Mechanics of Fluids

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z direction and therefore the same streamline pattern could at any instant be found in all planes in the fluid perpendicular to the z direction. The flow past a wing of uniform cross-section and infinite span, for instance, is the same in all planes perpendicular to its span. Water flow over a weir of uniform crosssection and infinite width is likewise two-dimensional. In practice, where the width is not infinite it may be satisfactory to assume two-dimensional flow over most of the wing or weir, that is, to consider flow in a single plane; the results may then be modified by end corrections accounting for the three-dimensional flow at the ends. Axi-symmetric flow, although not two-dimensional in the sense just defined, may be analysed more simply with the use of two cylindrical coordinates (x and r). 1.9 CLASSIFICATION AND DESCRIPTION OF FLUID FLOW It is helpful to the understanding of fluid mechanics to broadly classify certain different types of fluid flow, and here we introduce some of the terminology used. 1.9.1 Internal and external flows The distinction between internal and external flows often needs to be made. When the motion of a fluid is between bounding surfaces the flow is described as internal flow. Airflow management systems are widely used to control the quality of air within buildings and vehicles; the movement of air within the ducting which forms part of such a system is an example of an internal flow. Conversely, when a body is surrounded by a fluid in motion, the flow around the immersed body is described as external flow. Examples of external flows are the flows surrounding an aircraft wing, around an entire aircraft, around a road vehicle such as a car or lorry or around a building. 1.9.2 Laminar and turbulent flows From about 1840, it had been realized that the flow of a fluid could be of two different kinds. The distinction between them is most easily understood by reference to the work undertaken in the early 1880s by Osborne Reynolds (1842–1912), Professor of Engineering at Manchester University. The apparatus used by Reynolds was as shown in Fig. 1.11. A straight length of circular glass tube with a smoothly rounded, flared inlet was placed in a large glass-walled tank full of water. The other end of the tube passed through the end of the tank. Water from the tank could thus flow out along the tube at a rate controlled by a valve at the outlet end. A fine nozzle connected to a small reservoir of a liquid dye discharged a coloured filament into the inlet of the glass tube. By observing the behaviour of the stream of dye, Reynolds was able to study the way in which the water was flowing along the glass tube. Classification and description of fluid flow 33
34 Fundamental concepts Fig. 1.11 Fig. 1.12 If the velocity of the water remained low and especially if the water in the tank had previously been allowed to settle for some time so as to eliminate all disturbances as far as possible, the filament of dye would pass down the tube without mixing with the water, and often so steadily as almost to seem stationary (Fig. 1.12a). As the valve was opened further and the velocity of the water thereby increased, this type of flow would persist until the velocity reached a value at which the stream of dye began to waver (Fig. 1.12b). Further increase in the velocity of the water made the fluctuations in the stream of dye more intense, particularly towards the outlet end of tube, until a state was reached, quite suddenly, in which the dye mixed more or less completely with the water in the tube. Thus, except for a region near the inlet, the water in the tube became evenly coloured by the dye. Still further increases of velocity caused no more alteration in the type of flow, but the dye mixed even more readily with the water and complete mixing was achieved nearer the inlet of the tube. The original type of flow, in which the dye remained as a distinct streak, could be restored by reducing the velocity. It is of particular interest that the disturbed flow always began far from the inlet (in Reynold’s tests, usually at a length from the inlet equal to about 30 times the diameter of the tube); also that the complete mixing occurred suddenly.
Page 2 and 3: Mechanics of Fluids
Page 4 and 5: Mechanics of Fluids Eighth edition
Page 6 and 7: Contents Preface to the eighth edit
Page 8 and 9: 8 Boundary Layers, Wakes and Other
Page 10 and 11: Preface to the eighth edition In th
Page 12 and 13: Fundamental concepts 1 The aim of C
Page 14 and 15: 1.1.1 Molecular structure The diffe
Page 16 and 17: therefore, to have in place systems
Page 18 and 19: has been a strong movement in favou
Page 20 and 21: Notation, dimensions, units and rel
Page 22 and 23: is nearer 2 m, rather than 1 m. Her
Page 24 and 25: Properties of fluids 13 The relativ
Page 26 and 27: that is, p1 − p3 = 1 2BC ϱax (1.
