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106 Chapter 3 ■ Elementary Fluid Dynamics—The Bernoulli Equation Stagnation streamline Stagnation point Stagnation point (a) F I G U R E 3.5 Stagnation points. (b) V3.7 Stagnation point flow Hence, the pressure at the stagnation point is greater than the static pressure, p 1 , by an amount 2 rV 12, the dynamic pressure. It can be shown that there is a stagnation point on any stationary body that is placed into a flowing fluid. Some of the fluid flows “over” and some “under” the object. The dividing line 1or surface for two-dimensional flows2 is termed the stagnation streamline and terminates at the stagnation point on the body. 1See the photograph at the beginning of Chapter 3.2 For symmetrical objects 1such as a baseball2 the stagnation point is clearly at the tip or front of the object as shown in Fig. 3.5a. For other flows such as a water jet against a car as shown in Fig. 3.5b, there is also a stagnation point on the car. If elevation effects are neglected, the stagnation pressure, p rV 2 2, is the largest pressure obtainable along a given streamline. It represents the conversion of all of the kinetic energy into a pressure rise. The sum of the static pressure, hydrostatic pressure, and dynamic pressure is termed the total pressure, p T . The Bernoulli equation is a statement that the total pressure remains constant along a streamline. That is, p 1 2rV 2 gz p T constant along a streamline (3.15) Again, we must be careful that the assumptions used in the derivation of this equation are appropriate for the flow being considered. F l u i d s i n t h e N e w s Pressurized eyes Our eyes need a certain amount of internal pressure in order to work properly, with the normal range being between 10 and 20 mm of mercury. The pressure is determined by a balance between the fluid entering and leaving the eye. If the pressure is above the normal level, damage may occur to the optic nerve where it leaves the eye, leading to a loss of the visual field termed glaucoma. Measurement of the pressure within the eye can be done by several different noninvasive types of instruments, all of which measure the slight deformation of the eyeball when a force is put on it. Some methods use a physical probe that makes contact with the front of the eye, applies a known force, and measures the deformation. One noncontact method uses a calibrated “puff” of air that is blown against the eye. The stagnation pressure resulting from the air blowing against the eyeball causes a slight deformation, the magnitude of which is correlated with the pressure within the eyeball. (See Problem 3.29.) Knowledge of the values of the static and stagnation pressures in a fluid implies that the fluid speed can be calculated. This is the principle on which the Pitot-static tube is based [H. de Pitot (1695–1771)]. As shown in Fig. 3.6, two concentric tubes are attached to two pressure gages 1or a differential gage2 so that the values of p 3 and p 4 1or the difference p 3 p 4 2 can be determined. The center tube measures the stagnation pressure at its open tip. If elevation changes are negligible, p 3 p 1 2rV 2
3.5 Static, Stagnation, Dynamic, and Total Pressure 107 (3) (4) V (1) p (2) F I G U R E 3.6 The Pitot-static tube. where p and V are the pressure and velocity of the fluid upstream of point 122. The outer tube is made with several small holes at an appropriate distance from the tip so that they measure the static pressure. If the effect of the elevation difference between 112 and 142 is negligible, then p 4 p 1 p Pitot-static tubes measure fluid velocity by converting velocity into pressure. By combining these two equations we see that which can be rearranged to give p 3 p 4 1 2rV 2 V 221 p 3 p 4 2r (3.16) The actual shape and size of Pitot-static tubes vary considerably. A typical Pitot-static probe used to determine aircraft airspeed is shown in Fig. 3.7. (See Fig. E3.6a also.) Four static pressure ports Heated outer case Stagnation pressure port Mounting flange Stagnation pressure fitting Heater leads (a) Static pressure fitting V3.8 Airspeed indicator (b) F I G U R E 3.7 Airplane Pitot-static probe. (a) Schematic, (b) Photograph, (Photograph courtesy of SpaceAge Control, Inc.)
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Achieve Positive Learning Outcomes
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WileyPLUS combines robust course ma
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Sixth Edition F undamentals of Flui
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About the Authors vii Bruce R. Muns
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P reface This book is intended for
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Preface xi Life Long Learning Probl
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Preface xiii WileyPLUS offers today
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Featured in this Book xv LEARNING O
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C ontents 1 INTRODUCTION 1 Learning
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Contents xix 6.4.3 Irrotational Flo
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12.6 Axial-Flow and Mixed-Flow Pump
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10 8 Jupiter red spot diameter 2 Ch
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4 Chapter 1 ■ Introduction and do
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6 Chapter 1 ■ Introduction up the
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8 Chapter 1 ■ Introduction the SI
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10 Chapter 1 ■ Introduction E XAM
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12 Chapter 1 ■ Introduction TABLE
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14 Chapter 1 ■ Introduction TABLE
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16 Chapter 1 ■ Introduction Crude
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18 Chapter 1 ■ Introduction Visco
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20 Chapter 1 ■ Introduction Kinem
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22 Chapter 1 ■ Introduction As th
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24 Chapter 1 ■ Introduction Boili
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26 Chapter 1 ■ Introduction E XAM
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28 Chapter 1 ■ Introduction The r
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30 Chapter 1 ■ Introduction fluid
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32 Chapter 1 ■ Introduction Secti
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34 Chapter 1 ■ Introduction avail
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36 Chapter 1 ■ Introduction compr
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2 Fluid Statics CHAPTER OPENING PHO
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40 Chapter 2 ■ Fluid Statics in a
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42 Chapter 2 ■ Fluid Statics p Fo
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44 Chapter 2 ■ Fluid Statics the
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46 Chapter 2 ■ Fluid Statics p 2
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48 Chapter 2 ■ Fluid Statics wher
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50 Chapter 2 ■ Fluid Statics F l
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52 Chapter 2 ■ Fluid Statics E XA
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54 Chapter 2 ■ Fluid Statics diff
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156 Chapter 4 ■ Fluid Kinematics
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188 Chapter 5 ■ Finite Control Vo
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264 Chapter 6 ■ Differential Anal
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) 266 Chapter 6 ■ Differential An
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7 Dimensional Analysis, Similitude,
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334 Chapter 7 ■ Dimensional Analy
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384 Chapter 8 ■ Viscous Flow in P
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WARNING Stand clear of Hazard areas
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462 Chapter 9 ■ Flow over Immerse
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∋ 530 Chapter 9 ■ Flow over Imm
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10 Open-Channel Flow CHAPTER OPENIN
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536 Chapter 10 ■ Open-Channel Flo
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646 Chapter 12 ■ Turbomachines Ex
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648 Chapter 12 ■ Turbomachines Q
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650 Chapter 12 ■ Turbomachines E
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652 Chapter 12 ■ Turbomachines Th
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656 Chapter 12 ■ Turbomachines Co
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702 Appendix A ■ Computational Fl
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A ppendix B Physical Properties of
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716 Appendix B ■ Physical Propert
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720 Appendix C ■ Properties of th
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722 Appendix D ■ Compressible Flo
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724 Appendix D ■ Compressible Flo
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Answers to Selected Even-Numbered H
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I ndex Absolute pressure, 13, 48 Ab
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Index I-3 guidelines for selection,
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Index I-5 hydrostatic: on curved su
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Index I-7 piezometer tube, 50, 105,
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Index I-9 Pressure force, 40 Pressu
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Index I-11 Stress field, 277 Strouh
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Index I-13 Weber, M., 29, 350 Weber
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V4.8 Jupiter red spot V4.9 Streamli
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V10.3 Water strider V10.13 Triangul
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■ TABLE 1.4 Conversion Factors fr
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■ TABLE 1.7 Approximate Physical
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