Index
Note: Page number followed by f indicate figures and t indicate tables.
A
Adams-Moulton formula 500
Aerostatics 315–317
Amine-based cationic surfactants 436
Anionic surfactants 431
carboxylic acids 434–435
sulfonic acids 435–436
Artificial compressibility method 702–703
Aspect ratio 263
Atom 93
gases 244
liquids 244
phase changes 244–245
plasmas 245
solids 243–244
Average flow velocity 333–334
Avogadro constant 96
Axisymmetric drop shape analysis (ADSA) 493
B
Balanced rectangular function 411
Fourier series expansion 411–413
frictionless flow field 277–278
Stokes' friction approach 278–280
Balancing viscous dissipation 358
Bashforth–Adams schemes 499
Bashforth and Adams curvature method 480–481
Basis functions, FEM 658–660
modified 38–39
roots of 39–40
standard 35–38
Bilayer and micelle formation 432–434
Bingham fluids 250
Biochemical passivation 441
Blast furnace process 111
Boltzmann constant 100
finite difference method in Microsoft Excel 627–629
nonperiodic 619–620
one-dimensional heat equation 645
periodic 619
pressure 256–257
shear stress 256
temperature 256
velocity 255–256
Brownian motion 128
Buoyancy forces 317
Burgers' equation 680
C
curvature 48–49
denominators containing sums 48
infinitesimal changes 49–50
variable and derivative 48
length 446–447
measurement by 459
meniscus formation 447–451
pressure 445–446
Capillary heights 446
normalized 446–447
capillary number 447
meniscus depth 447
negative capillary pressure 447
negative 447
Cartesian/cylindrical coordinates conversion 148–150
coordinate system transformation 148–150
Jacobian matrix 157–159
Cartesian/polar coordinates conversion 150
coordinate system transformation 150
Jacobian matrix 159
Cartesian/spherical coordinates conversion 151–154
coordinate system transformation 151–154
Jacobian matrix 159–160
Central finite distance scheme 561–562
Central processing unit (CPU) 4
Chain rule 27–28
complex roots 196–197
recurring complex roots 197–198
recurring roots 196
Chebyshev differential equation 620
average flow velocity 333–334
Darcy friction factor 334–335
Euler equation 329–332
force balance approach 332–333
hydraulic diameter 335
Bernoulli equation 311–312
energy conservation 312
Euler equation 311–313
mass conservation 309
momentum conservation 310
SOR implementation in 601–603
SOR solver in 605f
Classical LU-decomposition 525
Closed integral 147
Colebrook-White equation 335
Compactness factor C 363–367
Compressible continuity equation 303–306
Compressible fluids 316–317
Computational fluid dynamics (CFD) 609
Concentration jump 45
Conservative form notation 633–634
derivation 634
integration 635
interpretation 636
right-hand side terms 635–636
volume-averaged conservative form 636
hysteresis 430
microscope 430
static 430
Continuity equation 309
flowrate 268–269
flow velocity 268
geometry 268
mass and volume flow 268
Continuum hypothesis 246–247
continuum hypothesis 246–247
Lennard-Jones potential 245–246
dense and dilute gases 247–248
gas molecules 247
water molecules 247
Convection constant, pressure for 717
Taylor-Aris dispersion 401–405
ratio test 51
root test 51
Convergent series 51
Convolution property 82
Cartesian coordinates 303
cylindrical coordinates 304–305
polar coordinates 305–306
spherical coordinates 306–308
Cartesian/cylindrical coordinates conversion 148–150
Cartesian/polar coordinates conversion 150
Cartesian/spherical coordinates conversion 151–154
Correction factor α 362–363
elliptical cross-sections 362–364
Cosine function 31
in one dimension 72–74
in two dimensions 74
derivation 318
flow rate 318
shear stress profile 318
Critical micelle temperature (CMT) 432
Critical surface tension 463
Cross-section hydraulic resistance 361–362
cylindrical coordinates 163–166
spherical coordinates 176–181
Curl-free vector fields 140
curl 163–166
divergence 161–163
gradient 160–161
material derivative 168–170
scalar Laplacian 166
vector Laplacian 166–167
D
Dampening property 86
Darcy friction factor 334–335
Darcy-Weisbach equation 334
