Mechanics of Fluids

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Mach angle 498 intersection 510–1 reflection 510 Mach cone 497–8 Mach line 498 Mach number 21, 166, 496, 501, 518 Mach waves 508 Mach–Zehnder interferometer 550 Magnus effect 395 Manning’s formula 422–3 Manning’s roughness coefficient 422 Manometers 50–5 Manometric efficiency 629 head 629 Mass flow parameter 521 Mean density 12 Mean steady flow 37 Metacentre 72 Metacentric height 73 Metacentric radius 75 Michell bearing 224 Micro-manometers 54 Micropoise 26 Mild slope 435 Millibar 14 Minor losses 260 Mixed-flow machines 596 Mixed-flow pump 634, 645 Mixing length 343 Molecular structure 3 Moment of inertia 59 Momentum correction factor 138 Momentum equation 134–54 applications 138–54 boundary layer 303–6 Momentum integral equation, boundary layer 306 Momentum theory propeller 150–4 wind turbine 154 Momentum thickness of boundary layer 302 Moody diagram 252 Moving fluids, equilibrium of 80–3 Multi-stage pumps 639 Nappe 126–7 Navier–Stokes equations 354–6 numerical procedures for solving 356–8 Net Positive Suction Head (NPSH) 644 Neutral equilibrium 71 Newtonian fluid 24 Newton’s First Law 92 Newton’s Law of Universal Gravitation 12 Newton’s laws of motion 138 Newton’s Second Law 14, 95, 134 Newton’s Third Law 22, 136, 440 Nikuradse’s experiments 250–2 Nodes 478–9 Non-Newtonian liquids 26–7 laminar flow in circular pipe 196–8 Non-uniform flow 30 Non-uniform velocity distribution effects 100, 138, 632–4 Normal depth 419 Normal flow 419 Normal shock waves 500–5 Notches 126–30 rectangular 127 V 129 Nozzle, force at 144–8 Numeric 5 Oblique shock waves 505–12 intersection 510–2 reflection 510–2 One-dimensional flow 32 with negligible friction 522–4 Open channels 414 boundary layer in 423–4 occurrence of critical conditions 443–54 optimum cross-section 425–6 simple waves and surges in 427–31 specific energy and alternative depths of flow 431–7 steady-flow energy equation for 416–8 types of flow 415–6 Index 693 Orifice flow through sharp-edged 112–9, 268–9 quasi-steady flow through 119 submerged 118–9 Orifice meter 123–5 Oscillatory waves see Waves Oseen’s formula 331 Overturning couple 73 Parabolic velocity profile 193 Parallel axes theorem 59 Particle mechanics 332 Pascal (unit) 7, 9, 14 Pascal’s Law 15 Path-line 31 Pelton wheel 598–605 Percentage slip 596 Perfect gas 18, 489 Period of oscillation 77 Perpendicular axes theorem 59 Petroff’s law 233 Phase velocity 468 Physical constants 667 Physical similarity 159–70 see also specific types Piezo–electric gauges 56 Piezometer tube 45–6 Piezometric head 46 Piezometric pressure 46 Pipe bend force caused by flow round 141–4 head loss due to 266 Pipe fittings, losses in 267–8 Pipe networks 280–1 Pipe with side tappings 281–2 Pipes branched 278–80 in parallel 277–8 in series 277 Pi theorem 172 Pitometer 112 Pitot-static tube 110–2, 519 Pitot tube 110–2 in flow with variable density 517–20 Plasticity 27 Plastic solids 2 Poise 26 Poiseuille (unit) 25 Poiseuille’s equation 194
694 Index Polar diagram 337 Porosity 239, 270 Positive-displacement machines 591 Potential flow 368 Prandtl-Meyer angle 515 Prandtl-Meyer expansion 515 Prandtl-von Kármán theory 343 Pressure 13–6 absolute 13, 46, 49 centre of 61–3 gauge 13, 45, 49 measurement of 48–57 piezometric 46 variation with position in fluid 43–8 Pressure coefficient 168 Pressure diagrams 565 Pressure drag 324 Pressure forces 162, 168 Pressure gauges 55–7 Pressure gradient 320–2 adverse 321 favourable 320–1 Pressure head 46 Pressure line 271–5 Pressure losses 245–55, 259, 260–71 Pressure