Submitted By shegzydee

Words 50231

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Words 50231

Pages 201

A Mathematical Introduction to Fluid Mechanics

Alexandre Chorin

Department of Mathematics University of California, Berkeley Berkeley, California 94720-3840, USA

Jerrold E. Marsden

Control and Dynamical Systems, 107-81 California Institute of Technology Pasadena, California 91125, USA

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A Mathematical Introduction to Fluid Mechanics

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Library of Congress Cataloging in Publication Data Chorin, Alexandre A Mathematical Introduction to Fluid Mechanics, Third Edition

(Texts in Applied Mathematics) Bibliography: in frontmatter Includes. 1. Fluid dynamics (Mathematics) 2. Dynamics (Mathematics) I. Marsden, Jerrold E. II. Title. III. Series. ISBN 0-387 97300-1 American Mathematics Society (MOS) Subject Classiﬁcation (1980): 76-01, 76C05, 76D05, 76N05, 76N15 Copyright 1992 by Springer-Verlag Publishing Company, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher, Springer-Verlag Publishing Company, Inc., 175 Fifth Avenue, New York, N.Y. 10010. Typesetting and illustrations prepared by June Meyermann, Gregory Kubota, and Wendy McKay The cover illustration shows a computer simulation of a shock diﬀraction by a pair of cylinders, by John Bell, Phillip Colella, William Crutchﬁeld, Richard Pember, and Michael Welcome.

The corrected fourth printing, April 2000.

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Series Preface Page (to be inserted)

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This is page vii Printer: Opaque this

Preface

This book is based on a one-term course in ﬂuid mechanics originally taught in the Department of Mathematics of the University of California, Berkeley, during the spring of 1978. The goal of the course was not to provide an…...

... It is generally known that water boils at 100°C, but really that is only true if the water is at a pressure of If the water pressure increases so too does its boiling point, and vice versa. The relationship between pressure and temperature of boiling water can be plotted on a so-called transition or phase diagram (Figure 1). In this lab, the boiling of water will be studied and plotted on a transition diagram. Liquid 9 s \ 1- Solid A J Vapour / i! ! 0.01 Temperature Figure 1: Phase Diagram of Water TM PDF Editor Lab Worksheet Apparatus The apparatus comprises a boiler and a heating element (Figure 2). The boiler (and the fluid it contains) is separated from the surroundings by isolating valves. Pressure Saturated vapour \ valve T1 Filling point Isolating valve Viewing port Boiler Heating elements Figure 2 TH3 Saturation Pressure Apparatus In setting up the laboratory, the water is boiled with the isolating valves open. Vapour leaves the system, pushing out any atmospheric gases (oxygen and nitrogen). During your experimental work, you will heat the vessel and then allow it to cool, monitoring the temperature and pressure as you do so. Measurement Devices Temperature measurement Platinum resistance thermometers are......

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...Fluid Mechanics Learning Objectives Outcomes • Explain the pressure-depth relationship. Pressure increases with depth. • Define Pascal’s Principle. Pascal's Principle states that the pressure is transmitted evenly through a liquid. • Describe how to use Pascal’s Principle in practical application. When you inflate a balloon with air, it expands evenly in all directions, this is an example. • Describe Archimedes Principle. States that the mass of a liquid displaced by a floating body is equal to the mass of that body. • Determine if an object will float in a fluid based on its relative densities. So if you fill a tumbler up with water to the brim, put an object into it, weigh the water that has been pushed out of the tumbler, and compare that with the weight of the object, you'll know whether it floats or not. • Use the continuity equation and Bernoulli’s equation to explain common effects of ideal fluid flow. The pressure in a fluid moving steadily without friction or outside energy input decreases when the fluid velocity increases Assignment Requirements 3. Mass is the same, so if the whale is taking up less volume, the density must have increased. The whale has displaced a greater mass of water at the depth, so the buoyant force is greater. 20. Ice cubes float in water, and sink in alcohol. Anything with less density than the liquid that it's in will float. 22. It will increase 35. It would be harder on the top of a mountain because the pressure of the......