Debye and Keesom interaction 463
Decelerating Poiseuille flow 372
de/ds method 492
Fourier series of 41–42
Fourier transformation of 43
mathematical treatment 40–41
shifting 42–43
two-dimensional 43
usage 40
Denominator factor 716
Dense gases 247–248
Density 98
Dependent variable 22
Detergents 441–442
chain rule 27–28
integration by parts 28
product rule 27
Differential equation 22–24
classification 194
diffusion equation 190
Fourier transform 224–230
for Hagen-Poiseuille flow 615f
harmonic oscillation ODE, Laplace transform for 200–204
heat conduction 190
Helmholtz equation 194
Laplace equation 193–194
Laplace transform 230–235
linear differential equations with nonconstant coefficients 198–200
ordinary differential equations, solutions to 194–198
partial differential equations, solutions to 205–213
Poisson equation 192
time-dependent diffusion equation, in three dimensions 238–240
transport equation 189
two-dimensional partial differential equations 235–238
variables approaches, substitution and separation of 214–219
wave equation 190–192
Differentials 25–26
Differentiation property 82
convection and 134
Fick's second law 134
Diffusion equation 190
Digital microfluidics 441
Dilatant fluids 250–251
Dilute gases 247–248
Dilution factor 247
for diffusion and convection 289
dimensionless Navier-Stokes equation 284–285
Froude number 288
Reynolds number 285–288
scaled variables 283–284
Dimensionless analysis 405
Dimensionless convection-diffusion equation 289
Dimensionless Navier-Stokes equation 284–285
aspect ratio 263
capillary number 261–262
Eckert number 259
Eötvös/Bond number 261
Froude number 262
Knudsen number 257–259
Lewis number 260
Mach number 257
Marangoni number 262
Ohnesorge number 262
Péclet number 260–261
Prandtl number 259
Reynolds number 261
Schmidt number 259–260
Weber number 262
accelerating flow 378
decelerating flow 378
visualization 378–379
Dimensionless volume forces 288
Discontinuities 66
Discretization 636–637
finite volume method 636–637
relation 474–475
Dissipation coefficient 300
cylindrical coordinates 161–163
spherical coordinates 173–176
Divergence-free vector fields 140
Divergent Navier-Stokes equation 705–707
Dominating effects 243
Dot product 138
Droplet microfluidics 441
Drop-shape analysis 492–493
Duality theorem 82
du Noüy ring 453
Dynamically linked library (DLL) 20
Dynamic contact angle measurement 430–431
Dynamic surface tension 457
Dynamic viscosity 249–250
E
Eckert number 259
Educt 97
Eigenfunctions 211–212
Electron 93
assignment 9
built-in functions 10
command termination 9
comments 11
equations 10
functions 10
plotting 11
vectors and matrices 10–11
working with units 13
analytical solution 323–324
flow rate for circular cross-sections 325
flow rate for elliptical cross-sections 324–325
velocity profile for circular cross-sections 325
velocity profile for elliptical cross-sections 324
visualization 325–326
balance of contributions 296–300
by convection 292–293
by conduction 293–294
by volume effects 295
state, thermodynamic equation of 300–301
by boundary forces 294–295
by volume forces 295–296
modified 38–39
roots of 39
standard 35–38
curvature 48–49
denominators containing sums 48
infinitesimal changes 49–50
variable and derivative 48
delta function 39–43
boiled egg 23
boundary conditions (BC) 29–30
degree of 23
differential calculus, important rules in 27–28
differential equations, systems of 30
differentials 25–27
functions 21–22
homogeneous and inhomogeneous 24
initial values 30
obtaining 24
ordinary and partial 24–25
solving 24
error function 47
heaviside function 43–46
signum function 46–47
conversion from cosine 34
conversion from cotangent 34
conversion from sine 34
conversion from tangent 34
conversions for half-angles 34
Euler's formula 35
inverse functions 33
Pythagorean theorem 33
cosine 31
derivatives 32–33
hyperbolic cosine 32
hyperbolic sine 32
hyperbolic tangent 32
sine 31
tangent 32
interpretation 398
Navier-Stokes equation 394–395
