Table Of ContentMICROFLUIDICS:
MODELING,
MECHANICS, AND
MATHEMATICS
MICROFLUIDICS:
MODELING,
MECHANICS, AND
MATHEMATICS
BASTIAN E. RAPP
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Preface
Many people have not heard about microfluidics. This is somewhat astonishing as microfluidics surrounds us.
Microfluidicsisthescientificdisciplinesthatstudiesandemploysthefluidphysicsatthemicro-andnanometer
scale. Fluids do interesting things when probed in such miniaturized systems. Capillarity is one of these effects
wearefamiliarwith.Ifyoueverdippedatissuepaperinwaterandwatchedthewaterpenetrateityoumayhave
wondered what forces drive it. This is microfluidics. However, nowadays, there is an ample number of technical
systems which make use of microfluidics ranging from analytical devices in biomedical engineering and the life
sciences all the way to systems such as inkjet printers.
Microfluidics shares many of its fundamentals with classical fluid mechanics. Many students struggled with
fluid mechanics during their studies because it is a discipline that requires a quite sound understanding of
engineeringmathematics.Asaconsequence,students(aswellasestablishedscientists)oftentakeratherheuristic
approacheswhendesigningmicrofluidicsystems.Asystemischaracterizedandoptimizedsolelyonexperimental
data. Very often, the findings are published omitting pretty much all fluid mechanical fundamentals and (very
often) without any theoretical model. Equations are often copied from textbooks and simply applied without
real understanding of what the equations actually describe. Whenever a more complex fluidic system is to be
designedmanyresearchersdefaulttonumericalsoftwarepackagesforprovidinganassessmentofthesystemprior
to device manufacturing. Very often, these extensive calculations are not necessary as there would have been a
very simple theoretical model which would have been sufficient to understand the fluid mechanics of the system.
This book is intended for all students and researchers who want to understand the fundamentals of fluid me-
chanics.Thebookdoesnotsimplystatethemostimportantequations,itderivesthem.Ibelievethatbyproviding
thederivationofanequation,itwillbesignificantlysimplertounderstanditsmeaninganditsapplicability.The
reader will require very little prior knowledge when starting with this book. All mathematical concepts, tricks
and methods used will be introduced and explained. The book will elucidate analytical techniques as well as
numerical methods. It serves as a practical coursebook for those who want to deepen their knowledge, e.g., of
numerical methods as well as for those who are already very experienced in fluid mechanics but simply want
to understand where a specific formula (such as, e.g., the velocity distribution in a given channel cross-section)
actually comes from.
The book comes with a number of worksheets written in Maple. These worksheets serve as templates to
experiment with the equations derived and visually check which changes in the fluid mechanics are induced by
changing certain variables. They can be used as basis for theoretical assessment of many microfluidic systems.
This book contains many listings written in C which elucidate the numerical methods commonly used in fluid
mechanics and microfluidics. In the supplementary you will also find a compiled DLL with exports functions
thatcanbeusedtosolvethree-dimensionalflowproblemsnumerically.Thesefunctionsaredevelopedwithinthis
booktogetherwillallnecessarynumericalfundamentals.Byfollowingthebookchapter-by-chapteryouwillgain
a very detailed understanding of engineering mechanics, fluid mechanics and numerical methods.
Have fun exploring the fluid mechanics of microfluidics!
Bastian E. Rapp
November 2016
xxix
Na gut Kollegen!
Justus Jonas
Acknowledgement
As anyone who ever wrote a book will agree on, there are many people that help bringing the book to life.
The first and foremost thanks goes out to all the members of my group, the NeptunLab who supported me
duringthemonthsittooktofinishthisbook.Mysincerethanksgoesouttomymentorandlong-timesupporter
Volker Saile for all his help and encouragement.
A warm thank you goes out to my dear friend and colleague Matthias Worgull who first encouraged me
to write this book and then helped me sticking to this decision. I also want to thank Mohammad for all the
Flammkuchen evenings and philosophical discussions.
I would also like to thank my publisher Elsevier and, especially, Simon Holt for his support and his patience
despite the fact that (to quote Dave Barry) “the deadline is months over and he has still not received what the
publishing industry generally refers to as ’the book’”.