transducer 57 Pressure transients 558–80 Pressure variation perpendicular to streamlines 107–8 Pressure waves 560 magnitude 560–4 velocity 560–4 Principle of conservation of mass 90–2 Profile drag 324 three-dimensional bodies 331–5 two-dimensional bodies 329–31 Propeller, momentum theory 150–4 Propeller turbine 607 Pseudo-plastic liquids 27, 197 Pumps 591, 625–51 characteristic curves for 644–5 performance characteristics 644–6 selection 650–1 Quasi-steady flow through orifice 119 through pipes 284–6 Radial blades 629 Radial-flow machine 596 Radial-flow turbine 605 Rankine–Hugoniot relation 502 Rankine oval 389 Rapid flow 416 approaching weir 449–51 in open channel 435 Rapidly varied flow 456 Rate of shear 23 Rayleigh step 228–9 Reaction turbine 597 cavitation limits for 621 net head across 607–9 Reciprocating pump 592–6 Rectilinear flow 374, 400–1 Region of influence 497–8 Relative density 13 Relative equilibrium 80 Restoring couple 72 Reversible adiabatic process 503 Reversible process 488 Reynolds number 35 local 137 significance of 163–4 Reynolds stress 342–3 Rheology 28 Rheopectic liquids 27 Rigid-body rotation 381 Ripples 469 Robins effect 396 Rocket propulsion 146–8 Rotameter 303 Rotational flow 366, 381 Rotodynamic machines 592 basic equations 609–13 Rotodynamic pumps 625–51 Rotor 592 Rough zone of flow 251 Runner 592 Salt-dilution method 291 Salt-velocity method 290 Saturation pressure 16 Saturation vapour pressure of water 667 Scale effect 182 Scale factor 160 Schlieren method 549 Secondary flow, losses due to 266 Second Law of Thermodynamics 441 Second moment of area 58–9 Seiches 479 Semi-perfect gas 19 Separation 320–1 from aerofoil 335–7 position of 339 predicting in laminar boundary layer 322–3 Separation point 321 Separation streamline 321 Shadowgraph method 548 Shear rate 23 Shear stress 1, 22, 191 distribution in circular pipe 257–8 Ship resistance 182–8 Shock 499 Shock losses 631 Shock phenomena 340 Shock stall 546 Shock wave 499–511 definition 499–500 intersection of 510–12 normal 499–505 oblique 505–11 reflection of 510–12 Shooting flow 435 SI units 6 internationally agreed names 6–7 prefixes for multiples and submultiples 8 Similarity 160 chemical 162 dynamic 161–7 geometric 160 kinematic 160–1 of machines 639–40 physical 159–60 thermal 161 Similarity laws pumps 639–40 turbines 613–7 Single suction pump 627 Singular point 375 Sink 376 Siphon 272 Skin friction 307 Skin-friction coefficient 168, 307–8 Slant depth 61
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Mechanics of Fluids
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Mechanics of Fluids Eighth edition
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Contents Preface to the eighth edit
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8 Boundary Layers, Wakes and Other
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Preface to the eighth edition In th
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Fundamental concepts 1 The aim of C
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1.1.1 Molecular structure The diffe
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therefore, to have in place systems
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has been a strong movement in favou
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Notation, dimensions, units and rel
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is nearer 2 m, rather than 1 m. Her
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Properties of fluids 13 The relativ
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that is, p1 − p3 = 1 2BC ϱax (1.