separation variables 395–397
substitution variables 395–397
transient solution 397–398
Eötvös/Bond number 261
Equivalent circuit theory 367–370
parallel connection 369–370
serial connection 369
Equivalent molar concentration 98
Error function 47
Essentially non-oscillatory (ENO) reconstruction 640
Euler's formula 35
Even expansion 230
Exponential function 76
Extended Fowkes model 462
F
Fast numerical solver 498–499
Fast radial diffusion 407
Field variables 265
buildestimation 14–17
extracttoplot 14
includes 14
plotode 17
quickplot 14
quickplot3d 14
in Microsoft Excel 624–631
alternative use of the spreadsheet 629–631
boundary conditions 627–629
derivation 624–625
different profiles 629
numerical scheme implementation 626–627
spreadsheet preparation 625–626
Finite-difference scheme 663
approximation functions in 659f
basis functions 658–660
pyramid functions 659
Finite element method (FEM)-Galerkin method 659–660
discretization 636–637
Flow channel geometries 359–361
Flow rate 627
Flow tubes 268
contact angle 425–427
temperature dependency 424
Young-Laplace pressure 427–430
Fluid friction 248
characteristic breakup time 476
standing waves on 475–476
boundary conditions 255–257
control volumes 245–248
critical shear stress 243
dimensionless numbers 257–264
dominating effects 243
heat transport 254–255
mass transport 255
momentum transport 248–256
properties 248
scaling law 243
gases 244
liquids 244
phase changes 244–245
plasmas 245
solids 243–244
Fluorinated anionic surfactants 435
Fluorinated surfactants 435
Fluorination 424
analytical solution 333
derivation 332–333
Formula translating system (FORTRAN) 492
Forward Euler scheme 680–681
Forward finite distance approximation 60
Fourier expansion 68–77
cosine series 72–74
sine series 69–72
Fourier series 374–375
Bessel series 77–80
building 63–68
delta function 41–42
heaviside function 45–46
nonperiodic functions 68–77
heat flows, balance of 127
instationary heat conduction 128
stationary heat conduction 127
in three dimensions 126–127
definition 81
one-dimensional diffusion equation 224–230
pairs 84
properties 82–83
application 85
delta function 43
heaviside function 46
free surface energy 462
OCG model 463
Free surface boundary condition 256
Free-surface condition 29
estimation of 422–423
extended Fowkes model 462
Fowkes model 460–462
low and high 422
Marangoni effect 442
OCG model 463
OWRK model 462
Wu model 462
Friction forces 310
Frictionless flow field 277–278
Full pivoting 515
Fully developed flow profile 266
Function approximation methods 590–593
piecewise-constant profile 639–640
piecewise-linear profile 640–641
piecewise-parabolic profile 641–642
WENO schemes 642–644
Fused deposition modeling (FDM) 624
G
Galerkin method 609
algorithmic implementation 619
example 612–614
multiple independent variables 615–618
systems of differential equations 618–619
weighting function 611
Gas pressure 102–103
Gauβ-Jordan elimination 501–503
Gauβ-Jordan inversion 505–507
Gauβ-Seidel method 512–513
Geometric hydraulic resistance 361
for electrochemical reactions 118
interpretation of 120–121
pressure dependency of 123–124
temperature dependency of 121–122
Gibbs-Helmholtz equation 122
Gibbs phenomenon 64–65
Global positioning system (GPS) 538
Global truncation error 564
cylindrical coordinates 160–161
spherical coordinates 171–173
Gravity-driven flow 318–320
Gravity forces 310
Green's function 227–228
H
Hagen-Poiseuille flow 78, 320, 329–335, 329f
, 371, 372f
, 402, 409, 414, 605, 606f
, 612, 615f
, 629, 687
Heat conduction equation 190
Heat transfer 124–125
Heaviside function 228–229
Fourier series of 45–46
Fourier transformation 46
usage 45
Helmholtz equation 194
Hess's Law 111
High-level languages 601
Homogeneous differential equations 24
Homogeneous solution 195–196
Hydraulic diameter 335
Hydraulic resistance 368–369
equivalent circuit theory 367–370
flow channel geometries 359–361
simplification approaches 361–367
viscous dissipation 351–358
Hydrophilic surface 425–426
Hydrostatics 315–317
