Mechanics of Fluids

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Absolute pressure 46 Absolute viscosity 24–6 Acceleration 81, 89 convective 90 of fluid particle 89–90 substantial 89 temporal 90 Acoustic velocity 493, 496 Actuator disc 151 Added mass 393 Adhesion 28 Adiabatic flow in pipe 531–7 Adiabatic frictionless conditions 522 Adiabatic process 19, 488 Adiabatic temperature lapse rate 79–80 Aerofoils 403–9 definitions 403 finite span 406–9 in high-speed flow 544–6 infinite span 404–6 separation 335–8 span of 403 vortex starting 405 Affinity laws for pumps 640 Air cavitation 620 Air locks 107 Airy waves 467 Alternative depths 432 Anemometer 288 Aneroid barometer 50 Angle of attack 403 Angle of heel 72 Angle of incidence 403 Angular velocity 9 Antinodes 478 Archimedes, Principle of 70 Area coefficient 580 Aspect ratio 403 Atmosphere equilibrium of 46–8, 79 stability of 79 (unit) 14 Atmospheric properties 670–1 Attitude angle (of bearing) 234 Avogadro’s hypothesis 18 Axial-flow machine 596 Axial-flow pumps 634–5 Axial-flow turbine 596, 607 Axi-symmetric flow 33 Backward difference 356 Backward-facing blades 629 Backwater curve 457, 462 Bar (unit) 9, 14 Barometer 49–50 Bearings inclined slipper 222–8 of infinite length 231 journal 230–9 very short 235 Bend-meter 290 Bends, losses in 266–8 Bernoulli constant 381 Bernoulli’s equation 92–6, 107, 391 applications 109–30 significance of terms in 95–6 Bingham plastic 197 Blade element theory 637 Blasius’s formula (friction in smooth pipes) 254 Blasius’s solution for laminar boundary layer 308–9 Bluff body 325 Boiling 16 Index Borda–Carnot head loss 262 Bore 428, 482 Boundary-element method (BEM) 356, 358 Boundary layer 298–352 control 338–9 definition 298 description of 299 displacement thickness 301 laminar 300, 306–9 momentum equation 303–6 momentum integral equation 306 momentum thickness 302 in open channels 423–4 transition region 299 see also Laminar boundary layer; Turbulent boundary layer Bourdon gauge 55–6 Boyle’s Law 490 Broad-crested weir 444–7 Bulk modulus of elasticity 20 Buoyancy 69–71 centre of 70, 72 Calorically perfect gas 18 Capillary depression, capillary rise 29 Capillary waves 183, 469 Cascade 267 Cauchy number 166 Cauchy–Riemann equations 400 Cavitation 16, 107, 619–22 in centrifugal pumps 643–4 damage 619–20
690 Index Cavitation limits for reaction turbines 621 Cavitation number 170, 622 Celerity 563–4 Centipoise 26 Centistokes 26 Central difference 356 Centred expansion 514–5 Centre of buoyancy 70 Centre of pressure 61 Centrifugal pumps 626–7 basic equations 627–32 diffuser-type 627 volute-type 627 Centroid 57, 59 Centroidal axis 57 Changes of state 19–20 Characteristic curve (of pump) 631, 646 Characteristic equations 577–8 Characteristics 578 method of 577–80 Chézy equation 419–23 Chézy’s coefficient 421, 459 Chézy’s formula 459 Choking 107, 525, 535, 541 Chord (of aerofoil) 403 Chord line 403 Circulation 364–7 Classical hydrodynamics 361 Closed conduits only partly full 426–7 Coanda effect 108–9 Coefficient of contraction 114, 116 Coefficient of discharge 114, 168–70 for orifice 117 for venturi-meter 120 Coefficient of friction 227 Coefficient of velocity 114 Coefficient of viscosity 23 Cohesion 28 Colebrook’s equation 351 Complex potential 400 Complex variables 399–402 Compressibility (quantity) 20 Compressibility effects aerofoils 544 drag 340–1 elastic forces 166 Compressibility factor 518 Compressible flow of gases 487–550 Compressible fluids 20, 487, 517 Compressor 591 Computational fluid dynamics (CFD) 353–8 Conformal transformation 404 Conjugate depths 440 conjugate functions 399 Conservation of energy 95, 96–101 Conservation of matter 90 Continuity 90–2 Continuity equation 354, 458, 576 Continuum 4 Contraction, loss at abrupt 262–4 Control volume 139, 419 Convection, free 79 Convective acceleration 90 Convergent-divergent nozzle 522, 524–9 Conversion factors 663–6 Corresponding velocity 180 Couette flow 205 Creeping motion 331 Critical depth 432, 437 Critical flow 416 in open channel 432–5, 443–7 Critical pressure ratio 524 Critical Reynolds number 247, 317 Critical slope 435, 462 Critical velocity in open channel 435 Current meters 288 d’Alembert’s Paradox 392 Darcy’s equation 248, 531 Darcy’s Law (flow through porous media) 239 Dashpot 207–9 Deflection angle 506, 508 de Laval nozzle 523–4 Density 12 at a point 12 Design pressure ratio 527 Deviation angle 633 Differential equations, of fluid dynamics 354–6 Diffuser 264–5 in centrifugal pump 626 Diffuser pump 627 Dilatancy 27 Dilatant liquids 197 Dimensional analysis 170–9 application 179–82 methods 172 process 172–3 Dimensional formulae 11–2, 679–83 Dimensional homogeneity 12 Dipole 390 Discharge 114, 123 measurement of 290–1 Discretization errors 356–7 Dispersive waves 468 Displacement thickness of boundary layer 301–2 Displacement work 95–6 Double suction machine 627 Doublet 390–1, 402 Downdrop curve 457 Downwash velocity 407 Draft tube 608 Drag 324–35 form 324 induced 408 normal pressure 324 profile 324 vortex 407–9 wave 340 Drag coefficient 325, 637 of bodies of revolution 341 effect of compressibility 544–6 of three-dimensional bodies 331–5 of two-dimensional bodies 329–31 Drag force 290, 314, 325 Drain-hole vortex 379 Drowned weir 448–9 Dynamic pressure 110 Dynamic similarly 161–7 application 179–82 flow with elastic forces acting 166–7 flow with gravity forces acting 164–5 flow with surface tension forces acting 165–6 flow with viscous forces acting 163–4 principal ratios 167 ratios of forces arising in 162–7 Dynamic viscosity 23–6
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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(
Page 650 and 651: 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 702 and 703: Eccentricity 230 Eccentricity ratio
Page 704 and 705: Mach angle 498 intersection 510-1 r
Page 706 and 707: Slip in fluid couplings 652 in reci
Page 708: eBooks - at www.eBookstore.tandf.co
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