Index
A
Absolute pressure, 8
Aggregatively fluidized beds, 560, 566
fluid flow in, 569
particle flow in, 468
pressure distribution in, 567
Analogies, between momentum and heat transfer, 509
Angular momentum, 81
Angular velocity, 4, 39, 258, 355
Annular die, flow through, 322
A.P.I., degrees, 11
Archimedes, biographical sketch, 36
Archimedes’ law, 37
Archimedes number, 200
Axially symmetric irrotational flow, 378
B
Basis or shape function, in finite-element methods, 681
Bearing, journal, 443
thrust, 443
flow in, using COMSOL, 448
Bernoulli, Daniel, biographical sketch, 68
Bernoulli’s equation, 67, 355, 384, 533
compressible flow, 161
generalized, 64
Bézier curves in COMSOL, 720
Bingham plastic fluids, 594, 600
in pipe flow 600
Blake-Kozeny equation, 207
Blasius equation, 129, 428, 493, 495, 511
Blasius solution for boundary layer flow, 425
Blow molding, 313
Blunt-nosed object, flow past, 358, 383
Bob-and-cup viscometer, 627
Body-force potential, 321
Boltzmann distribution, 649
Boltzmann’s constant, 131
Boolean operations in COMSOL, 722
Boundary conditions, 293
Boundary layers, 414
application to turbulent jets, 513
dimensional analysis of, 430
laminar, 415
simplification of equations of motion for, 422
solution using COMSOL, 435
turbulent, 428
Boundary settings in COMSOL, 712, 724, 727
Bourdon-tube pressure gauge, 89
Brinkman equation with COMSOL, 729
Brownian motion, 129
Bubble caps, dynamics of, in distillation columns, 216
Bubble flow in vertical pipes, 543, 545
Bubbles, in fluidized beds, 560
formation at an orifice, 563
Bubbles, rise velocity of, 531, 545
Bubbles, terminal velocity, 532
Buckingham Pi theorem, 227
Buffer region, 490
Buoyancy, 36
Burke-Plummer equation, 207
C
Cake, in a filter, 210
pressure distribution in, 455
Capillary pressure, in porous medium, 393
Capillary tube, for surface tension, 19
Capillary viscometer, 623
Caprock, 395
Cascade process in turbulence, 474
Centrifugal filter, 214
Characteristic time, 596
Charge number, 644
“Choking” of the throat, 163
Churchill, S.W., Reynolds stress correlation, 496
interpolation between two asymptotic limits, 498
Coating a moving substrate, 461
Coating or spreading, 313
Coaxial cylinder rheometer, 627
Coefficient of contraction, 71
Coefficient of discharge, 73
Coefficient of thermal expansion, 12
Coions, 649
Colebrook and White equation, 136, 494
Commercial pipe, sizes, 138
Complex piping systems, 163
Composite object, COMSOL, 712, 723
Compressibility factor, 12
Compressibility, isothermal, 12
Compressible flow of gases,
in a nozzle, 159
in a pipeline, 156
with COMSOL, 729
Compressive stress, 6
Computational fluid dynamics (CFD), 473, 671
applications in chemical engineering, 672
COMSOL Multiphysics, examples involving,
boundary-layer flow, 435
die flow, non-Newtonian, 606
jet flow and mixing, 505
lake flow, 373
lubricated bearing, 448
momentum diffusion, 307
orifice plate, 501
parallel-plate flow, 435
porous-medium flow, 705
screw extruder, 318
COMSOL Multiphysics, capabilities of, 703
axes and grid settings, 710
Bézier curves, 720
Boolean operations, 722
documentation, 705
draw mode, 719
draw toolbar, 711
equations solvable by, 704
graphical user interface, 708
how to run, 705
interior boundaries
menus and toolbars, 709
model library
model navigator, 706
physics modes, 703
plot parameters, 718
postprocessing, 717
problems solvable by, 725
surface plot, 716
Cone-and-plate viscometer, 328, 626
Connate water, 392
Constitutive equations,
Carreau model, 599
generalized Newtonian fluids, 598
general viscoelastic fluids, 615
Maxwell model, 615
power-law model, 599