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that there is an almost perfect vac
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calorically perfect. (Some writers
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As a liquid is compressed its molec
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or τ = µ ∂u ∂y (1.9) where µ
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shared among the occupants of bb, a
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the origin with slope equal to µ (
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The interplay of these various forc
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non-uniformity is always encountere
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z direction and therefore the same
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Although Reynolds used water in his
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cylinder. For a certain range of Re
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The roles of experimentation and th
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A flow model which assumes steady,
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2.1 INTRODUCTION Fluid statics 2 Fl
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3. dp/dz =−ϱg. (Since the pressu
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that is, dp p =−g dz R T Variatio
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atmosphere it is termed vacuum or s
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at its centreline be p. Then, provi
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possible, and a sloping manometer m
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The amount of liquid B on each side
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In a pressure transducer the pressu
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where the suffixes C and Oy indicat
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surface (where the pressure is atmo
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which has different depths of water
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Hydrostatic thrusts on submerged su
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plane and may then be combined into
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Since all the elemental thrusts are
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neglected when a body is weighed on
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couple acting on the body in its di
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centre of buoyancy on to the transv
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move, but also the centre of gravit
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that is, W(OG) = (W + F)(OB + BM) =
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is given by ∂p ∂x =−ϱax Equi
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Then ∂p/∂ξ = 0 by definition o
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2.5 Two small vessels are connected
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2.18 In the vertical end of an oil
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The principles governing fluids in
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planes B and C (Fig. 3.2), δs bein
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an actual fluid is thus often remar
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density are small. Then eqn 3.9 has
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General energy equation for steady
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− Energy loss to friction/time Ma
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a decrease of pressure (provided th
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in the neighbourhood of B results i
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it is difficult to measure the heig
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The diagram illustrates an orifice
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� depth h below the free surface
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Solution (a) We consider first the
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of the same liquid. A vena contract
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To ensure that the pressure measure
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Since �h = p1 − p2 ϱwg the flo
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For consistency with the mathematic
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showing the essential form of the r
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With the same assumptions used in d
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∴ Velocity of approach = 0.0660 m
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of the base of the notch. Assume th
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Our aim now is to derive a relation
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The velocity, in general, varies fr
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the control volume is � ϱ1u1ux1
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Applications of the momentum equati
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that the resultant force on the flu
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The existence of the reaction may b
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our reference axes attached to the
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of increase of x-momentum is � D
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This is a simplified picture of wha
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since u4 −u1 is the velocity of t
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(1) u 1 (2) (3) (4) Assume a sea le
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and frictional effects to be neglig
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Physical similarity and dimensional
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A well-known example of kinematic s
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polygon. Now a polygon can be compl
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The condition for dynamic similarit
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equality of the Mach numbers is not
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Consider the incompressible flow al
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moving in a straight line with cons
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of the analysis, and its correct im
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Note: a suffix has been attached to
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For example, the parameter Fϱ/µ 2
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The application of dynamic similari
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atmospheric air, the dynamic viscos
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gives rise to waves on the surface.
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To be tested, the model must theref
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it follows that Hence V2 V1 = P2 P1
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5.5 A torpedo-shaped object 900 mm
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Laminar flow between solid boundari
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the place of y: τ = µ ∂u ∂r A
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compensate for the reduction in vel
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The expression for the total discha
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etween the resisting viscous forces
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must be zero so as to meet the requ
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Example 6.2 Two stationary, paralle
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Thus for any value of y the velocit
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Steady laminar flow between moving
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in eqn 6.24 we obtain � D Vp 2
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and the passage of the liquid level
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this book, it is important to give
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6.6.4 Rotary viscometers A simple m
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The torque T on the cylinder of rad
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stationary when the outer cylinder
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Fundamentals of the theory of hydro
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part of the bearing then gives when
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e determined by our analysis.) From
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where a = 6µ�R h3 dh pc dx Funda
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Integrating this between θ = 0 and
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Rearrangement of eqn 6.71 and subst
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6.4 A cylindrical drum of length l
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7.1 INTRODUCTION Flow and losses in
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a maximum value of 2.0. The line CD
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America the friction factor commonl
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course, quite different from the ro
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Variation of friction factor 253 Fi
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straightforward manner without iter
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Distribution of shear stress in a c
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7.5 FRICTION IN NON-CIRCULAR CONDUI
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estimated by simple theory. The pip
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the junction, the curvature of the
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particular, the performance of a di
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A pipe bend thus causes a loss of h
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cross-sectional area available in t
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correlating experimental data on th
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to do this. But sub-atmospheric pre
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For systems comprising only short r
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7.8 PIPES IN COMBINATION 7.8.1 Pipe
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2. There can be only one value of h
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So For two pipes in parallel, and s
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7.9 CONDITIONS NEAR THE PIPE ENTRY