Hyperbolic sine function 32
Hyperbolic tangent function 32
I
Ideal gas 110
definition 100–101
gas pressure 102–103
kinetic energy 103
Maxwell speed distribution 106–108
mean free path 101–102
one-dimensional velocity probability distribution 104–106
Idealized thermodynamic processes 108
Immiscible flow 429
Incompressible continuity equation 303–306
Incompressible flow 266
Incompressible fluids 315–316
Incompressible Newtonian fluids 280–282
Independent variable 21–22
Infinitesimal negative change 59
Infinitesimal positive change 59
Inhomogeneous differential equations 24
Inhomogeneous partial differential equations 239
Inhomogeneous solution 196
Initial boundary value problem (IBVP) 30
Initial value problem (IVP) 30
Initial values (IV) 30
Initiating Poiseuille flows 372
Inkjet printing 4
Integrated circuit technology 4
Integrated development environment (IDE) 20
Integrated gas chromatography 4
Integration, by parts 28
Integration factor 198–199
Interface stabilization 441
Interpretation 398
Inverse functions 33
Ion 93
Ionic surfactants 432
Ionization energy 93
J
Cartesian/cylindrical coordinates conversion 157–159
Cartesian/polar coordinates conversion 159
Cartesian/spherical coordinates conversion 159–160
definition and usage 157
Jacobi method 508–512
iterative scheme 508–510
numerical stability 511–512
special matrices 508–509
Joule-Lenz heating 368
K
Killer application 5
Kinematic boundary condition 473
Kinematic viscosity 251–252
Knudsen number 257–259
L
Laminar Hagen-Poiseuille flow 334
Langmuir–Blodgett film 433
Laplace constant, for velocities 717
application 91–92
definition 85
for harmonic oscillation ODE 200–204
pairs 87–91
properties 86–87
Leibniz integral rule 91
Length scales 3
Lewis number 260
Lifshitz/Van der Waals contribution 463
constant coefficients 195–198
nonconstant coefficients 198–200
Gauβ-Jordan elimination 501–503
Gauβ-Jordan inversion 505–507
Gauβ-Seidel method 512–513
Jacobi method 508–512
LU-decomposition 517–525
sequential dense LU-decomposition 525–530
SOR 513–517
substitution 530–531
tridiagonal matrix algorithm (Thomas algorithm) 531–534
Liquid/gas interface 441
capillarity measurement 459
du Nouy ring 453
pendant drop analysis 455–456
Local truncation error 564
Loschmidt number 101
Low free surface energy 422
Low-level languages 601
LU-decomposition 517–518
calculation pattern 519–520
derivation 518–519
performing 599
pivoting 523–524
procedure 520–521
sequential dense 596–598
substitution 522–523
M
Mach number 257
matrix evaluation 543
matrix/vector multiplication 542–543
vector evaluation 543–544
Galerkin method 614–615
one-dimensional example 665–666
two-dimensional example 672–677
tears of wine 443
temperature 442
Marangoni number 262
Mass concentration 97
Mass concentration function 402–403
continuity equation 268–269
convection-diffusion equation 270–271
flowrate, integral representation 269
fluid flow, in bulk 265–268
Gauss's theorem 270
mass balance 269–270
Mass fraction 98
Mass transfer 125
Mass transport 255
Material derivative 304
Matrix 10
Matrix notation 138
Maximum bubble pressure method 456–457
mean free velocity 107
partial pressure 108
viscosity 107
Mean free path 101–102
Mean square velocity 103
Meniscus depth 447
contours 450–451
linearization 448–450
surface curvature 448
Micelle formation 432–434
reaction compartments 441
Microelectromechanical system (MEMS) 3
analytical applications 4–5
commercial aspects 5
definition 3
inkjet printing 4
integrated circuit technology 4
microfluidics today 5
Microoptoelectromechanical system (MOEMS) 3
Microsoft Excel, finite difference method (FDM) in 624–631
Miniaturized total analysis systems (μTAS) 4
Minimal surfaces 429
Modified Bessel functions 38–39
Modulation property 82
Molality 98
Molar concentration 97
Molar volume 101
Molecular weights 96
Mole fraction 98
balance of 276–280
control volume, transfer into and out of 273
dimensional analysis 283–289
Navier-Stokes equation 280–282
by volume forces 276