Page 28 and 29: that there is an almost perfect vac
Page 30 and 31: calorically perfect. (Some writers
Page 32 and 33: As a liquid is compressed its molec
Page 34 and 35: or τ = µ ∂u ∂y (1.9) where µ
Page 36 and 37: shared among the occupants of bb, a
Page 38 and 39: the origin with slope equal to µ (
Page 40 and 41: The interplay of these various forc
Page 42 and 43: non-uniformity is always encountere
Page 46 and 47: Although Reynolds used water in his
Page 48 and 49: cylinder. For a certain range of Re
Page 50 and 51: The roles of experimentation and th
Page 52 and 53: A flow model which assumes steady,
Page 54 and 55: 2.1 INTRODUCTION Fluid statics 2 Fl
Page 56 and 57: 3. dp/dz =−ϱg. (Since the pressu
Page 58 and 59: that is, dp p =−g dz R T Variatio
Page 60 and 61: atmosphere it is termed vacuum or s
Page 62 and 63: at its centreline be p. Then, provi
Page 64 and 65: possible, and a sloping manometer m
Page 66 and 67: The amount of liquid B on each side
Page 68 and 69: In a pressure transducer the pressu
Page 70 and 71: where the suffixes C and Oy indicat
Page 72 and 73: surface (where the pressure is atmo
Page 74 and 75: which has different depths of water
Page 76 and 77: Hydrostatic thrusts on submerged su
Page 78 and 79: plane and may then be combined into
Page 80 and 81: Since all the elemental thrusts are
Page 82 and 83: neglected when a body is weighed on
Page 84 and 85: couple acting on the body in its di
Page 86 and 87: centre of buoyancy on to the transv
Page 88 and 89: move, but also the centre of gravit
Page 90 and 91: that is, W(OG) = (W + F)(OB + BM) =
Page 92 and 93: is given by ∂p ∂x =−ϱax Equi
Page 94 and 95:
Then ∂p/∂ξ = 0 by definition o
Page 96 and 97:
2.5 Two small vessels are connected
Page 98 and 99:
2.18 In the vertical end of an oil
Page 100 and 101:
The principles governing fluids in
Page 102 and 103:
planes B and C (Fig. 3.2), δs bein
Page 104 and 105:
an actual fluid is thus often remar
Page 106 and 107:
density are small. Then eqn 3.9 has
Page 108 and 109:
General energy equation for steady
Page 110 and 111:
Page 112 and 113:
Page 114 and 115:
Page 116 and 117:
− Energy loss to friction/time Ma
Page 118 and 119:
a decrease of pressure (provided th
Page 120 and 121:
in the neighbourhood of B results i
Page 122 and 123:
it is difficult to measure the heig
Page 124 and 125:
The diagram illustrates an orifice
Page 126 and 127:
� depth h below the free surface
Page 128 and 129:
Solution (a) We consider first the
Page 130 and 131:
of the same liquid. A vena contract
Page 132 and 133:
To ensure that the pressure measure
Page 134 and 135:
Since �h = p1 − p2 ϱwg the flo
Page 136 and 137:
For consistency with the mathematic
Page 138 and 139:
showing the essential form of the r
Page 140 and 141:
With the same assumptions used in d
Page 142 and 143:
∴ Velocity of approach = 0.0660 m
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of the base of the notch. Assume th
Page 146 and 147:
Our aim now is to derive a relation
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The velocity, in general, varies fr
Page 150 and 151:
the control volume is � ϱ1u1ux1
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Applications of the momentum equati
Page 154 and 155:
that the resultant force on the flu
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The existence of the reaction may b
Page 158 and 159:
our reference axes attached to the
Page 160 and 161:
of increase of x-momentum is � D
Page 162 and 163:
This is a simplified picture of wha
Page 164 and 165:
since u4 −u1 is the velocity of t
Page 166 and 167:
(1) u 1 (2) (3) (4) Assume a sea le
Page 168 and 169:
and frictional effects to be neglig
Page 170 and 171:
Physical similarity and dimensional
Page 172 and 173:
A well-known example of kinematic s
Page 174 and 175:
polygon. Now a polygon can be compl
Page 176 and 177:
The condition for dynamic similarit
Page 178 and 179:
equality of the Mach numbers is not
Page 180 and 181:
Consider the incompressible flow al
Page 182 and 183:
moving in a straight line with cons
Page 184 and 185:
of the analysis, and its correct im
Page 186 and 187:
Note: a suffix has been attached to
Page 188 and 189:
For example, the parameter Fϱ/µ 2
Page 190 and 191:
The application of dynamic similari
Page 192 and 193:
atmospheric air, the dynamic viscos
Page 194 and 195:
gives rise to waves on the surface.
