Basics of Fluid Mechanics, 2014a

More documents

Recommendations

Info

13.9. COUNTER–CURRENT FLOW 561 For this shear stress, the critical upward interface velocity is ( 2 3 − 1 2) {}}{ ( 1 ρL gh 2 ) U critical | interface = (13.71) 6 μ L The wall shear stress is the last thing that will be done on the liquid side. The wall shear stress is ⎛ ( dU τ L | @wall = μ L dx ∣ = μ L ⎜ ρ L g ⎝ ✟ ✟✯ 0 ) 2 x − h + x=0 μ L τ i { }} { 2 ghρ L 3 ⎞ 1 ⎟ μ L ⎠ x=0 (13.72) Simplifying equation (13.72) 12 becomes (notice the change of the sign accounting for the direction) τ L | @wall = ghρ L 3 (13.73) Again, the gas is assumed to be in a laminar flow as well. The shear stress on gas side is balanced by the pressure gradient in the y direction. The momentum balance on element in the gas side is dτ xyG dx = dP dy (13.74) The pressure gradient is a function of the gas compressibility. For simplicity, it is assumed that pressure gradient is linear. This assumption means or implies that the gas is incompressible flow. If the gas was compressible with an ideal gas equation of state then the pressure gradient is logarithmic. Here, for simplicity reasons, the linear equation is used. In reality the logarithmic equation should be used ( a discussion can be found in “Fundamentals of Compressible Flow” a Potto project book). Thus, equation (13.74) can be rewritten as dτ xyG dx = ΔP Δy = ΔP L (13.75) Where Δy = L is the entire length of the flow and ΔP is the pressure difference of the entire length. Utilizing the Newtonian relationship, the differential equation is d 2 U G dx 2 = ΔP μ G L (13.76) 12 Also noticing that equation (13.70) has to be equal ghρ L to support the weight of the liquid.
562 CHAPTER 13. MULTI–PHASE FLOW Equation (13.76) can be integrated twice to yield U G = ΔP μ G L x2 + C 1 x + C 2 (13.77) This velocity profile must satisfy zero velocity at the right wall. The velocity at the interface is the same as the liquid phase velocity or the shear stress are equal. Mathematically these boundary conditions are and U G (x = D) =0 (13.78) U G (x = h) =U L (x = h) (a) or (13.79) τ G (x = h) =τ L (x = h) (b) Applying B.C. (13.78) into equation (13.77) results in U G =0= ΔP μ G L D2 + C 1 D + C 2 (13.80) ↩→ C 2 = − ΔP μ G L D2 + C 1 D Which leads to U G = ΔP μ G L ( x 2 − D 2) + C 1 (x − D) (13.81) At the other boundary condition, equation (13.79)(a), becomes ρ L gh 2 = ΔP ( h 2 − D 2) + C 1 (h − D) (13.82) 6 μ L μ G L The last integration constant, C 1 can be evaluated as C 1 = ρ L gh 2 ΔP (h + D) − 6 μ L (h − D) μ G L (13.83) With the integration constants evaluated, the gas velocity profile is U G = ΔP μ G L ( x 2 − D 2) + ρ L gh 2 (x − D) 6 μ L (h − D) − ΔP (h + D)(x − D) μ G L (13.84) The velocity in Equation (13.84) is equal to the velocity equation (13.64) when (x = h). However, in that case, it is easy to show that the gas shear stress is not equal to the liquid shear stress at the interface (when the velocities are assumed to be the equal). The difference in shear stresses at the interface due to this assumption, of the equal velocities, cause this assumption to be not physical.
Page 2 and 3:
Basics of Fluid Mechanics Genick Ba
Page 4 and 5:
iii ‘We are like dwarfs sitting o
Page 6 and 7:
CONTENTS Nomenclature xxiii GNU Fre
Page 8 and 9:
CONTENTS vii 4 Fluids Statics 69 4.