Bingham fluids 250
Dilatant fluids 250–251
dynamic viscosity 249–250
kinematic viscosity 251–252
Newtonian fluids 250
plastic fluids 251
pseudo-plastic fluids 251
time-dependent viscosity 251
viscosimeters 252–254
N
Navier-Stokes equation 24, 303–308, 310, 312, 338–339, 352, 371, 373, 379–380, 471, 498, 609, 679–680, 693–694, 701, 705–707
aerostatics 315–317
for circular cross-section 585–586
gravity-driven flow 318–320
hydrostatics 315–317
Poiseuille flows 320–321
pressure-driven flow 320–321
shear force-driven flow 317–318
Lagrangian frame of reference 282
Nabla operator 281–282
compiler options 20
linker options 20
Neutron 93
Newtonian movement 128
derivation 540
finite difference 540–541
iterative scheme 542
listing for 544–546
nonlinear systems 537
quasi-Newton method 541
Non-ionic surfactants 438–440
Nonperiodic boundary condition 619–620
Fourier expansion 68–77
period length 68
Nonzero Dirichlet boundary conditions 628
Normalized capillary heights 446–447
Normalized hydraulic resistance 359–361
Adams–Bashforth methods 562–568
Adams–Moulton methods 568–572
backward Euler method 555–559
central finite distance scheme 561–562
coupled ordinary differential equations 575–585
Crank-Nicolson method 560–561
Euler's method 549–555
forward and backward finite distance 559–560
function approximation methods 590–593
higher-order ordinary differential equations 575–585
infinitesimal changes 549
ordinary differential equations with boundary conditions 585–590
Runge-Kutta methods 572–575
advantages and disadvantages 497–498
concepts in 498–499
historical overview 499–500
linear systems 500–534
Newton's method 568–570
Numerical solver 725–743
O
Odd expansion 229
Ohm's law 368
Oleophilic surface 425–426
One-dimensional diffusion 190
general notation 667–668
hat functions 661
integration 661–662
matrix notation 662–665
problem definition 661
One-dimensional heat conduction 190
One-dimensional velocity probability distribution 104–106
cross product 140
Laplace 140
material derivative 142
nabla 139
scalar Laplacian 141
scalar multiplication 139–140
vector field potentials 142
vector Laplacian 141
via curl and divergence 141–142
vector multiplication 140
constant coefficients 195–198
nonconstant coefficients 198–200
variables and partial integration, separation of 194–195
Ostwald approximation 250
Owens, Wendt, Rabel, and Kaelble (OWRK) model 462
P
Parasitic solutions 555
Parseval's theorem 82
Particular solution 195–196
Péclet number 260–261
Pendant drop analysis 465
Perfect gas 110
Periodic boundary condition 619
Periodicity 71
element 93–94
group 95
nuclide 95
period 95
Period length 68
Permutating LU-decomposition (PLU) 523
Phase changes 244–245
Phospholipids 432
derivation 326–327
flow rate 328
shear stress profile 328
velocity profile 327
visualization 328–329
Plasmas 245
Plastic fluids 251
characteristic breakup time 476
fluid jets 468–470
stability considerations 467
Point notation 137
Reynolds numbers in 286–287
circular cross-sections 329–335
elliptical and circular profiles 323–326
planar infinitesimally extended channel cross-sections 326–329
rectangular cross-sections 335–348
sequential dense LU-decomposition 596–598
Poisson equation using SOR 600–606
calling SolvePoissonSOR from Maple 603–604
Polysorbate surfactants 440
Pressure 256–257
convection constant for 717
volume force constant for 717
Pressure constant, for pressure 717
Pressure dependency 111
Pressure-driven flow 320–321
Pressure forces 310
in velocities 716
Pressure, in bubbles 430
Pressure-Poisson equation 710–715
artificial compressibility method 702–703
divergent Navier-Stokes equation 705–707
problem with incompressible fluids 702
SIMPLE algorithm 703–705
Product rule 27
Proton 93
Pseudo-plastic fluids 251
Pyranoside surfactant 440
Pythagorean theorem 33
Q
Quasi-Newton method 541
R
Radial average mass concentration 403
Radial diffusion 407
Ratio test 51
Reactant 97
Reaction enthalpy 110–111
Receding contact angle 430–431
Reconstruction stencil 640