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...UNIVERSITI TUNKU ABDUL RAHMAN |Centre |: Centre for Foundation Studies (CFS) |Unit Code |: FHSC1014 | |Course |: Foundation in Science |Unit Title |: Mechanics | |Year/ Trimester |: Year 1 / Trimester 1 | | | |Session |: 201401 | | | Tutorial 4: Application of Newton’s Laws. 1. The distance between two telephone poles is 50.0 m. When a 1.00 kg bird lands on the telephone wire midway between the poles, the wire sags 0.200 m. Draw a free-body diagram of the bird. How much tension does the bird produce in the wire? Ignore the weight of the wire. [614 N] 2. A 40 kg crate rests on a horizontal floor, and a 75 kg person is standing on the crate. Determine the magnitude of the normal force that (a) the floor exerts on the crate and (b) the crate exerts on the person. [(a) 1.13 x 103 N, (b) 735 N] 3. A worker stands still on a roof sloped at an angle of 45° above the horizontal. He is prevented from slipping by a static frictional force of 450 N. Find the mass of the worker. [85 kg] 4. A 4.0-kg bucket of water is raised from a well by a rope. If the upward acceleration of the......

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... Plumb line 5. Steel ball 6. Wooden board 7. Carbon paper 8. Meter rule 9. Plasticine Setup: 2. A ramp has been set up at the edge of a bench as shown in the Figure 4-1. 3. Suspend a plum-line from the edge of the bench as shown in Figure 4-2. 4. Mount a wooden board horizontally using two clamps so that the board is situated about the bottom of the ramp. 5. Place a sheet of blank paper on top of the board. 6. Place a piece of carbon paper on the top of the blank paper. The ink-side of the carbon paper should be facing down. 7. When a ball is released at the top of the ramp, the ball will travel through a trajectory as shown in Figure 4-2. Theory: Figure 4-1 Figure 4-2 UTAR FHSC1014 Mechanics Semester 1 14 Let: g = 9.8 ms-2 u = speed of the ball as it leaves the ramp k = constant y = vertical distance (between the bottom of the ramp and the top of the board) x = horizontal distance (between the plum-line and mark on the paper] The equation which relates x and y is k u g k x x y + + = 2 ² 1( ²) Procedure: 1. Position the ball at the top of the ramp. Release the ball so that it rolls down the ramp and onto the board below. 2. Remove the carbon paper and observe that the ball makes a small mark on the blank paper. 3. Measure and record the vertical distance y and the horizontal distance x. 4. Reduce the value of y and repeat the steps above to obtain eight (8) sets of...

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...UNIVERSITI TUNKU ABDUL RAHMAN CentreCourseYear/ TrimesterSession | : Centre for Foundation Studies (CFS): Foundation in Science: Year 1 / Trimester 1: 201405 | Unit CodeUnit Title | : FHSC1014: Mechanics | Tutorial 1: Introduction. 1. How many significant figures do each of the following numbers have: (a) 214, (b) 81.60, (c) 7.03, (d) 0.03, (e) 0.0086, (f) 3236, and (g) 8700? 2. The diameter of the earth is about 1.27 x 107 m. Find its diameter in (a) Millimeters, (b) Megameters, (c) Miles 3. Express the following using the prefixes: (a) 1×106 volts, (b) 2×106 meters, (c) 6×103 days, (d) 18×102 bucks, and (e) 8×109 pieces. 4. The speed limit on an interstate highway is posted at 75 mi/h. (a) What is this speed in kilometers per hour? (b) In feet per second? (c) In meter per second? [(a) 121 km/h, (b) 110 ft/s, (c) 33.5 m/s] 5. Five length have been measured and recorded as follows: L1 = 3.427m L2 = 3.5m L3 = 0.333m L4 = 32.000m (a) What is the uncertainty is there in each measurement? (b) What is the result if L1 + L2? [6.9 m] (c) What is the result if L1 + L3? [3.760 m] (d) What is the result if L1 – L3? [3.094 m] (e) What is the result if L2 – L1? [0.1 m] (f) What is the result if L1L2? [12 m2] (g) What is the result if L4L3? [96.1] 6. What is the volume (with the......

