Mechanics of Fluids

Recommendations

Info

is given by ∂p ∂x =−ϱax Equilibrium of moving fluids 81 (2.23) Similarly, the net force in the (vertical) z direction due to pressure is given by − ∂p ∂z δxδyδz The weight of the element acting vertically downwards is ϱgδxδyδz, so − ∂p δxδyδz − ϱgδxδyδz = ϱδxδyδzaz ∂z ∴ ∂p =−ϱ(g + az) ∂z (2.24) In general there would also be a component of acceleration in the y direction and a corresponding pressure gradient in the y direction ∂p/∂y =−ϱay. For simplicity, however, we shall consider the total acceleration to be in the x–z plane. From eqns 2.23 and 2.24 the pressure variation throughout the fluid may be determined. A surface of constant pressure in the fluid is one along which that is, along which dp = ∂p ∂p dx + dz = 0 ∂x ∂z dz dx =−∂p/∂x −ax = ∂p/∂z g + az (2.25) For constant acceleration, therefore, dz/dx is constant and a surface of constant pressure has a constant slope relative to the x direction of −ax/(g +az). One such surface is a free surface; other constant pressure planes are parallel to it. For example, consider the tank illustrated in Fig. 2.28. It contains a liquid and is given a uniform horizontal acceleration ax (the vertical acceleration az is zero). Once the liquid has adjusted itself to uniform conditions the free surface settles at a slope as shown. (During the period when the liquid is moving into its new position shear forces are involved, so the methods of hydrostatics do not then apply.) Here tan θ = dz/dx =−ax/g. If, however, the acceleration is only in the vertical direction eqn 2.25 shows that dz/dx = 0, so planes of constant pressure are horizontal. The variation of pressure through the fluid is then given simply by eqn 2.24. A tank of fluid allowed to fall freely would have an acceleration in the z direction (upwards) of −g and thus uniform pressure would be obtained throughout the fluid. But if a tank of liquid were accelerated upwards the hydrostatic pressure variation would be intensified. Fluid completely filling a closed tank would have no free surface, but planes of constant pressure would still be inclined to the x direction at an
82 Fluid statics Fig. 2.28 Fig. 2.29 angle of arctan {−ax/(g +az)}. Pressures at particular points in the fluid may be determined by integrating eqns 2.23 and 2.24: � p = dp = � � ∂p ∂p dx + ∂x ∂z dz =−ϱaxx − ϱ(g + az)z + constant (2.26) for a constant-density fluid. The integration constant is determined by the conditions of the problem – for example, that p = patm at a free surface. All the foregoing results refer only to a horizontal x- (and y-) axis and a vertical z-axis; it should be remembered too that z is measured upwards from a suitable horizontal datum. Once the direction of the constant-pressure planes is known, alternative expressions may be obtained by considering, say, ξ and η axes, respectively parallel to and perpendicular to the constant-pressure planes (see Fig. 2.29).
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 44 and 45: z direction and therefore the same
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 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 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
Page 144 and 145:
of the base of the notch. Assume th
Page 146 and 147:
Our aim now is to derive a relation
Page 148 and 149:
The velocity, in general, varies fr
Page 150 and 151:
the control volume is � ϱ1u1ux1
Page 152 and 153:
Applications of the momentum equati
Page 154 and 155:
that the resultant force on the flu
Page 156 and 157:
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
Page 272 and 273:
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
Page 288 and 289:
7.8 PIPES IN COMBINATION 7.8.1 Pipe
Page 290 and 291:
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
Page 296 and 297:
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:
eBooks - at www.eBookstore.tandf.co
show all

Mechanics of Fluids

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?