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where u denotes the mean velocity i
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Assuming that entry and exit losses
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of one-per-revolution electrical co
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of the piping system, this meter is
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losses, calculate (a) the total pow
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How are the running costs altered i
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main. The depth of water in the tan
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the features of boundary layer flow
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the surface other considerations ar
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and θ = = = � δ u 0 um � δ 0
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The momentum equation applied to th
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1 = y η δ The laminar boundary la
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Table 8.1 Comparison of results for
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and � � ∂f (η) B = ∂η A =
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From equation 8.17, the frictional
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The turbulent boundary layer on a s
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Hence F = 0.037ϱu 2 m l(Re l) −1
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From eqn 8.29 Hence and xt − x0 x
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from it. This breakaway before the
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this expression into eqn 8.31 we ge
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When, for example, a ship moves thr
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separate vortices in an inviscid fl
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8.8.4 Profile drag of two-dimension
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are negligible. With reduction in t
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Fig. 8.14 Drag coefficients of smoo
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sphere is falling through the atmos
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At high Reynolds numbers, the varia
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this does not prevent the continual
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Eddy viscosity and the mixing lengt
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Eddy viscosity and the mixing lengt
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Velocity distribution in turbulent
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then, although at very small values
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as η → 1 and (1 − η) is then
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When f −1/2 − 4 log 10 (d/k) is
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comparable with the velocity of sou
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The x, y and z components of the mo
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y y x i-1, j+1 i, j+1 i+1, j+1 i-1,
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e completely covered by a turbulent
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9.1 INTRODUCTION The flow of an inv
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Now consider another point P ′′
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number of smaller ones of which M a
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distortion, the element is thus not
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Any function φ that satisfies Lapl
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ecome perfect squares - except wher
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fluid, and the flow net indicates t
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9.6.2 Flow from a line source A sou
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higher orders of small magnitudes b
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Combining eqns 9.16 and 9.17 we obt
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9.6.5 Forced (rotational) vortex Th
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For a free vortex qR = constant = K
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source is unable to move to the lef
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Also we note that the source is pos
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as shown in Fig. 9.23, encloses all
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eqn 9.23 becomes ψ =−U � r −
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Thus the magnitude of the velocity
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At a stagnation point, qt = 0 and t
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function of the combined flow is gi
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or, more approximately, that betwee
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Then dw dz =−U =−u + iv Hence u
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An introduction to elementary aerof
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ound the aerofoil, its magnitude be
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and lower layers. Since the vortex
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where Ɣ0 represents the circulatio
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9.7 An open cylindrical vessel, hav
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steadily south-east at 6 m · s −
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10.2 TYPES OF FLOW IN OPEN CHANNELS
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The steady-flow energy equation for
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where the hs represent vertical dep
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to the turbulent rough flow regime
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selection of an appropriate value r
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10.6 OPTIMUM SHAPE OF CROSS-SECTION
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oughness coefficient n is independe
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thrust divided by width of the chan
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which the waves follow one another
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The conditions for the critical dep
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y downstream conditions. In these c
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may be obtained by reducing eqn 10.
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this depth is greater than the crit
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The dissipation of energy is by no
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friction, the steady-flow momentum
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Fig. 10.28 Notice that, for a chann
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calculated, and then H + u2 1 /2g m
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Fig. 10.32 If the depth of flow ove
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(Section 10.11.1). Rapid flow can n
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where 0 ≤ θ ≤ 90 ◦ and sin
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(a) The continuity equation is The
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oundaries. It can occur when the fl
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into eqn 10.39 we obtain whence dh
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Table 10.2 Gradually varied flow 46
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The slope of a given channel may, o
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direction. The effect of any bounda
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This momentum term, however, is pro
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With surface tension effects ignore
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are for waves of small amplitude) w
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when the wave was formed. The lengt
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At a particular instant the crests
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Hence h = 2.65λ 2π = 2.65 × 225
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As the ratio of these velocity comp
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water is impulsively displaced and
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is relative movement of one layer o
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the channel bed a drop of 150 mm in
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11.1 INTRODUCTION Compressible flow
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energy ÷ mass, so that the dimensi
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Energy equation with variable densi
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Solution (a) From the equation of s
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whence (c − u)δρ = (ρ + δρ)
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the sonic velocity as supersonic (M
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velocity into the undisturbed fluid
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whence p2 p1 = 1 + γ M2 1 1 + γ M
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of eqns 11.2 and 11.20 T0 T u2 = 1
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the temperature rises greatly (say
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Fig. 11.10 Oblique shock relations
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however, that for a given upstream
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The deflection through the first wa
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As δθ is very small, cos δθ →
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eqn 11.41. Substituting M 2 − 1 =
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oundaries, both of which are curved
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For supersonic flow eqn 11.45 is no
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Some general relations for one-dime
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of minimum cross-section the veloci
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downstream of the minimum cross-sec
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‘telegraphed’ upstream and a pr
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the external pressure. This compres
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Compressible flow in pipes of const
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Table 11.2 Changes with distance al
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From the Fanno Tables at pc/pB = 0.