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...FHSC1014 MECHANICS FULL REPORT TITLE: TO INVESTIGATE THE TRAJECTORY OF A SMALL BALL AS IT ROOLS OFF A SURFACE WHICH IS INCLINED TO THE HORIZONTAL NAME: SAM WAI YEE STUDENT ID: 1403613 PRACTICAL GROUP: P1 LECTURER’S NAME: MR.ZOHEIR KHOEINIHA Title : To investigate the trajectory of a small ball as it rolls off a surface which is inclined to the horizontal Objective: To investigate the trajectory of a two dimensional motion. Apparatus and Materials: 1. Ramp 2. Wooden block 3. Pendulum bob 4. Plumb line 5. Steel ball 6. Wooden board 7. Carbon paper 8. Meter rule 9. Plasticine Setup: 1. A ramp has been set up at the edge of a bench as shown in the Figure 3.1. 2. Suspend a plum-line from the edge of the bench as shown in Figure 3.2. 3. A wooden board is mounted horizontally using two clamps so that the board is situated about the bottom of the ramp. 4. A sheet of blank paper is placed on top of the board. 5. A piece of carbon paper is placed on the top of the blank paper. The ink-side of the carbon paper should be facing down. 6. When a ball is released at the top of the ramp, the ball will travel through a trajectory as shown in Figure 3.2. Theory: Let: g = 9.80 msˉ² u = speed of the ball as it leaves the ramp k = constant y = vertical distance (between the bottom of the ramp and the top of the board) x = horizontal distance (between the plum-line and mark on the paper) The equation which relates x and y is ...

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...Variation of Pressure vertically in a fluid under gravity P_2 A P_1 Force due to P_1 on area A acting up = P_1 A Force due P_2 on area A acting down = P_2 A Force due to the weight of the element =mg= ρA(z_2- z_1 )g Since the fluid is at rest, there can be no shear forces and hence no vertical forces on the side of element due to surrounding fluid. Considering upward as positive, sum of all forces = 0, We have P_1 A-P_2 A - ρA(z_2- z_1 )g=0 Or P_2- P_1= - ρ(z_2- z_1 )g So, under the influence of gravity , pressure decreases with the increase height. Equality of Pressure at the same level in a static fluid Since the fluid is at rest, there are no horizontal shear stresses on the sides of the element. For static equilibrium the sum of horizontal forces must be zero, P_1 A=P_2 A P_1=P_2 Therefore the pressure at any two points at the same level in a body of fluid at rest will be the same If the (x,y) is the horizontal plane then ∂p/∂x=0 and ∂p/∂y=0 Variation of pressure due to gravity from point to point in a static fluid Since the element is in equilibrium, the resultant of the forces in any direction should be zero Resolving forces in the axial direction, pA-(p+δp)A-ρgAδs cosθ=0 δp=-ρgδs cosθ Putting above equation in differential form, dp/ds=-ρgcos......

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...Nurs 2820 Fluid, Electrolyte, and Acid-Base Balance Name: _______________________________________ Case Study Jimmy Lewis is brought to the hospital emergency room by some friends. He had been vomiting for several days and was complaining of heart palpitations. Mr. Lewis is a 58-year-old white male who is homeless. He has not had any health care for at least 10 years. He is an alcoholic and drinks a quart of gin or vodka every day. He does not have a job, and his family is all out of state. The emergency physician does an initial assessment and transfers him to a hospitalist, who admits him to a medical-surgical unit for further evaluation and treatment. Mr. Lewis has lab work drawn. His electrolytes are as follows: sodium 138 mEq/L, potassium 3.1 mEq/L (low), chloride 104 mEq/L, and magnesium 1.5 mEq/L (low). His arterial blood gas measurements are as follows: pH 7.48 (high), PaCO2 40 mm Hg, HCO3 29 (high). Jamie Taylor, a 22-year-old nursing student, is assigned to Mr. Lewis. She reviews Mr. Lewis’ medical record before going in to assess him. 1. After reviewing his chart and lab work, what fluid and electrolyte imbalances would Jamie determine? (Select all that Apply) A. Fluid volume deficit B. Hypokalemia C. Hypermagnesemia D. Hyperkalemia E. Hypomagnesemia 2. What acid-base imbalance is Mr. Lewis experiencing? A. Metabolic acidosis B. Respiratory acidosis C. Metabolic alkalosis D. Respiratory alkalosis 3. The hospitalist orders an IV of D5NS......