White-Metzner model, 619
Continuity equation, 59, 72, 267, 268
time-averaged, 477
Control surface, 55
for momentum transfer, 81
Convection of momentum, 81
Convective derivative, 266, 619
Converging/diverging nozzle, 159
Conversion factors, table of, inside front cover, 25
Coriolis mass-flow meter, 95
Couette flow, 294, 312, 316, 328
in lubrication, 447
Counterions, 649
Critical pressure for compressible gas flow, 158, 163
Cross product, 251
Curl of a vector, 259
expressions for, 266
Curved surface, change in pressure across, 18, 458
Cyclone separation, 219
Cylinder, flow past, computed by FlowLab, 195, 696
drag coefficient, 699
Cylindrical coordinates, 263
mass balance in, 268
momentum balances in, 322
solution of problems in, 322
D
d’Arcy’s law, 207, 388, 392, 706
Dam, force on, 32
Deborah number, 596
Debye-Huckel limit, 650
Debye length, 650
Deformation of a fluid element, 275, 357
Del (nabla) operator, in rectangular coordinates, 265
Density, 10
°A.P.I., 11
Derivative, definition of, 27, 257
Derivatives, 731
Derived quantities, 225
Diameter, hydraulic mean, 151
Die swell, 614
non-Newtonian, 606
Differential equations, solution of,
separation of variables, 733
Differential mass balance, 267
Differential momentum balance, 271
Diffuser, in a nozzle, 159
Diffusion coefficient, 15
Diffusion in microchannels, 642
Diffusion of momentum, 307
Dilatant fluids, 593, 599, 603
Dimensional analysis, 224
Dimensional analysis of boundary layer flow, 430
Dimensionless groups, for
drag force, 196
filtration, 224
flow through packed beds, 206
laminar sublayer, 230
pumps, 192
Dimensionless numbers, table of, 228
Dimensionless shear stress, 132, 491
Dimensions, 226
mass, length, and time, 10
Directional derivative, 252
Discharge coefficient, 73
Discretization, in numerical methods, 674
Dissipation, see frictional dissipation
Dissipation, turbulent, 499
transport equation for, 500
Distillation column, dynamics of bubble caps, 216
Dittus-Boelter equation, 511
Divergence of a vector, 254
expressions for, 265
Dot product, 250
Double-dot product, 597
Doublet, 384
Drag coefficient, 196
Drag coefficient on a flat plate, 415, 419, 428, 429, 432
Drag force, 194
Draw mode in COMSOL, 719
Draw toolbar in COMSOL, 711
Drawing or spinning, 312
Droplet, excess pressure inside, 18
Ducts, flow in noncircular, 150
Dynamical similarity, 229
E
formation of, 475
Eddy diffusivity, 483
Eddy kinematic viscosity, 132, 482, 483, 484
correlation for, 486
determination of, 485
in turbulent jets, 518
Eddy thermal diffusivity, 483
Eddy transport, 481
Elastic modulus, 621
Elastic recoil, 617
Electrical double layer, 647, 648, 649
Electric charge, 644
flux of, 645
Electric field, 644
Electric potential, 646, 649, 650
Electrokinetic forces, 664
measurement of, 659
Electroosmosis in a microchannel (COMSOL), 653, 657
Electroosmotic flow around a particle, 653
Electroosmotic mobility, 647
Electrophoretic mobility, 645
Electrostatic precipitator, 202
Electroviscosity, 661
Energy, conservation of, 9, 55
English units, 22
Entrance region between flat plates, 440
Eötvös number, in slug flow, 549
Equations of motion, 268, 281, 294, 322, 327
solutions of, 293
Equipment, visual encyclopedia of, 185
Equipotentials, 366
Equivalent length of fittings, 154
Ergun equation, 206
Euler’s method, 733
Eulerian viewpoint, 267
Excel spreadsheets, 143, 145, 146, 150, 167, 454
Extrusion of polymer, 312
F
Falling-sphere viscometer, 202
Fanning friction factor, 132, 133, 135, 136, 137
Faraday’s constant, 649
FEMLAB—see under its new name, COMSOL Multiphysics
Fick’s law, 644
Film flow, 456
Film, in lubrication, 443
