhysics 201 Homework 8
Physics 201 Homework 8 Feb 27, 2013 1. A ceiling fan is turned on and a net torque of 1.8 N-m is applied to the blades. 8.2 rad/s 2 The blades have a total moment of inertia of 0.22 kg-m 2. What is the
More informationScalar : Vector : Equal vectors : Negative vectors : Proper vector : Null Vector (Zero Vector): Parallel vectors : Antiparallel vectors :
ELEMENTS OF VECTOS 1 Scalar : physical quantity having only magnitue but not associate with any irection is calle a scalar eg: time, mass, istance, spee, work, energy, power, pressure, temperature, electric
More informationCE 3500 Fluid Mechanics / Fall 2014 / City College of New York
1 Drag Coefficient The force ( F ) of the wind blowing against a building is given by F=C D ρu 2 A/2, where U is the wind speed, ρ is density of the air, A the cross-sectional area of the building, and
More informationCh 2 Properties of Fluids - II. Ideal Fluids. Real Fluids. Viscosity (1) Viscosity (3) Viscosity (2)
Ch 2 Properties of Fluids - II Ideal Fluids 1 Prepared for CEE 3500 CEE Fluid Mechanics by Gilberto E. Urroz, August 2005 2 Ideal fluid: a fluid with no friction Also referred to as an inviscid (zero viscosity)
More informationUniformly Accelerated Motion
Uniformly Accelerated Motion Under special circumstances, we can use a series of three equations to describe or predict movement V f = V i + at d = V i t + 1/2at 2 V f2 = V i2 + 2ad Most often, these equations
More informationHeat Transfer Prof. Dr. Ale Kumar Ghosal Department of Chemical Engineering Indian Institute of Technology, Guwahati
Heat Transfer Prof. Dr. Ale Kumar Ghosal Department of Chemical Engineering Indian Institute of Technology, Guwahati Module No. # 04 Convective Heat Transfer Lecture No. # 03 Heat Transfer Correlation
More informationSOLUTIONS TO CONCEPTS CHAPTER 17
1. Given that, 400 m < < 700 nm. 1 1 1 700nm 400nm SOLUTIONS TO CONCETS CHATER 17 1 1 1 3 10 c 3 10 (Where, c = spee of light = 3 10 m/s) 7 7 7 7 7 10 4 10 7 10 4 10 4.3 10 14 < c/ < 7.5 10 14 4.3 10 14
More informationNewton s Laws. Physics 1425 lecture 6. Michael Fowler, UVa.
Newton s Laws Physics 1425 lecture 6 Michael Fowler, UVa. Newton Extended Galileo s Picture of Galileo said: Motion to Include Forces Natural horizontal motion is at constant velocity unless a force acts:
More informationSection 3.3. Differentiation of Polynomials and Rational Functions. Difference Equations to Differential Equations
Difference Equations to Differential Equations Section 3.3 Differentiation of Polynomials an Rational Functions In tis section we begin te task of iscovering rules for ifferentiating various classes of
More informationUniversity Physics 226N/231N Old Dominion University. Getting Loopy and Friction
University Physics 226N/231N Old Dominion University Getting Loopy and Friction Dr. Todd Satogata (ODU/Jefferson Lab) satogata@jlab.org http://www.toddsatogata.net/2012-odu Friday, September 28 2012 Happy
More informationEXAMPLE: Water Flow in a Pipe
EXAMPLE: Water Flow in a Pipe P 1 > P 2 Velocity profile is parabolic (we will learn why it is parabolic later, but since friction comes from walls the shape is intuitive) The pressure drops linearly along
More informationChapter 28 Fluid Dynamics
Chapter 28 Fluid Dynamics 28.1 Ideal Fluids... 1 28.2 Velocity Vector Field... 1 28.3 Mass Continuity Equation... 3 28.4 Bernoulli s Principle... 4 28.5 Worked Examples: Bernoulli s Equation... 7 Example
More informationNewton s Laws. Newton s Imaginary Cannon. Michael Fowler Physics 142E Lec 6 Jan 22, 2009
Newton s Laws Michael Fowler Physics 142E Lec 6 Jan 22, 2009 Newton s Imaginary Cannon Newton was familiar with Galileo s analysis of projectile motion, and decided to take it one step further. He imagined
More informationSimple Harmonic Motion
Simple Harmonic Motion 1 Object To determine the period of motion of objects that are executing simple harmonic motion and to check the theoretical prediction of such periods. 2 Apparatus Assorted weights
More informationoil liquid water water liquid Answer, Key Homework 2 David McIntyre 1
Answer, Key Homework 2 David McIntyre 1 This print-out should have 14 questions, check that it is complete. Multiple-choice questions may continue on the next column or page: find all choices before making
More informationMotion Graphs. It is said that a picture is worth a thousand words. The same can be said for a graph.
Motion Graphs It is said that a picture is worth a thousand words. The same can be said for a graph. Once you learn to read the graphs of the motion of objects, you can tell at a glance if the object in
More informationWhat is the most obvious difference between pipe flow and open channel flow????????????? (in terms of flow conditions and energy situation)
OPEN CHANNEL FLOW 1 3 Question What is the most obvious difference between pipe flow and open channel flow????????????? (in terms of flow conditions and energy situation) Typical open channel shapes Figure
More informationy or f (x) to determine their nature.