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...UNIVERSITI TUNKU ABDUL RAHMAN CentreCourseYear/ Trimester | : Centre for Foundation Studies (CFS): Foundation in Science: Year 1 / Trimester 1 | Session Course CodeCourse Title | : 201605: FHSC1014: Mechanics | Tutorial 1: Introduction 1. What are the dimensions for the physical quantity expressed as , where m, g, d, T and l are the mass, acceleration due to gravity, distance, period and length respectively. A MT2 B ML2 C MLT2 D MLT–2 [Answer: B] 2. Suppose A = BC, where A has the dimension LM–1 and C has the dimension LT–1. What is the dimension of B? A TM–1 B L2T–1M–1 C TML–2 D L2TM–1 [Answer: A] 3. (a) State the dimension for (i) velocity (ii) force (b) Suppose an equation related with force F, viscosity ƞ, speed v, radius r is given by this formula , where k is a dimensionless constant; x, y, and z are the power to which ƞ, v and r. Given that the dimensions of viscosity ƞ is ML–1T–1. Use dimensional analysis to determine the value of x, y and z. [Answer: (a) (i) LT–1, (ii) MLT–2; (b) 1, 1, 1] 4. (a) Discuss briefly whether in general the method of dimensional checking can positively confirm that an equation is correct. (b) Newton’s law of universal gravitation is represented by where F is the gravitational force, M and m are masses, and r is a length. Given that the dimensions of force is MLT–2. What are the dimensions and SI unit of the proportionality...

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...UNIVERSITI TUNKU ABDUL RAHMAN |Centre |: Centre for Foundation Studies (CFS) |Unit Code |: FHSC1014 | |Course |: Foundation in Science |Unit Title |: Mechanics | |Year/ Trimester |: Year 1 / Trimester 1 |Lecturer | | |Session |: 201605 | | | Additional Tutorial 2: Vector and translational kinematics. 1. Find the x and y-components of: (a) a displacement of 200 km, at 30.0o. (b) a velocity of 40.0 km/h, at 120o; and (c) a force of 50.0 N at 330o. [(a) 173 km, 100 km, (b) -20.0 km/h, +34.6 km/h, (c) 43.3 N, -25.0 N] 2. Three forces are applied to an object, as indicated in the drawing. Force [pic] has a magnitude of 21.0 Newton (21.0 N) and is directed 30.0° to the left of the + y axis. Force [pic] has a magnitude of 15.0 N and points along the + x axis. What must be the magnitude and direction (specified by the angle ( in the drawing) of the third force [pic] such that the vector sum of the three forces is 0 N? [18.7 N, 76o] 3. A 200 N block rests on a 30o inclined plane as shown in figure below. If the weight of the block acts vertically downward, what are the components of......

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...FLUID DYNAMICS In physics, fluid dynamics is a subdiscipline of fluid mechanics that deals with fluid flow—the natural science of fluids (liquids and gases) in motion. Fluid dynamics is "the branch of applied science that is concerned with the movement of liquids and gases," according to the American Heritage Dictionary. Fluid dynamics is one of two branches of fluid mechanics, which is the study of fluids and how forces affect them. (The other branch is fluid statics, which deals with fluids at rest.) Scientists across several fields study fluid dynamics. Fluid dynamics provides methods for studying the evolution of stars, ocean currents, weather patterns, plate tectonics and even blood circulation. Some important technological applications of fluid dynamics include rocket engines, wind turbines, oil pipelines and air conditioning systems. FLOW The movement of liquids and gases is generally referred to as "flow," a concept that describes how fluids behave and how they interact with their surrounding environment — for example, water moving through a channel or pipe, or over a surface. Flow can be either steady or unsteady. In his lecture notes, "Lectures in Elementary Fluid Dynamics" (University of Kentucky, 2009) J. M. McDonough, a professor of engineering at the University of Kentucky, writes, "If all properties of a flow are independent of time, then the flow is steady; otherwise, it is unsteady." That is, steady flows do not change over time. An example......