Filtrate, 210
Filtration, 210
centrifugal, 213
plate-and-frame, 210
rotary-vacuum, 212
Finite-difference methods, 674
Finite-element methods, 680
Finite-volume methods, 676
Fittings, equivalent length, 154
Five-spot pattern, 391
Flooding, 555
Flow energy, 62
Flow, around sphere, 194
in noncircular ducts, 150
in open channels, 151
through a porous medium, 207
through packed beds, 204
FlowLab, examples involving,
flow in pipe entrance, 687
flow past a cylinder, 696
sudden expansion, 690
two-dimensional mixing, 692
FlowLab, CFD software, 682
geometry panel, 684
graphical user interface, 683
mesh and solve panels, 685
operation toolpad buttons, 683
physics, boundary condition, and materials panels, 684
reports and postprocessing panels, 686
Flow rates, 9
Flow rate, measurement of, 94
by Coriolis meter, 95
by orifice plate, 71
by rotameter, 89
Flow regimes in two-phase flow,
horizontal pipes, 541
vertical pipes, 543
Fluent, Inc., 682
Fluid, definition of, 9
Fluid mechanics, laws of, 9
particulate, 559
Fluidized bed, 559
reaction in, 572
Force, 22
as a rate of momentum transfer, 79
on arbitrary surfaces, 33
on dam, 32
power for displacement of, 64
units of, 21
Form drag, 194
Fox, T.R.C., xvii, 456 footnote
of rotating fluid, 39
Friction factor, 124
analogy with the Stanton number, 510
as a dimensionless group, 132, 210
Friction-factor plot, 135
Friction velocity, 489
Frictional dissipation, 63, 598
noncircular ducts, 151
open channels, 152
packed beds, 207
Froude number, in slug flow, 549
Fundamental dimensions, 225
G
Galerkin’s method, 681
Gas constant, values of, 12
Gas law, 11
Gas-lift pump, 550
Gas, pressure variations in, 31
Gas, underground storage of, 395
Gases, 5
flow of compressible, 156, 159
viscosity of, 131
Gauge pressure, 8
Gate valve, 154
gc, conversion factor, 22
General linear viscoelastic fluids, 615, 618
Generalized Maxwell model, 618
Geometrical shapes, 731
Geometrical similarity, 229
Globe valve, 154
Gradient of a scalar, 252
expressions for, 265
Graphical interface, for COMSOL, 708
H
Hagen-Poiseuille law, 125
Harrison, D., bubble formation in fluidized beds, 565
Head, of fluid, 68
Head/discharge curve for centrifugal pump, 192
Heat transfer, analogy with momentum transfer, 509
Hookean solid, 616
Hoop stress in pipe wall, 139
Hydraulic mean diameter, 151
Hydraulically smooth pipe, 136
Hydrostatics, 26
multiple fluids, 30
I
Incipient fluidization, 215, 559
Infinite-shear viscosity, 599
Injection molding, 312
Integrals, 731
Intensity of turbulence, 477
Internal energy, 61
Invariants of the strain-rate tensor, 597
Inviscid fluid, motion of, 321
Irrigation ditch, 152
axially symmetric, 378
in cylindrical coordinates, 363
in rectangular coordinates, 361
line source, 370
past a blunt-nosed object, 358, 383
past a cylinder, 367
past a sphere, 386
point source, 382
Irrotationality condition, for axially symmetric flow, 379
for cylindrical flow, 364
for rectangular flow, 361
Isentropic expansion, 160
Isothermal compressibility, 12
Isothermal flow of gas in pipe, 156
J
Jet mixing, COMSOL computation of, 505
FlowLab computation of, 692
Journal bearing, 443
K
Karamanev, D.G., method for terminal velocities, 200
Kármán vortex street, 475, 697
k/ε method for turbulent flows, 499
with COMSOL, 726
with FlowLab, 690
for pipe flow, 127
Kinetic energy, turbulent, 499
transport equation for, 500
Kronecker delta, 251
L
Lagrangian viewpoint, 266
Lake flow, with COMSOL, 373
Lamb, Horace, feelings about turbulence, 473
Laminar flow, friction in,
frictional dissipation, 126
friction factor for, 134
Laminar flow, unstable, 475, 700