Rectangular cross-sections 385–393
flow rate 344–348
Naive approach 336
single λ approach 336–340
substitution and separation of variables 336
two λ approach 340–342
Reversible processes 108
Reynolds' transport theorem 145–146
Riemann zeta function 52
Robust numerical solver 498–499
Root-mean-square position 237–238
Root test 51
Rule of Sarrus 504
S
Saponification 434–435
cylindrical coordinates 166
operators 141
spherical coordinates 181
Scalar product 138
Scalar quantities 137
operators 139–142
vector operations 138
Scaled lengths 284
Scaled operators 284
Scaled pressure 283–284
Scaled time 284
Scaled velocity 283
Scaling law 243
Separation variables 395–397
derivation 526–528
pivoting 529–530
substitution 530–531
convergence 51
definition 51
Fourier series 62–80
Taylor series 53–61
derivation 481–482
numerical solution 482–486
numerics 493
Shear force-driven flow 317–318
Shear stress 256
Shift property 82
boundary value 586–587
implementation 587–588
pendant drop revisited 588–590
rationale 587
speeding up 590
Signum function 46–47
normalized 84
unnormalized 84
Sine function 31
in one dimension 69–70
in two dimensions 70–72
Single λ approach 336–340
Sodium dodecyl sulfate polyacrylamide gel electrophoresis (SDS-PAGE) 435
coordinate system transformation 405
fast radial diffusion assumption 407
time-dependency 405–407
Source-free vector fields 140
Spectral methods 619
example 621
nonperiodic boundary condition 619–620
periodic boundary condition 619
curl 176–181
divergence 173–176
gradient 171–173
material derivative 183–188
scalar Laplacian 181
vector Laplacian 181–183
Spherical coordinate system 147
Spinning-drop method 457–459
Square wave function 63–65
Stabilization, of suspensions 440
Stable drops 490–491
Stable numerical solver 498–499
Stalagmometer 455
Standard formation enthalpy 111
Static contact angle measurement 430
boundary values 715–716
constants 716–717
implementation 718–725
structures 717–718
Stencils 640
WENO 643–644
Stoichiometry 97
Stoke' drag law 132
Stokes drag 253
Stokes' friction approach 278–280
Stoke's theorem 144
Substitution variables 395–397
Successive over-relaxation (SOR) 513–517
constants 716
Poisson equation using 600–606
disadvantages 601
Sugar-based surfactants 438–440
Surface integrals 645–646
contact angle measurement 430–431
fluid effects, at interfaces 421–430
at liquids 423
Marangoni effect 442–443
at solids 421
surfactants 431–442
capillarity measurement 459
du Noüy ring 453
free surface energy 459–464
maximum bubble pressure method 456–457
pendant drop analysis 455–456
spinning-drop method 457–459
Marangoni effect 442–443
T
Tangent function 32
Tate's law 455
convection-diffusion equation 401–405
on moving fluid plug 401–402
on static fluid plug 401
example 409–416
mass concentration function 402–403
solution validity 409
solving for p 405–407
building 53–56
exponential function 54–56
rationale 54
sine function 56
central finite difference 59
forward and backward finite difference 60
implications 61
infinitesimal negative change 59
infinitesimal positive change 59
second derivative 60
finite changes as 58–61
Taylor series expansion 54–56
Temperature 256
Temperature dependency 111
Tensiometric method 453
Thermodynamic potential 120
Thermodynamic process 99
Thermodynamics 99
control volume 99
density 98
electrons 93
first law 109–112
heat and mass transfer 124–134
definition 100–101
gas pressure 102–103
kinetic energy 103
Maxwell speed distribution 106–108
mean free path 101–102
one-dimensional velocity probability distribution 104–106
idealized thermodynamic processes 108
neutrons 93
periodic table 93–95
protons 93
second law 112–124
third law 124
weights and concentrations 96–98
Thermodynamics state variables 99–100
Thermodynamic state 99
derivation 701–715
numerical solver 725–743
pressure recovery 702–707
stationary flow numerical solver implementation 715–725
Three-dimensional grids 658
Three-dimensional numerical solver 725–743
modifications 741–743