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...shall begin by studying incompressible flow problems. Of course all fluids are, to some extent, compressible but under steady flow conditions we may assume that the effects of changes in fluid density are small. In fact, it is the velocity of the fluid that dictates whether changes in density are significant and must be accounted for. In Chapter 6 we shall quantify the velocity limit, below which may assume that the fluid is incompressible; however, the majority of fluid flow problems that you are likely to encounter may be assumed to be incompressible. We shall focus in this chapter on incompressible flow, and on problems in which the fluid is bounded by a surface (we shall call this internal flow); the next chapter will focus on unbounded (or external) fluid flow problems. Both chapters will study real fluid flows and do this by taking into account the effects of viscosity. To do this we must examine how fluids interact with boundaries and here the concept of zero fluid velocity on a surface (boundary) is important. Once we have an understanding of how real fluid flows behave – and see how difficult it is to analyses turbulent flows –then in Chapters 4 and 5 we shall turn our attention to modelling techniques useful for examining simple fluid flow problems commonly found in engineering. Accordingly, this chapter will look at laminar and turbulent bounded fluid flows. We shall focus on pipe flow as this represents a classic......

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...Fundamentals of Fluid Mechanics Fluid mechanics is the study of fluids and the forces on them. (Fluids include liquids, gases, and plasmas.) Fluid mechanics can be divided into fluid kinematics, the study of fluid motion, and fluid dynamics, the study of the effect of forces on fluid motion, which can further be divided into fluid statics, the study of fluids at rest, and fluid kinetics, the study of fluids in motion. Fluid mechanics is very important to engineers when observing flow in pipes, viscous effects of fluids, and the forces that act on a fluid. As a student, I am suppose to demonstrate an adequate understanding of many properties involved fluid mechanics. Some learning outcomes that must be accomplished by taking this class are: * Demonstrate understanding of fluid mechanics fundamentals, fluid and flow properties such as compressibility, viscosity, buoyancy, hydrostatic pressure and forces on surfaces * Apply Bernoulli equation to solve problems in fluid mechanics * Solve fluid mechanics problem using control volume analysis using conservation of mass, energy equation and irreversible flow * Use differential analysis of fluid flow, potential flow theory, viscous flow, Navier Stokes equations to solve problems * Perform modeling and similitude using Buckingham Pi theorem, correlation of experimental data. * Analyze flow in pipes to determine laminar and turbulent flow behaviors. * Apply energy and momentum equations to......

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...CVEN 3100: Fluid Mechanics Fluid Properties: Review Questions 1. What is the definition of a fluid? A substance that deforms continuously when acted on by a shearing stress of any magnitude. 2. Normal force per unit area in a fluid is called what? Pressure 3. True or False: - Static fluids are not subjected to shear force at any time. T 4. True or False: - Normal forces can occur in a fluid whether it is static or Moving T 5. What is the relation between absolute pressure and gage pressure? Absolute pressure can be found from the gage pressure by adding the value of the atmospheric pressure. 6. What formula is used to calculate density of gases? Identify the parameters in the formula p/RT 7. Define specific weight. What is relation to density? Weight per unit volume. Multiply by gravity 8. Because of viscosity, what happens when a fluid tries to flow? It resists and does not flow quick 9. What is kinematic viscosity? The ratio of absolute viscosity to density 10. State the Newton’s law of viscosity and express it mathematically. Change in velocity over distance which velocity changes. Du/dy 11. What is the purpose of lubricating metal hinges? 12. Why does viscosity of liquids decrease with temperature? Molecules are spread further apart 13. Why does viscosity of gases increase with temperature? Molecular activity increases 14. What is an ideal fluid? A fluid that......

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...the earth is roughly 7000˚C. The two reasons why the earth is hot is when the earth was formed, the interior was heated rapidly due to the gravitational forces being converted into heat and radioactive isotopes within the earth liberate heat as they continue to decay. Our ability to drill into the earth is restricted to the upper few kilometers of the earth’s crust. We must look for a location where the earth’s interior heat is brought within our reach. This is most common at plate boundaries. Direct heating systems are designed to supply hot water only with no electricity generation. Borehole drilled to depth of 1800 meters beneath the city of Southampton, UK. Near the bottom of the hole is water at 70˚C. The fluid contains dissolved salts. The fluid is more accurately described as Brine. The Brine is pressurized and so rises unaided to within 100 m of the surface. A turbine pumps the Brine up to the surface through a heat exchanger that transfers heat to clean water. There are four main types of installations, Dry Steam Power Plant, Single Flash Steam Power Plant, Binary Cycle Power Plant, and Double Flash Power Plant....

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