dimensional analysis of, 229
Laminar velocity profile, 124, 155
Laplace, Pierre Simon, Marquis de, biographical sketch, 362
Laplace’s equation, 262
in irrotational flow, 362, 364, 379
for axially symmetric flow, 379
with COMSOL, 727
Laplacian operator, 262
expressions for, 266
Laws of fluid mechanics, 9
Leibnitz’s rule, 618 footnote, 736
Leung, L.S., bubble formation in fluidized beds, 565
Linear viscoelasticity, 615
Line source, 370
Liquids, 5
Lockhart/Martinelli correlation, 539, 552
Logarithmic velocity profile, 487, 490
Lorentz force, 646
Loss angle, 621
Loss modulus, 621
Lubrication approximation, 444
Lubrication flow, with COMSOL, 448
M
Macintosh computer for COMSOL, 706
Magnetic settling, 662
Manometer, 93
Mass, 21
steady state, 57
Mass flow rate, 9
Mass velocity, 157
Material types, 591
MATLAB, xvi, 613, 703, 704, 706
Maxwell, James Clerk, biographical sketch, 616
Maxwell model, 615
Memory function, 618
Mesh refinement in COMSOL, 715, 725
Microfluidics, 639
chips for, 640
Microscale fluid mechanics, 640
Mist flow, 475
Mixing length, correlation for, 485
determination of, 484
Mixing-length theory, 481
for turbulent jets, 515
Model navigator, in COMSOL, 706
Moment of inertia, 90
Momentum, 78
angular, 90
diffusion of, 307
Momentum balance, for bubble formation at an orifice, 564
in film flow, 405
shell, 301
time-averaged, 478
Momentum transfer,
by force, 79
in laminar flow, 129
in turbulent flow, 131
Momentum transfer, analogy with heat transfer, 509
Moody friction factor, 132
N
Natural gas, underground storage of, 395
Navier, Claude-Louis-Marie-Henri, biographical sketch, 281
Navier-Stokes equations, 278, 281
in microfluidics, 646
with COMSOL, 725
Needle valve, 154
Newton, Sir Isaac, biographical sketch, 131
Newtonian fluid, 4, 14, 124, 130, 275, 276, 279, 591, 598
Nicklin, D.J., correlation for two-phase slug flow, 548
Nikuradse, pipe friction experiments, 136
Nonlinear simultaneous equations, 149, 166
Non-Newtonian flow in a die, with COMSOL, 606
viscosity profiles, 611
Non-Newtonian flows using COMSOL, 728
viscoelastic, 613
Normal-stress difference, 613
No-slip boundary condition, 273
Nozzle, gas flow in, 159
Numerical methods for solving fluid mechanics problems, 673
O
Oldroyd derivative, 620
One-seventh power law, 493, 495
Open-channel flow, 151
Order-of-magnitude analysis,
for boundary-layer flow, 423
for turbulent jets, 513
Ordinary differential equations, solution of, 733
Orifice, flow through, compressible, 159
incompressible, 70
Orifice-plate “meter,” 71
Orifice plate, COMSOL solution, 88, 501
pressure recovery, 504
Oscillatory shear, with COMSOL, 309
Ostwald-de-Waele model, 599
P
Packed beds, 204
Packed-bed reactor, pressure drop in, 208
Packed column, flooding of, 556
Paint films, leveling of, 463
Parabolic velocity profile, 124, 155
Parallel-plate rheometer, 627
Particle motion in microfluidic channels, 661
Particles, settling of, 199, 201, 222
Particulate phase (“emulsion”), in fluidization, 560
PC for COMSOL, 706
Péclet number, 643
Permeability, 208, 387, 391, 396
Physical properties, 10
Piezoelectric and piezoresistive effects in pressure transducer, 93
Piezometer, 93
Pipe fittings, pressure drop, 154
Pipe flow, Bingham plastic, 604
power-law fluid, 600
Pipe flow, pressure drop in, 123, 133, 139
Pipe roughness, 136
Pipeline, for gas, 156
Pipes, flow through, 120
Pitot tube, 74
Pitot-static tube, 74
Plate-and-frame filter, 210
Plot parameters in COMSOL, 718
Point source, 381
Poiseuille flow, 294, 312, 316
in lubrication, 447
Poisson’s equation, in lubrication, 445
Poisson’s equation, solution of,