Level C5 of challenge: D C5 Fining stationar points of cubic functions functions Mathematical goals Starting points Materials require Time neee To enable learners to: fin the stationar points of a cubic
More informationDesign Considerations for Water-Bottle Rockets. The next few pages are provided to help in the design of your water-bottle rocket.
Acceleration= Force OVER Mass Design Considerations for Water-Bottle Rockets The next few pages are provided to help in the design of your water-bottle rocket. Newton s First Law: Objects at rest will
More informationLecture 17. Last time we saw that the rotational analog of Newton s 2nd Law is
Lecture 17 Rotational Dynamics Rotational Kinetic Energy Stress and Strain and Springs Cutnell+Johnson: 9.4-9.6, 10.1-10.2 Rotational Dynamics (some more) Last time we saw that the rotational analog of
More informationPhysics Labs with Computers, Vol. 2 P38: Conservation of Linear Momentum 012-07001A
Name Class Date Activity P38: Conservation of Linear Momentum (Motion Sensors) Concept DataStudio ScienceWorkshop (Mac) ScienceWorkshop (Win) Newton s Laws P38 Linear Momentum.DS P16 Cons. of Momentum
More informationLecture 6. Weight. Tension. Normal Force. Static Friction. Cutnell+Johnson: 4.8-4.12, second half of section 4.7
Lecture 6 Weight Tension Normal Force Static Friction Cutnell+Johnson: 4.8-4.12, second half of section 4.7 In this lecture, I m going to discuss four different kinds of forces: weight, tension, the normal
More informationIf A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C?
Problem 3 If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Suggested Questions to ask students about Problem 3 The key to this question
More informationA Determination of g, the Acceleration Due to Gravity, from Newton's Laws of Motion
A Determination of g, the Acceleration Due to Gravity, from Newton's Laws of Motion Objective In the experiment you will determine the cart acceleration, a, and the friction force, f, experimentally for
More informationStack Contents. Pressure Vessels: 1. A Vertical Cut Plane. Pressure Filled Cylinder
Pressure Vessels: 1 Stack Contents Longitudinal Stress in Cylinders Hoop Stress in Cylinders Hoop Stress in Spheres Vanishingly Small Element Radial Stress End Conditions 1 2 Pressure Filled Cylinder A
More informationNewton s Second Law. ΣF = m a. (1) In this equation, ΣF is the sum of the forces acting on an object, m is the mass of
Newton s Second Law Objective The Newton s Second Law experiment provides the student a hands on demonstration of forces in motion. A formulated analysis of forces acting on a dynamics cart will be developed
More informationNumerical integration of a function known only through data points
Numerical integration of a function known only through data points Suppose you are working on a project to determine the total amount of some quantity based on measurements of a rate. For example, you
More informationSolving Simultaneous Equations and Matrices
Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering
More informationPhysics 40 Lab 1: Tests of Newton s Second Law
Physics 40 Lab 1: Tests of Newton s Second Law January 28 th, 2008, Section 2 Lynda Williams Lab Partners: Madonna, Hilary Clinton & Angie Jolie Abstract Our primary objective was to test the validity
More information1. Fluids Mechanics and Fluid Properties. 1.1 Objectives of this section. 1.2 Fluids
1. Fluids Mechanics and Fluid Properties What is fluid mechanics? As its name suggests it is the branch of applied mechanics concerned with the statics and dynamics of fluids - both liquids and gases.
More informationChapter 5 Using Newton s Laws: Friction, Circular Motion, Drag Forces. Copyright 2009 Pearson Education, Inc.
Chapter 5 Using Newton s Laws: Friction, Circular Motion, Drag Forces Units of Chapter 5 Applications of Newton s Laws Involving Friction Uniform Circular Motion Kinematics Dynamics of Uniform Circular
More informationConservation of Energy Physics Lab VI
Conservation of Energy Physics Lab VI Objective This lab experiment explores the principle of energy conservation. You will analyze the final speed of an air track glider pulled along an air track by a
More informationExperiment 3 Pipe Friction
EML 316L Experiment 3 Pipe Friction Laboratory Manual Mechanical and Materials Engineering Department College of Engineering FLORIDA INTERNATIONAL UNIVERSITY Nomenclature Symbol Description Unit A cross-sectional
More informationdu u U 0 U dy y b 0 b
BASIC CONCEPTS/DEFINITIONS OF FLUID MECHANICS (by Marios M. Fyrillas) 1. Density (πυκνότητα) Symbol: 3 Units of measure: kg / m Equation: m ( m mass, V volume) V. Pressure (πίεση) Alternative definition:
More informationDetermining the Acceleration Due to Gravity
Chabot College Physics Lab Scott Hildreth Determining the Acceleration Due to Gravity Introduction In this experiment, you ll determine the acceleration due to earth s gravitational force with three different
More informationLinear functions Increasing Linear Functions. Decreasing Linear Functions
3.5 Increasing, Decreasing, Max, and Min So far we have been describing graphs using quantitative information. That s just a fancy way to say that we ve been using numbers. Specifically, we have described
More informationForce on Moving Charges in a Magnetic Field
[ Assignment View ] [ Eðlisfræði 2, vor 2007 27. Magnetic Field and Magnetic Forces Assignment is due at 2:00am on Wednesday, February 28, 2007 Credit for problems submitted late will decrease to 0% after
More informationACCELERATION DUE TO GRAVITY
EXPERIMENT 1 PHYSICS 107 ACCELERATION DUE TO GRAVITY Skills you will learn or practice: Calculate velocity and acceleration from experimental measurements of x vs t (spark positions) Find average velocities
More information