Page 10 and 11:
CONTENTS ix 9.2.1 Construction of t
Page 12 and 13:
CONTENTS xi 12.5 The Working Equati
Page 14 and 15:
LIST OF FIGURES 1.1 Diagram to expl
Page 16 and 17:
LIST OF FIGURES xv 4.32 Polynomial
Page 18 and 19:
LIST OF FIGURES xvii (a) Streamline
Page 20 and 21:
LIST OF FIGURES xix 12.22The angles
Page 22 and 23:
LIST OF TABLES 1 Books Under Potto
Page 24 and 25:
NOMENCLATURE ¯R Universal gas cons
Page 26 and 27:
LIST OF TABLES xxv w Work per unit
Page 28 and 29:
The Book Change Log Version 0.3.4.0
Page 30 and 31:
LIST OF TABLES xxix ˆ Add discussi
Page 32 and 33:
LIST OF TABLES xxxi ˆ Add the open
Page 34 and 35:
Notice of Copyright For This Docume
Page 36 and 37:
GNU FREE DOCUMENTATION LICENSE xxxv
Page 38 and 39:
GNU FREE DOCUMENTATION LICENSE xxxv
Page 40 and 41:
GNU FREE DOCUMENTATION LICENSE 7. A
Page 42 and 43:
CONTRIBUTOR LIST How to contribute
Page 44 and 45:
CREDITS Typo corrections and other
Page 46 and 47:
About This Author Genick Bar-Meir i
Page 48 and 49:
Prologue For The POTTO Project This
Page 50 and 51:
CREDITS xlix process. Someone has t
Page 52 and 53:
CREDITS li have a new version every
Page 54 and 55:
Prologue For This Book Version 0.3.
Page 56 and 57:
VERSION 0.1 APRIL 22, 2008 lv and t
Page 58 and 59:
How This Book Was Written This book
Page 60 and 61:
Preface "In the beginning, the POTT
Page 62 and 63:
To Do List and Road Map This book i
Page 64 and 65:
CHAPTER 1 Introduction to Fluid Mec
Page 66 and 67:
1.2. BRIEF HISTORY 3 lay is one of
Page 68 and 69:
1.3. KINDS OF FLUIDS 5 results. The
Page 70 and 71:
1.4. SHEAR STRESS 7 For cases where
Page 72 and 73:
1.5. VISCOSITYVISCOSITY 9 The veloc
Page 74 and 75:
1.5. VISCOSITYVISCOSITY 11 always p
Page 76 and 77:
2.5 2.0 1.5 1.0 0.5 0 100 200 300 4
Page 78 and 79:
1.5. VISCOSITYVISCOSITY 15 Chemical
Page 80 and 81:
1.5. VISCOSITYVISCOSITY 17 2 Reduce
Page 82 and 83:
1.5. VISCOSITYVISCOSITY 19 In very
Page 84 and 85:
1.6. FLUID PROPERTIES 21 The total
Page 86 and 87:
1.6. FLUID PROPERTIES 23 straight.
Page 88 and 89:
1.6. FLUID PROPERTIES 25 Table -1.5
Page 90 and 91:
1.6. FLUID PROPERTIES 27 Two layers
Page 92 and 93:
1.6. FLUID PROPERTIES 29 1.6.2.1 Bu
Page 94 and 95:
1.7. SURFACE TENSION 31 terials. Th
Page 96 and 97:
1.7. SURFACE TENSION 33 or Or after
Page 98 and 99:
1.7. SURFACE TENSION 35 The maximum
Page 100 and 101:
1.7. SURFACE TENSION 37 kipenii,”
Page 102 and 103:
1.7. SURFACE TENSION 39 Equation (1
Page 104 and 105:
1.7. SURFACE TENSION 41 Example 1.1
Page 106 and 107:
1.7. SURFACE TENSION 43 Table -1.7.
Page 108 and 109:
CHAPTER 2 Review of Thermodynamics
Page 110 and 111:
2.1. BASIC DEFINITIONS 47 Since the
Page 112 and 113:
2.1. BASIC DEFINITIONS 49 when the
Page 114 and 115:
2.1. BASIC DEFINITIONS 51 Utilizing
Page 116 and 117:
CHAPTER 3 Review of Mechanics This
Page 118 and 119:
3.2. CENTER OF MASS 55 3.2 Center o
Page 120 and 121:
3.3. MOMENT OF INERTIA 57 The momen
Page 122 and 123:
3.3. MOMENT OF INERTIA 59 Equation
Page 124 and 125:
3.3. MOMENT OF INERTIA 61 Example 3
Page 126 and 127:
3.3. MOMENT OF INERTIA 63 or x (1 a
Page 128 and 129:
3.5. ANGULAR MOMENTUM AND TORQUE 65
Page 130 and 131:
3.5. ANGULAR MOMENTUM AND TORQUE 67
Page 132 and 133:
CHAPTER 4 Fluids Statics 4.1 Introd
Page 134 and 135:
4.3. PRESSURE AND DENSITY IN A GRAV
Page 136 and 137:
Page 138 and 139:
Page 140 and 141:
Page 142 and 143:
Page 144 and 145:
Page 146 and 147:
Page 148 and 149:
Page 150 and 151:
Page 152 and 153:
Page 154 and 155:
Page 156 and 157:
4.4. FLUID IN A ACCELERATED SYSTEM
Page 158 and 159:
Page 160 and 161:
Page 162 and 163:
Page 164 and 165:
4.5. FLUID FORCES ON SURFACES 101 T
Page 166 and 167:
4.5. FLUID FORCES ON SURFACES 103 x
Page 168 and 169:
Page 170 and 171:
4.5. FLUID FORCES ON SURFACES 107 I
Page 172 and 173:
Page 174 and 175:
4.5. FLUID FORCES ON SURFACES 111 C
Page 176 and 177:
4.5. FLUID FORCES ON SURFACES 113 A
Page 178 and 179:
4.5. FLUID FORCES ON SURFACES 115 F
Page 180 and 181:
4.6. BUOYANCY AND STABILITY 117 In
Page 182 and 183:
4.6. BUOYANCY AND STABILITY 119 ρ
Page 184 and 185:
4.6. BUOYANCY AND STABILITY 121 ( )
Page 186 and 187:
4.6. BUOYANCY AND STABILITY 123 The
Page 188 and 189:
4.6. BUOYANCY AND STABILITY 125 on
Page 190 and 191:
4.6. BUOYANCY AND STABILITY 127 A w
Page 192 and 193:
4.6. BUOYANCY AND STABILITY 129 It
Page 194 and 195:
4.6. BUOYANCY AND STABILITY 131 Thu
Page 196 and 197:
4.6. BUOYANCY AND STABILITY 133 The
Page 198 and 199:
4.6. BUOYANCY AND STABILITY 135 A n
Page 200 and 201:
4.6. BUOYANCY AND STABILITY 137 of
Page 202 and 203:
4.7. RAYLEIGH-TAYLOR INSTABILITY 13
Page 204 and 205:
4.7. RAYLEIGH-TAYLOR INSTABILITY 14
Page 206 and 207:
4.8. QUALITATIVE QUESTIONS 143 can
Page 208:
Part I Integral Analysis 145
Page 211 and 212:
148 CHAPTER 5. MASS CONSERVATION di
Page 213 and 214:
150 CHAPTER 5. MASS CONSERVATION It
Page 215 and 216:
152 CHAPTER 5. MASS CONSERVATION 5.
Page 217 and 218:
154 CHAPTER 5. MASS CONSERVATION Th
Page 219 and 220:
156 CHAPTER 5. MASS CONSERVATION Wh
Page 221 and 222:
158 CHAPTER 5. MASS CONSERVATION Th
Page 223 and 224:
Page 225 and 226:
162 CHAPTER 5. MASS CONSERVATION ri
Page 227 and 228:
Page 229 and 230:
166 CHAPTER 5. MASS CONSERVATION En
Page 231 and 232:
168 CHAPTER 5. MASS CONSERVATION Ex
Page 233 and 234:
170 CHAPTER 5. MASS CONSERVATION So
Page 235 and 236:
172 CHAPTER 5. MASS CONSERVATION In
Page 237 and 238:
174 CHAPTER 6. MOMENTUM CONSERVATIO
Page 239 and 240:
Page 241 and 242:
Page 243 and 244:
Page 245 and 246:
Page 247 and 248:
Page 249 and 250:
Page 251 and 252:
Page 253 and 254:
Page 255 and 256:
Page 257 and 258:
Page 259 and 260:
Page 261 and 262:
198 CHAPTER 7. ENERGY CONSERVATION
Page 263 and 264:
Page 265 and 266:
Page 267 and 268:
Page 269 and 270:
Page 271 and 272:
Page 273 and 274:
Page 275 and 276:
Page 277 and 278:
Page 279 and 280:
Page 281 and 282:
Page 283 and 284:
Page 285 and 286:
Page 287 and 288:
Page 290 and 291:
CHAPTER 8 Differential Analysis 8.1
Page 292 and 293:
8.2. MASS CONSERVATION 229 The firs
Page 294 and 295:
8.2. MASS CONSERVATION 231 Combinin
Page 296 and 297:
8.2. MASS CONSERVATION 233 Equation
Page 298 and 299:
8.2. MASS CONSERVATION 235 The dens
Page 300 and 301:
8.2. MASS CONSERVATION 237 Example
Page 302 and 303:
8.3. CONSERVATION OF GENERAL QUANTI
Page 304 and 305:
8.4. MOMENTUM CONSERVATION 241 8.4
Page 306 and 307:
8.4. MOMENTUM CONSERVATION 243 The
Page 308 and 309:
8.5. DERIVATIONS OF THE MOMENTUM EQ
Page 310 and 311:
Page 312 and 313:
Page 314 and 315:
Page 316 and 317:
Page 318 and 319:
8.6. BOUNDARY CONDITIONS AND DRIVIN
Page 320 and 321:
8.6. BOUNDARY CONDITIONS AND DRIVIN
Page 322 and 323:
8.7. EXAMPLES FOR DIFFERENTIAL EQUA
Page 324 and 325:
Page 326 and 327:
Page 328 and 329:
Page 330 and 331:
Page 332 and 333:
Page 334 and 335:
Page 336 and 337:
CHAPTER 9 Dimensional Analysis This
Page 338 and 339:
9.1. INTRODUCTORY REMARKS 275 the r
Page 340 and 341:
9.1. INTRODUCTORY REMARKS 277 End S
Page 342 and 343:
9.1. INTRODUCTORY REMARKS 279 Deriv
Page 344 and 345:
9.2. BUCKINGHAM-π-THEOREM 281 para
Page 346 and 347:
9.2. BUCKINGHAM-π-THEOREM 283 magn
Page 348 and 349:
9.2. BUCKINGHAM-π-THEOREM 285 Tabl
Page 350 and 351:
9.2. BUCKINGHAM-π-THEOREM 287 End
Page 352 and 353:
9.2. BUCKINGHAM-π-THEOREM 289 Equa
Page 354 and 355:
9.2. BUCKINGHAM-π-THEOREM 291 the
Page 356 and 357:
9.2. BUCKINGHAM-π-THEOREM 293 than
Page 358 and 359:
9.2. BUCKINGHAM-π-THEOREM 295 Unde
Page 360 and 361:
9.2. BUCKINGHAM-π-THEOREM 297 Solu
Page 362 and 363:
9.3. NUSSELT’S TECHNIQUE 299 The
Page 364 and 365:
9.3. NUSSELT’S TECHNIQUE 301 boun
Page 366 and 367:
9.3. NUSSELT’S TECHNIQUE 303 Noti
Page 368 and 369:
9.3. NUSSELT’S TECHNIQUE 305 In t
Page 370 and 371:
9.3. NUSSELT’S TECHNIQUE 307 Wher
Page 372 and 373:
9.4. SUMMARY OF DIMENSIONLESS NUMBE
Page 374 and 375:
Page 376 and 377:
Page 378 and 379:
Page 380 and 381:
Page 382 and 383:
9.5. SUMMARY 319 It can be noticed
Page 384 and 385:
9.6. APPENDIX SUMMARY OF DIMENSIONL
Page 386 and 387:
BIBLIOGRAPHY [1] Buckingham, E. On
Page 388 and 389:
CHAPTER 10 Inviscid Flow or Potenti
Page 390 and 391:
10.1. INTRODUCTION 327 where in thi
Page 392 and 393:
10.1. INTRODUCTION 329 With the ide
Page 394 and 395:
10.1. INTRODUCTION 331 The partial
Page 396 and 397:
10.2. POTENTIAL FLOW FUNCTION 333 T
Page 398 and 399:
10.2. POTENTIAL FLOW FUNCTION 335 t
Page 400 and 401:
Page 402 and 403:
Page 404 and 405:
10.2. POTENTIAL FLOW FUNCTION 341 9
Page 406 and 407:
10.3. POTENTIAL FLOW FUNCTIONS INVE
Page 408 and 409:
Page 410 and 411:
Page 412 and 413:
Page 414 and 415:
Page 416 and 417:
Page 418 and 419:
Page 420 and 421:
Page 422 and 423:
Page 424 and 425:
Page 426 and 427:
Page 428 and 429:
Page 430 and 431:
Page 432 and 433:
10.4. CONFORMING MAPPING 369 flow d
Page 434 and 435:
10.4. CONFORMING MAPPING 371 The un
Page 436 and 437:
10.5. UNSTEADY STATE BERNOULLI IN A
Page 438 and 439:
10.6. QUESTIONS 375 Table -10.2. Di
Page 440 and 441:
CHAPTER 11 Compressible Flow One Di
Page 442 and 443:
11.3. SPEED OF SOUND 379 to be cont
Page 444 and 445:
11.3. SPEED OF SOUND 381 Solution T
Page 446 and 447:
11.3. SPEED OF SOUND 383 consider a
Page 448 and 449:
11.4. ISENTROPIC FLOW 385 Now, subs
Page 450 and 451:
11.4. ISENTROPIC FLOW 387 Static Pr
Page 452 and 453:
11.4. ISENTROPIC FLOW 389 that the
Page 454 and 455:
11.4. ISENTROPIC FLOW 391 Using the
Page 456 and 457:
11.4. ISENTROPIC FLOW 393 Solution
Page 458 and 459:
11.4. ISENTROPIC FLOW 395 Expressin
Page 460 and 461:
11.4. ISENTROPIC FLOW 397 The tempe
Page 462 and 463:
11.4. ISENTROPIC FLOW 399 Table -11
Page 464 and 465:
Page 466 and 467:
Page 468 and 469:
11.4. ISENTROPIC FLOW 405 Example 1
Page 470 and 471:
11.5. NORMAL SHOCK 407 In a shock w
Page 472 and 473:
11.5. NORMAL SHOCK 409 Energy equat
Page 474 and 475:
11.5. NORMAL SHOCK 411 The density
Page 476 and 477:
11.5. NORMAL SHOCK 413 or in a dime
Page 478 and 479:
11.5. NORMAL SHOCK 415 which differ
Page 480 and 481:
11.5. NORMAL SHOCK 417 is different
Page 482 and 483:
11.5. NORMAL SHOCK 419 Table -11.3.
Page 484 and 485:
11.6. ISOTHERMAL FLOW 421 Table -11
Page 486 and 487:
11.6. ISOTHERMAL FLOW 423 Rearrangi
Page 488 and 489:
11.6. ISOTHERMAL FLOW 425 dT 0 T 0
Page 490 and 491:
11.6. ISOTHERMAL FLOW 427 10 2 0 1
Page 492 and 493:
11.6. ISOTHERMAL FLOW 429 The fact
Page 494 and 495:
11.6. ISOTHERMAL FLOW 431 Calculati
Page 496 and 497:
11.6. ISOTHERMAL FLOW 433 maximum M
Page 498 and 499:
11.6. ISOTHERMAL FLOW 435 From the
Page 500 and 501:
11.7. FANNO FLOW 437 The energy con
Page 502 and 503:
11.7. FANNO FLOW 439 Differentiatin
Page 504 and 505:
11.7. FANNO FLOW 441 11.7.3 The Mec
Page 506 and 507:
11.7. FANNO FLOW 443 pressure on th
Page 508 and 509:
11.7. FANNO FLOW 445 10 2 0 1 2 3 4
Page 510 and 511:
11.7. FANNO FLOW 447 End Solution A
Page 512 and 513:
11.7. FANNO FLOW 449 M x M y T y T
Page 514 and 515:
11.7. FANNO FLOW 451 11.7.5.1 Maxim
Page 516 and 517:
11.7. FANNO FLOW 453 At the startin
Page 518 and 519:
11.7. FANNO FLOW 455 Fanno Flow 5 4
Page 520 and 521:
11.7. FANNO FLOW 457 11.7.7.1 Choki
Page 522 and 523:
11.7. FANNO FLOW 459 back pressure
Page 524 and 525:
11.7. FANNO FLOW 461 The solution i
Page 526 and 527:
11.7. FANNO FLOW 463 11.7.8 The Pra
Page 528 and 529:
11.7. FANNO FLOW 465 0.9 0.8 0.7 0.
Page 530 and 531:
11.7. FANNO FLOW 467 3.0 2.5 2.0 Co
Page 532 and 533:
11.8. THE TABLE FOR FANNO FLOW 469
Page 534 and 535:
11.9. RAYLEIGH FLOW 471 Table -11.6
Page 536 and 537:
11.10. INTRODUCTION 473 or in simpl
Page 538 and 539:
11.10. INTRODUCTION 475 or explicit
Page 540 and 541:
11.10. INTRODUCTION 477 Table -11.7
Page 542 and 543:
11.10. INTRODUCTION 479 Example 11.
Page 544 and 545:
11.10. INTRODUCTION 481 Example 11.
Page 546 and 547:
11.10. INTRODUCTION 483 what the ma
Page 548 and 549:
CHAPTER 12 Compressible Flow 2-Dime
Page 550 and 551:
12.2. OBLIQUE SHOCK 487 12.2 Obliqu
Page 552 and 553:
12.2. OBLIQUE SHOCK 489 The relatio
Page 554 and 555:
12.2. OBLIQUE SHOCK 491 Where S = 3
Page 556 and 557:
12.2. OBLIQUE SHOCK 493 by evaluati
Page 558 and 559:
12.2. OBLIQUE SHOCK 495 Figure (12.
Page 560 and 561:
12.2. OBLIQUE SHOCK 497 be made. Th
Page 562 and 563:
12.2. OBLIQUE SHOCK 499 demonstrate
Page 564 and 565:
12.2. OBLIQUE SHOCK 501 or explicit
Page 566 and 567:
12.2. OBLIQUE SHOCK 503 Fig. -12.9.
Page 568 and 569:
12.2. OBLIQUE SHOCK 505 Table -12.1
Page 570 and 571:
12.2. OBLIQUE SHOCK 507 Example 12.
Page 572 and 573:
12.2. OBLIQUE SHOCK 509 In the obli
Page 574 and 575: 12.2. OBLIQUE SHOCK 511 M x M ys M
Page 576 and 577: 12.2. OBLIQUE SHOCK 513 M x M y T y
Page 578 and 579: 12.2. OBLIQUE SHOCK 515 M x M yw θ
Page 580 and 581: 12.2. OBLIQUE SHOCK 517 The pressur
Page 582 and 583: 12.2. OBLIQUE SHOCK 519 From these
Page 584 and 585: 12.3. PRANDTL-MEYER FUNCTION 521 as
Page 586 and 587: 12.3. PRANDTL-MEYER FUNCTION 523 Th
Page 588 and 589: 12.3. PRANDTL-MEYER FUNCTION 525 No
Page 590 and 591: 1 2 12.7. FLAT BODY WITH AN ANGLE O
Page 592 and 593: 12.8. EXAMPLES FOR PRANDTL-MEYER FU
Page 594 and 595: 12.9. COMBINATION OF THE OBLIQUE SH
Page 596 and 597: 12.9. COMBINATION OF THE OBLIQUE SH
Page 598 and 599: CHAPTER 13 Multi-Phase Flow 13.1 In
Page 600 and 601: 13.4. KIND OF MULTI-PHASE FLOW 537
Page 602 and 603: 13.5. CLASSIFICATION OF LIQUID-LIQU
Page 604 and 605: 13.5. CLASSIFICATION OF LIQUID-LIQU
Page 606 and 607: 13.6. MULTI-PHASE FLOW VARIABLES DE
Page 608 and 609: 13.6. MULTI-PHASE FLOW VARIABLES DE
Page 610 and 611: 13.7. HOMOGENEOUS MODELS 547 13.7 H
Page 612 and 613: 13.7. HOMOGENEOUS MODELS 549 There
Page 614 and 615: 13.8. SOLID-LIQUID FLOW 551 where
Page 616 and 617: 13.8. SOLID-LIQUID FLOW 553 So far
Page 618 and 619: 13.9. COUNTER-CURRENT FLOW 555 The
Page 628 and 629: 13.10. MULTI-PHASE CONCLUSION 565 1
Page 630 and 631: APPENDIX A The Mathematics Backgrou
Page 632 and 633: A.1. VECTORS 569 where θ is the an
Page 634 and 635: A.1. VECTORS 571 Divergence The sam
Page 636 and 637: A.1. VECTORS 573 R · S =(xî + y 2
Page 638 and 639: A.1. VECTORS 575 The curl is writte
Page 640 and 641: A.1. VECTORS 577 The divergence of
Page 642 and 643: A.2. ORDINARY DIFFERENTIAL EQUATION
Page 656 and 657: A.3. PARTIAL DIFFERENTIAL EQUATIONS
Page 658 and 659: A.4. TRIGONOMETRY 595 A.4 Trigonome
Page 660 and 661: SUBJECTS INDEX 597 Subjects Index S
Page 662 and 663: SUBJECTS INDEX 599 Ideal gas, 91 Re
Page 664 and 665: SUBJECTS INDEX 601 Slip condition r
Page 666 and 667: AUTHORS INDEX 603 Authors Index B B
show all

Basics of Fluid Mechanics, 2014a

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

Delete template?

Save as template?