by finite-element methods, 674
by finite-difference methods, 674
microfluidics, 646
Pores, flow through, 205
Porous medium, flow through, 207, 566
two-phase, 390
with COMSOL, 728
Potential, for porousmedium flow, 392
Potential energy, 61
Power,
for flowing stream, 64
for force displacement, 64
for pump, 64
for rotating shaft, 64
Power-law velocity profile, 495, 603
Prandtl hypothesis, 486
Prandtl, Ludwig, biographical sketch, 434
Prandtl mixing length, 483
Prandtl-Taylor analogy, 510
Pressure, 6
absolute, 8
gauge, 8
Pressure as a function of height, 26
Pressure change caused by rotation, 39
Pressure distribution, in calendering, 454
in fluidized beds, 567
Pressure drop, across pipe fittings, 154
Pressure drop in two-phase flow
horizontal pipes, 536
Pressure forces on submerged objects, 36
Pressure head, 68
Pressure measurement, 92
Pressure transducer, 93
Primary recovery of oil, 390
Projected area, 196
Pseudoplastic fluids, 593, 599, 603
positive displacement, 188, 189
Pumps in series and parallel, 193
R
Rabinowitsch equation, 624
Radius of curvature, 735
Rate laws, 57
Rate-of-deformation tensor, 279
Rate-of-strain tensor, 279, 596
invariants of, 597
Reaction in fluidized bed, 572
Reciprocating pumps, 188
Recirculation in sudden expansion,
in jet mixing, with FlowLab, 694
using FlowLab, 691
Rectangular coordinates, 249
mass balance in, 268
momentum balances in, 272, 281
problems in, 294
Rectangular duct, flow through, 150, 294
Reference quantities, 430
Relaxation modulus, 618
Relaxation time, 616
Residual oil, 392
Reynolds analogy, 509
Reynolds experiment, 121
for boundary-layer flow, 415, 428
for drag force, 196
in microfluidics, 641
in pipe flow, 134, 135, 137, 149
Reynolds, Osborne, biographical sketch, 121
Reynolds stresses, 479
correlation for, 496
Rheometers, 625
Rheopectic fluids, 594
Richardson-Zaki correlation, 222, 560
Rod-climbing effect, 614
Rotameter, 89
Rotary pumps, 189
Rotary-vacuum filter, 212
Rotating fluid, 39
Rotational flow, 356
Rotational rheometers, 625
Roughness, pipe, 136
S
Saturation, in porous medium, 391
Scalars, 249
Schedule number for pipe, 137
Screw extruder, 313
with COMSOL, 318
Secondary recovery of oil, 390
Sedimentation, 222
Separation of variables, 733
Settling of particles, 199, 201, 222
Shacham equation, for turbulent friction factor, 137, 150
Shear stress, 3, 6, 14, 271, 274
dimensionless groups for pipe flow, 225
models for, 129
Shear-thickening fluids, 593, 604
Shear-thinning fluids, 593, 599, 604
Shell momentum balance, 301
Sign convention for stresses, 271
Similar velocity profiles, 417
Simpson’s rule, 733
Simultaneous nonlinear equations, 149, 166
Slug flow in vertical pipes, 543, 547
Slurry, 210
Smooth pipe, flow in, 490
Solenoidal flow, 256
Solution procedure, for viscous-flow problems, 293
Source in a uniform stream, 382
Specific gravity, 13
Sphere, drag force on, 194, 434
Sphere, flow past, 194, 386, 434
Spherical coordinates, 263
mass balance in, 268
momentum balances in, 283
solution of problems in, 327
Spherical-cap bubbles, 533
Sphericity, particle, 197, 198
Spinning of fibers, 325
Spray drier, 201
Spreadsheets, 143, 145, 146, 150, 167, 454, 734
Spreadsheet solution of differential equations, 734
Spring/dashpot model, 617
Stagnation point, 369, 383, 533
Static head, 68
Steady-state energy balance, 62
Steady-state mass balance, 59
Steady-state problems, 57
Stokes, Sir George Gabriel, biographical sketch, 198
Stokes’ law, 198
Storage modulus, 621
for non-Newtonian flow in a die, 613
Stream function, 362, 364, 378
for boundary layers, 426
physical interpretation of, 364, 378