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Therefore and M1 = u1 a 24.5 m · s
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drag. The abrupt pressure rise thro
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Analogy between compressible flow a
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is, on ∂2n ∂y2 + ∂2 � n ∂
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and the mass flow rate in the duct.
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11.18 Air flows isothermally at 15
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12.2 INERTIA PRESSURE Any volume of
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Equation 12.3 indicates that u beco
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would come to rest later. Although
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a pressure wave in a gas; here we r
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value in eqn 12.7 gives that is, wh
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the moment disregarded. An increase
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at this point remains constant for
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In addition to the reflection of wa
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is closed so that the area coeffici
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The pressure diagrams show that for
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12.3.4 The method of characteristic
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Pressure transients 577 Multiplying
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and velocities are known, then for
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For pipe 2, and hence K ′ = c2 =
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(instantaneously, we will assume) a
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The one-dimensional continuity rela
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to accommodate instantaneous comple
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12.2 A turbine which normally opera
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13.1 INTRODUCTION Fluid machines13
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flow. By way of illustration we sha
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would obviate the changes of pressu
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demand for electric power - for exa
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Turbines 599 Fig. 13.7 Runner of Pe
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determined, multiplication by the r
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Substituting for �vw from eqn 13.
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ut the larger its value the more bu
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ate of flow possible is achieved wh
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the turbine. For the high heads nor
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The integrals in eqns 13.4 and 13.5
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diagram of Fig. 13.16 we have Simil
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would be sufficient to describe the
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Table 13.1 Type of turbine Approxim
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∴ Hydraulic efficiency = Overall
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If either z (the height of the turb
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∴ Hydraulic efficiency = u1vw1/gH
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shows the general effect of change
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an unmixed blessing and, except for
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orbiting space-craft, for example,
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Losses are also possible at the inl
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Those particles next to the forward
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It should be noted that the values
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∴ (β1)A = arctan(va/u) = arctan(
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so much that the aerofoil stalls (a
Page 652 and 653:
Use these data to deduce the pump c
Page 654 and 655: Manometric efficiency = 0.75 = gH/u
Page 656 and 657: and CP = P/ϱω 3 D 5 in place of Q
Page 658 and 659: yields h 1 = h f = 4fl d ≈ (100Q)
Page 660 and 661: (iii) the power dissipated by pipe
Page 662 and 663: Large numbers of test data for pump
Page 664 and 665: educed by rounding the inlet edges
Page 666 and 667: further: the turbine part then remo
Page 668 and 669: for reaction turbines: 1 − η = 1
Page 670 and 671: 75 mm wide at outlet. The blades oc
Page 672 and 673: Losses at valves, etc. are estimate
Page 674 and 675: APPENDIX Units and conversion 1 fac
Page 676 and 677: Table A1.4 (contd.) Force 32.17 pdl
Page 678 and 679: APPENDIX Physical constants and 2 p
Page 680 and 681: Kinematic viscosity mm 2 ·s -1 App
Page 682 and 683: Range II: 11 000 m ≤ z ≤ 20 000
Page 684 and 685: Table A3.1 (contd.) M 1 M 2 p 2/p 1
Page 686 and 687: Table A3.2 Isentropic flow M p/po
Page 688 and 689: Table A3.3 Adiabatic flow with fric
Page 690 and 691: APPENDIX 4 Algebraic symbols Table
Page 692 and 693: Table A4.1 (contd.) Symbol Definiti
Page 694 and 695: Table A4.1 (contd.) Symbol Definiti
Page 696 and 697: Answers to problems 1.1 56.2 m 3 1.
Page 698 and 699: 8.14 0.002955, 2.866 m · s −1 ,
Page 700 and 701: Absolute pressure 46 Absolute visco
Page 702 and 703: Eccentricity 230 Eccentricity ratio
Page 706 and 707: Slip in fluid couplings 652 in reci
Page 708: eBooks - at www.eBookstore.tandf.co
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