Three-dimensional point source diffusion 236
Time and space, two-dimensional problems of first order in 648–652
Time-dependent Couette flow 373–379
accelerating flow 374
decelerating flow 377
dimensionless velocity profiles 377–379
transient solution 381–382
Time-dependent diffusion equation, in three dimensions 238–240
Time-dependent Hagen-Poiseuille flow 379–385
boundary conditions 381–382
derivation 381
Navier-Stokes equation 379–380
transient and steady-state solutions 380–381
Time-dependent transient effects 371–373
Navier-Stokes equation 371
steady-state solutions 371–372
Time-dependent transient flows 679
Time-dependent transient solution 372
Time-dependent viscosity 251
Time-independent steady-state solution 402
Total differential 26
Fourier transform 81–85
Laplace transform 85–92
Transient and steady-state solutions 380–381
entrance flow 393–399
rectangular cross-sections 385–393
time-dependent Couette flow 373–379
time-dependent Hagen-Poiseuille flow 379–385
time-dependent transient effects 371–373
Transient flow, time-dependent 679
Transient solution 427–428
characteristic lines 205
solution 206
Triangular-like function 65–66
derivation 531–533
numerical stability 534
conversion from cosine 34
conversion from cotangent 34
conversion from sine 34
conversion from tangent 34
conversions for half-angles 34
Euler's formula 35
inverse functions 33
Pythagorean theorem 33
cosine 31
derivatives 32–33
hyperbolic cosine 32
hyperbolic sine 32
hyperbolic tangent 32
sine 31
tangent 32
Truncation error function 54
Two-dimensional entrance flow problems 687–699
considering convection 693–699
higher Reynolds numbers 695–699
implementation 695
Navier-Stokes equation 693–694
neglecting convection 688–693
numerical scheme 694–695
derivation 669–672
problem definition 668
using the worksheet to solve the Poisson equation 677–678
Two-dimensional partial differential equations 235–238
Two-dimensional point source diffusion 236
Two-dimensional problems of first order, in time and second order in space 652–654
comment on the grid 654
flux approximations 652–653
numerical scheme 653–654
regular axes-incident grid 653
Two-dimensional problems of first order, in time and space 648–652
Green's theorem 649–650
integration 649
linearization 650–651
numerical scheme 652
regular axes-incident grid 651
Two-dimensional time-dependent flow problems 680–687
Backward Euler scheme 681
Forward Euler scheme 680–681
grid preparation 681–682
implementation 682–684
time-dependent transient flows in circular channel profiles 685–687
time-dependent transient flows in rectangular channel profiles 684–685
Two λ approach 340–342
U
Unbalanced intermolecular forces 421
Uncharged surfactants 431
Uneven atom mass 97
V
van Oss, Chaudhury, and Good (OCG) model 463
Van't Hoff equation 123–124
acceleration 154–157
coordinate system transformation 147–154
cylindrical coordinates 160–170
Jacobian matrix 157–160
polar coordinates 170
position 154–157
scalars and vectors 137–142
spherical coordinates 171–188
theorems in 143–147
velocity 154–157
Vector Laplacian 304
cylindrical coordinates 166–167
spherical coordinates 181–183
Vector notation 137
cross product 139
norm 138
scalar multiplication 138
vector multiplication 138–139
Vector quantities 137
Viscous dissipation 351–358
Gauss's theorem 356–357
kinetic energy 352
in microfluidic channel 356
Navier-Stokes equation 352
Visualization, Taylor-Aris dispersion 414
for pressure 717
for velocity 717
W
force balance 191
Newton's second law of motion 191–192
one-dimensional 192
three-dimensional 192
variables, substitution and separation of 219–224
Weber function 35
Weber number 262
reconstruction 640
schemes 642–644
stencils 643–644
approximation function 611
differential operator 610
linear independent functions 610
weighting function 611
Wu model 462
Y
Young-Laplace pressure 448
Z
Zwitterionic surfactants 431
miscellaneous 437
phospholipids 438
quaternary ammonium 436–437
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