Streaming potential, 659
Stream tube, 62
Strength, of a doublet, 384
of a line source, 370
of a point source, 381
Stress and strain, for viscoelastic fluid, 622
Stress, compressive, 6
tensile, 5
Stress, sign convention for, 271
Stress relaxation, 617
Strong conservation form, 673, 676
Strouhal number, 699
Subdomain settings in COMSOL, 714, 724
Substantial derivative, 266
Sudden expansion, after orifice plate, 86
in a pipe, 88
solved by FlowLab, 690
Superficial velocity, 205
Surface energy, 17
Surface plot in COMSOL, 716
Surface roughnesses, 136
Surface tension, 16
in thin-film flow, 456
methods for measuring, 19
Surface waves, 396
T
Tangential stresses, 6, 272, 274
Tank draining, 70
evacuation, 58
filling, 76
Taylor dispersion, 642
Taylor, Geoffrey Ingram, biographical sketch, 534
Tensile stress, 5
“divergence” of, 741
“Laplacian” of, 741
Terminal velocities of spheres, 199
Karamanev method, 200
Tertiary recovery of oil, 390
Thermal diffusivity, 512
Thermal expansion, coefficient of, 12
Thin films, 456
Thixotropic fluids, 594
Thrust bearing, 443
Time-averaged continuity equation, 477
momentum balance, 478
Time-averaging, 476
power for rotation of, 64
Total head, 69
Transducer, for measuring pressure, 93
Transient problems, 57
Transient viscous diffusion of momentum (COMSOL), 307
Transition flow, 121
computation by the k/εmethod, 499
intensity of, 477
k/ε method for, 499
mixing-length theory, 481
momentum transport in, 131, 480, 509
orifice-plate flow, 501
solved by COMSOL, 726
velocity profiles, 155, 487, 490, 492
Turbulent boundary layers, 428
Turbulent core, 155, 229, 490, 492
dissipation rate ε, 499
Turbulent jets, 513
axisymmetric, 519
plane, 514
Turbulent properties computed by COMSOL, dissipation, 503, 508
Turbulent transport, summary of, 483
Two-phase flow in porous media, 390
Two-phase flow in pipes, horizontal pipes, 536
vertical pipes, 543
U
Underground flow of water, 364, 388
Underground storage of natural gas, 395
Unit vectors, 250
Universal velocity profile, 488
Unstable laminar flow, 475, 700
Unsteady-state problems, 57
Usagi, R., interpolation
between two asymptotic limits, 498
V
Valve, for pipeline, 154
Vanes, of centrifugal pump, 190
Variable-viscosity momentum balance, 284
Vector components, 250
Vector differentiation, 251
Vectors, 249
addition and subtraction, 250
dyadic product, 740
multiplication, 250
Velocity, 8
Velocity head, 68
Velocity, no-slip boundary condition, 293
Velocity potential, 260, 361, 364, 379
Velocity profiles, boundary-layer flow, 417, 418, 426, 428
concentric cylinders, 4
development in entrance region, 440
lubrication, 444
parallel plates, 14
viscous flow, 124, 297, 305, 316, 324, 331
Vena contracta, 71
constitutive equations, 613
Viscometers, 625
of gases, 131
Viscous dissipation function, 598
Viscous drag, 194
Viscous-flow problems, 292
Viscous modulus, 621
Visual encyclopedia of chemical engineering equipment, 185
Void fraction, 205
in two-phase flow, 536, 544, 549
Volumetric flow rate, 9
Volute chamber, 190
Von Kármán hypothesis, 487
Vortex formation during jet mixing, 506
Vortex lines, 355
Vortex shedding past a cylinder, 698
for non-Newtonian flow in a die, 612
source term for, 373
W
Waterflooding, 391
Wave motion in deep water, 396
paths followed by particles, 399
Weight, 21
Weir, in distillation column, 217
Weissenberg effect, 614
Weissenberg, lectures of, 592
Weissenberg rheogoniometer, 328, 626
Wetted perimeter, 150
Weymouth equation, 157
White-Metzner model, 619
Y
Z
Zajic, S.C., Reynolds stress correlation, 496
Zero-shear boundary condition, 294
Zero-shear viscosity, 599