Most importantly, I want to thank my wife Emily and my family for their everlasting love and support. For
mybrotherHolger,Ihavehiddensomethinginthisbookwhichyouaresupposedtopickupattheverybeginning
because it will make your life so much easier at the very end.
Bastian E. Rapp
xxxi
List of Figures
1.1 Number of papers and patents published in the field of microfluidics since 1990 . . . . . 5
2.1 The function f(x)=x2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.1 A barrier lake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 Graphical representation of differentials and derivatives . . . . . . . . . . . . . . . . . . 26
3.3 Boundary conditions and initial values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4 Important trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.5 The right triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.6 Bessel functions of first kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.7 Gamma function Γ(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.8 Bessel functions of second kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.9 Approximation of the delta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.10 Soluble solid as an example for a delta function . . . . . . . . . . . . . . . . . . . . . . . 41
3.11 Fourier series of the delta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.12 Shifted delta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.13 Fourier series of a two-dimensional delta function . . . . . . . . . . . . . . . . . . . . . . 44
3.14 Soluble solid as an example for a Heaviside function . . . . . . . . . . . . . . . . . . . . 45
3.15 Fourier series of the Heaviside function . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.16 Error and complementary error function . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.17 Curvature of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.18 Derivation of the curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.1 The Riemann zeta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2 Taylor series expansion of the exponential function . . . . . . . . . . . . . . . . . . . . . 55
4.3 Taylor series expansion in wrong interval . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.4 Taylor series expansion of the exponential function around a=−3 . . . . . . . . . . . . 56
4.5 Taylor series expansion of the sine function . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.6 Graphical explanation of the Fourier series weighting factors . . . . . . . . . . . . . . . 63
4.7 Fourier series expansion of the square wave function . . . . . . . . . . . . . . . . . . . . 65
4.8 Fourier series expansion of the triangular-like function . . . . . . . . . . . . . . . . . . . 66
4.9 Approximating a function by a Fourier sine series. . . . . . . . . . . . . . . . . . . . . . 68
4.10 Fourier sine series expansion of the constant function. . . . . . . . . . . . . . . . . . . . 70
4.11 One-dimensional Fourier expansion interpreted in two dimensions. . . . . . . . . . . . . 71
4.12 Fourier expansion of a constant to a sine series in two dimensions. . . . . . . . . . . . . 72
4.13 Fourier series expansion of the constant function . . . . . . . . . . . . . . . . . . . . . . 73
4.14 Fourier expansion of a constant to a cosine series in two dimensions . . . . . . . . . . . 74
4.15 Fourier expansion of the exponential function . . . . . . . . . . . . . . . . . . . . . . . . 76
4.16 Exponential function expanded to a purely sine or cosine series . . . . . . . . . . . . . . 76
4.17 Example of a Fourier-Bessel series expansion . . . . . . . . . . . . . . . . . . . . . . . . 79
5.1 Normalized and unnormalized sinc function . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.1 Periodic table of the elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.2 Thermodynamic control volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.3 Maxwell speed distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.4 Reversible and irreversible processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
xxxiii
xxxiv List of Figures
6.5 Fourier’s law of heat conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.6 Visualization of diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.7 Output of the digital diffusion experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.8 Derivation of the conservation of mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.9 Diffusion times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
7.1 Forming the cross product of two vectors . . . . . . . . . . . . . . . . . . . . . . . . . . 139
7.2 The theorems of Gauß, Stokes and Green . . . . . . . . . . . . . . . . . . . . . . . . . . 143
7.3 Control volume used for the Reynold’s transport theorem . . . . . . . . . . . . . . . . . 145
7.4 Common coordinate systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
8.1 Derivation of the wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
8.2 Sine wave function as a solution to the transport equation . . . . . . . . . . . . . . . . . 206
8.3 Pulse function as a solution to the transport equation . . . . . . . . . . . . . . . . . . . 207
8.4 Two waves colliding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
8.5 Analytical solution of the heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
8.6 Heat conduction examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
8.7 Half-wave sine function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
8.8 Half-wave sine function with overtone . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
8.9 Limited point source diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
8.10 Limited plane diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
8.11 Limited point source diffusion with boundary condition . . . . . . . . . . . . . . . . . . 230