Page 196 and 197:
To be tested, the model must theref
Page 198 and 199:
it follows that Hence V2 V1 = P2 P1
Page 200 and 201:
5.5 A torpedo-shaped object 900 mm
Page 202 and 203:
Laminar flow between solid boundari
Page 204 and 205:
the place of y: τ = µ ∂u ∂r A
Page 206 and 207:
compensate for the reduction in vel
Page 208 and 209:
The expression for the total discha
Page 210 and 211:
etween the resisting viscous forces
Page 212 and 213:
must be zero so as to meet the requ
Page 214 and 215:
Example 6.2 Two stationary, paralle
Page 216 and 217:
Thus for any value of y the velocit
Page 218 and 219:
Steady laminar flow between moving
Page 220 and 221:
in eqn 6.24 we obtain � D Vp 2
Page 222 and 223:
and the passage of the liquid level
Page 224 and 225:
this book, it is important to give
Page 226 and 227:
6.6.4 Rotary viscometers A simple m
Page 228 and 229:
The torque T on the cylinder of rad
Page 230 and 231:
stationary when the outer cylinder
Page 232 and 233:
Fundamentals of the theory of hydro
Page 234 and 235:
Page 236 and 237:
Page 238 and 239:
Page 240 and 241:
part of the bearing then gives when
Page 242 and 243:
e determined by our analysis.) From
Page 244 and 245:
Page 246 and 247:
Page 248 and 249:
where a = 6µ�R h3 dh pc dx Funda
Page 250 and 251:
Integrating this between θ = 0 and
Page 252 and 253:
Rearrangement of eqn 6.71 and subst
Page 254 and 255:
6.4 A cylindrical drum of length l
Page 256 and 257:
7.1 INTRODUCTION Flow and losses in
Page 258 and 259:
a maximum value of 2.0. The line CD
Page 260 and 261:
America the friction factor commonl
Page 262 and 263:
course, quite different from the ro
Page 264 and 265:
Variation of friction factor 253 Fi
Page 266 and 267:
straightforward manner without iter
Page 268 and 269:
Distribution of shear stress in a c
Page 270 and 271:
7.5 FRICTION IN NON-CIRCULAR CONDUI
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estimated by simple theory. The pip
Page 274 and 275:
the junction, the curvature of the
Page 276 and 277:
particular, the performance of a di
Page 278 and 279:
A pipe bend thus causes a loss of h
Page 280 and 281:
cross-sectional area available in t
Page 282 and 283:
correlating experimental data on th
Page 284 and 285:
to do this. But sub-atmospheric pre
Page 286 and 287:
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
Page 292 and 293:
So For two pipes in parallel, and s
Page 294 and 295:
7.9 CONDITIONS NEAR THE PIPE ENTRY
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where u denotes the mean velocity i
Page 298 and 299:
Assuming that entry and exit losses
Page 300 and 301:
of one-per-revolution electrical co
Page 302 and 303:
of the piping system, this meter is
Page 304 and 305:
losses, calculate (a) the total pow
Page 306 and 307:
How are the running costs altered i
Page 308 and 309:
main. The depth of water in the tan
Page 310 and 311:
the features of boundary layer flow
Page 312 and 313:
the surface other considerations ar
Page 314 and 315:
and θ = = = � δ u 0 um � δ 0
Page 316 and 317:
The momentum equation applied to th
Page 318 and 319:
1 = y η δ The laminar boundary la
Page 320 and 321:
Table 8.1 Comparison of results for
Page 322 and 323:
and � � ∂f (η) B = ∂η A =
Page 324 and 325:
From equation 8.17, the frictional
Page 326 and 327:
The turbulent boundary layer on a s
Page 328 and 329:
Hence F = 0.037ϱu 2 m l(Re l) −1
Page 330 and 331:
From eqn 8.29 Hence and xt − x0 x
Page 332 and 333:
from it. This breakaway before the
Page 334 and 335:
this expression into eqn 8.31 we ge
Page 336 and 337:
When, for example, a ship moves thr
Page 338 and 339:
separate vortices in an inviscid fl
Page 340 and 341:
8.8.4 Profile drag of two-dimension
Page 342 and 343:
are negligible. With reduction in t
Page 344 and 345:
Fig. 8.14 Drag coefficients of smoo
Page 346 and 347:
sphere is falling through the atmos
Page 348 and 349:
At high Reynolds numbers, the varia
Page 350 and 351:
this does not prevent the continual
Page 352 and 353:
Eddy viscosity and the mixing lengt
Page 354 and 355:
Eddy viscosity and the mixing lengt
Page 356 and 357:
Velocity distribution in turbulent
Page 358 and 359:
then, although at very small values
Page 360 and 361:
as η → 1 and (1 − η) is then
Page 362 and 363:
When f −1/2 − 4 log 10 (d/k) is
Page 364 and 365:
comparable with the velocity of sou
Page 366 and 367:
The x, y and z components of the mo
Page 368 and 369:
y y x i-1, j+1 i, j+1 i+1, j+1 i-1,
Page 370 and 371:
e completely covered by a turbulent
Page 372 and 373:
9.1 INTRODUCTION The flow of an inv
Page 374 and 375:
Now consider another point P ′′
Page 376 and 377:
number of smaller ones of which M a
Page 378 and 379:
distortion, the element is thus not
Page 380 and 381:
Any function φ that satisfies Lapl
Page 382 and 383:
ecome perfect squares - except wher
Page 384 and 385:
fluid, and the flow net indicates t
Page 386 and 387:
9.6.2 Flow from a line source A sou
Page 388 and 389:
higher orders of small magnitudes b
Page 390 and 391:
Combining eqns 9.16 and 9.17 we obt
Page 392 and 393:
9.6.5 Forced (rotational) vortex Th
Page 394 and 395:
For a free vortex qR = constant = K
Page 396 and 397:
source is unable to move to the lef
Page 398 and 399:
Also we note that the source is pos
Page 400 and 401:
as shown in Fig. 9.23, encloses all
Page 402 and 403:
eqn 9.23 becomes ψ =−U � r −
Page 404 and 405:
Thus the magnitude of the velocity
Page 406 and 407:
At a stagnation point, qt = 0 and t
Page 408 and 409:
function of the combined flow is gi
Page 410 and 411:
or, more approximately, that betwee
Page 412 and 413:
Then dw dz =−U =−u + iv Hence u
Page 414 and 415:
An introduction to elementary aerof
Page 416 and 417:
ound the aerofoil, its magnitude be
Page 418 and 419:
and lower layers. Since the vortex
Page 420 and 421:
where Ɣ0 represents the circulatio
Page 422 and 423:
9.7 An open cylindrical vessel, hav
Page 424 and 425:
steadily south-east at 6 m · s −
Page 426 and 427:
10.2 TYPES OF FLOW IN OPEN CHANNELS
Page 428 and 429:
The steady-flow energy equation for
Page 430 and 431:
where the hs represent vertical dep
Page 432 and 433:
to the turbulent rough flow regime
Page 434 and 435:
selection of an appropriate value r
Page 436 and 437:
10.6 OPTIMUM SHAPE OF CROSS-SECTION
Page 438 and 439:
oughness coefficient n is independe
Page 440 and 441:
thrust divided by width of the chan
Page 442 and 443:
which the waves follow one another
Page 444 and 445:
The conditions for the critical dep
Page 446 and 447:
y downstream conditions. In these c
Page 448 and 449:
may be obtained by reducing eqn 10.
Page 450 and 451:
this depth is greater than the crit
Page 452 and 453:
The dissipation of energy is by no
Page 454 and 455:
friction, the steady-flow momentum
Page 456 and 457:
Fig. 10.28 Notice that, for a chann
Page 458 and 459:
calculated, and then H + u2 1 /2g m
Page 460 and 461:
Fig. 10.32 If the depth of flow ove
Page 462 and 463:
(Section 10.11.1). Rapid flow can n
Page 464 and 465:
where 0 ≤ θ ≤ 90 ◦ and sin
Page 466 and 467:
(a) The continuity equation is The
Page 468 and 469:
oundaries. It can occur when the fl
Page 470 and 471:
into eqn 10.39 we obtain whence dh
Page 472 and 473:
Table 10.2 Gradually varied flow 46
Page 474 and 475:
The slope of a given channel may, o
Page 476 and 477:
direction. The effect of any bounda
Page 478 and 479:
This momentum term, however, is pro
Page 480 and 481:
With surface tension effects ignore
Page 482 and 483:
are for waves of small amplitude) w
Page 484 and 485:
when the wave was formed. The lengt
Page 486 and 487:
At a particular instant the crests
Page 488 and 489:
Hence h = 2.65λ 2π = 2.65 × 225
Page 490 and 491:
As the ratio of these velocity comp
Page 492 and 493:
water is impulsively displaced and
Page 494 and 495:
is relative movement of one layer o
Page 496 and 497:
the channel bed a drop of 150 mm in
Page 498 and 499:
11.1 INTRODUCTION Compressible flow
Page 500 and 501:
energy ÷ mass, so that the dimensi
Page 502 and 503:
Energy equation with variable densi
Page 504 and 505:
Solution (a) From the equation of s
Page 506 and 507:
whence (c − u)δρ = (ρ + δρ)
Page 508 and 509:
the sonic velocity as supersonic (M
Page 510 and 511:
velocity into the undisturbed fluid
Page 512 and 513:
whence p2 p1 = 1 + γ M2 1 1 + γ M
Page 514 and 515:
of eqns 11.2 and 11.20 T0 T u2 = 1
Page 516 and 517:
the temperature rises greatly (say
Page 518 and 519:
Fig. 11.10 Oblique shock relations
Page 520 and 521:
however, that for a given upstream
Page 522 and 523:
The deflection through the first wa
Page 524 and 525:
As δθ is very small, cos δθ →
Page 526 and 527:
eqn 11.41. Substituting M 2 − 1 =
Page 528 and 529:
oundaries, both of which are curved
Page 530 and 531:
For supersonic flow eqn 11.45 is no
Page 532 and 533:
Some general relations for one-dime
Page 534 and 535:
of minimum cross-section the veloci
Page 536 and 537:
downstream of the minimum cross-sec
Page 538 and 539:
‘telegraphed’ upstream and a pr
Page 540 and 541:
the external pressure. This compres
Page 542 and 543:
Compressible flow in pipes of const
Page 544 and 545:
Table 11.2 Changes with distance al
Page 546 and 547:
Page 548 and 549:
Page 550 and 551:
From the Fanno Tables at pc/pB = 0.
Page 552 and 553:
Page 554 and 555:
Therefore and M1 = u1 a 24.5 m · s
Page 556 and 557:
drag. The abrupt pressure rise thro
Page 558 and 559:
Analogy between compressible flow a
Page 560 and 561:
is, on ∂2n ∂y2 + ∂2 � n ∂
Page 562 and 563:
and the mass flow rate in the duct.
Page 564 and 565:
11.18 Air flows isothermally at 15
Page 566 and 567:
12.2 INERTIA PRESSURE Any volume of
Page 568 and 569:
Equation 12.3 indicates that u beco
Page 570 and 571:
would come to rest later. Although
Page 572 and 573:
a pressure wave in a gas; here we r
Page 574 and 575:
value in eqn 12.7 gives that is, wh
Page 576 and 577:
the moment disregarded. An increase
Page 578 and 579:
at this point remains constant for
Page 580 and 581:
In addition to the reflection of wa
Page 582 and 583:
is closed so that the area coeffici
Page 584 and 585:
The pressure diagrams show that for
Page 586 and 587:
12.3.4 The method of characteristic
Page 588 and 589:
Pressure transients 577 Multiplying
Page 590 and 591:
and velocities are known, then for
Page 592 and 593:
For pipe 2, and hence K ′ = c2 =
Page 594 and 595:
(instantaneously, we will assume) a
Page 596 and 597:
The one-dimensional continuity rela
Page 598 and 599:
to accommodate instantaneous comple
Page 600 and 601:
12.2 A turbine which normally opera
Page 602 and 603:
13.1 INTRODUCTION Fluid machines13
Page 604 and 605:
flow. By way of illustration we sha
Page 606 and 607:
would obviate the changes of pressu
Page 608 and 609:
demand for electric power - for exa
Page 610 and 611:
Turbines 599 Fig. 13.7 Runner of Pe
Page 612 and 613:
determined, multiplication by the r
Page 614 and 615:
Substituting for �vw from eqn 13.
Page 616 and 617:
ut the larger its value the more bu
Page 618 and 619:
ate of flow possible is achieved wh
Page 620 and 621:
the turbine. For the high heads nor
Page 622 and 623:
The integrals in eqns 13.4 and 13.5
Page 624 and 625:
diagram of Fig. 13.16 we have Simil
Page 626 and 627:
would be sufficient to describe the
Page 628 and 629:
Table 13.1 Type of turbine Approxim
Page 630 and 631:
∴ Hydraulic efficiency = Overall
Page 632 and 633:
If either z (the height of the turb
Page 634 and 635:
∴ Hydraulic efficiency = u1vw1/gH
Page 636 and 637:
shows the general effect of change
Page 638 and 639:
an unmixed blessing and, except for
Page 640 and 641:
orbiting space-craft, for example,
Page 642 and 643:
Losses are also possible at the inl
Page 644 and 645:
Those particles next to the forward
Page 646 and 647:
It should be noted that the values
Page 648 and 649:
∴ (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 700 and 701:
Absolute pressure 46 Absolute visco
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:
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