8.12 Laplace transform applied to solving the wave equation on the semi-infinite string . . . 231
8.13 Solution to the one-dimensional wave equation found using the Laplace transform . . . 233
8.14 Solution to the one-dimensional diffusion equation found using the Laplace transform . 234
8.15 Limited point source diffusion in two dimensions . . . . . . . . . . . . . . . . . . . . . . 237
9.1 Shear stress on solids and liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
9.2 Solids, liquids, and gases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
9.3 Lennard-Jones potential given as a function of the distance of two rigid particles . . . . 245
9.4 Momentum transport in fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
9.5 Thioxotropic and rheopexic fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
9.6 Momentum transport in water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
9.7 Measurement principles of viscosimeters . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
9.8 Setup of the Ostwald viscosimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
9.9 Momentum transport and heat conduction . . . . . . . . . . . . . . . . . . . . . . . . . 255
9.10 Slip and no-slip boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
10.1 Eulerian and Lagrangian frames of reference . . . . . . . . . . . . . . . . . . . . . . . . 266
10.2 Particle trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
10.3 Continuity equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
11.1 Momentum in-/outflux via mass flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
11.2 Momentum introduced by shear forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
11.3 Laminar and turbulent flow fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
11.4 Reynolds’ dye flow experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
11.5 Visualization of the Froude number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
12.1 Energy in- and outflux by convection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
12.2 Energy in- and outflux by conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
12.3 Work created by normal and shear forces . . . . . . . . . . . . . . . . . . . . . . . . . . 295
14.1 The fluid mechanics of the flow tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
15.1 Hydrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
15.2 Atmospheric pressure drop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
15.3 Couette flow in a slit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
List of Figures xxxv
15.4 Fluid flow under gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
15.5 Flow profiles for gravity driven flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320
15.6 Microfluidic channel with arbitrary cross-section . . . . . . . . . . . . . . . . . . . . . . 320
16.1 Poiseuille flow in elliptical cross-section . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
16.2 Calculated Poiseuille flow profiles for elliptical and circular channel cross-sections . . . . 326
16.3 Poiseuille flow in a planar infinitesimally extended channel . . . . . . . . . . . . . . . . 327
16.4 Velocity and shear force profile in a channel of 10µm height . . . . . . . . . . . . . . . . 328
16.5 Velocity and shear force profile in a channel of 50µm height . . . . . . . . . . . . . . . . 329
16.6 Hagen-Poiseuille flow in a circular tube . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
16.7 Calculated tube flow profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
16.8 Calculated shear stress profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
16.9 Alternative derivation of the Hagen-Poiseuille profile . . . . . . . . . . . . . . . . . . . . 333
16.10 Coordinate systems for Poiseuille flow in rectangular channels . . . . . . . . . . . . . . . 337
16.11 Flow profiles in rectangular channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342
16.12 Normalized flow profiles in rectangular channel cross-sections . . . . . . . . . . . . . . . 344
16.13 Approximations for the flow profiles in rectangular channel cross-sections . . . . . . . . 345
16.14 Simplified normalized velocity profiles for rectangular cross-sections . . . . . . . . . . . 346
16.15 Simplification errors for the flow rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
17.1 Viscous dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
17.2 Normalized hydraulic resistance in channels with elliptical and circular cross-sections
calculated for water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
17.3 Pressure drop in infinitesimally extended parallel channels calculated for water . . . . . 360
17.4 Pressure drop in channels with rectangular and square cross-sections calculated
for water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
17.5 Correction factor α over compactness factor C for elliptical cross-sections . . . . . . . . 364
17.6 Correction factor α over compactness factor C for planar infinitesimally extended
cross-sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
17.7 Correction factor α over compactness factor C for rectangular cross-sections . . . . . . . 367
17.8 Analogy between hydraulic and electrical resistance . . . . . . . . . . . . . . . . . . . . 369
18.1 Accelerating and decelerating flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
18.2 Fourier series expansion of the steady-state solution . . . . . . . . . . . . . . . . . . . . 376
18.3 Accelerating Couette flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
18.4 Decelerating Couette flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378
18.5 Dimensionless accelerating and decelerating Couette flow . . . . . . . . . . . . . . . . . 378
18.6 Dimensionless accelerating and decelerating Hagen-Poiseuille flow. . . . . . . . . . . . . 384
18.7 Accelerating and decelerating Hagen-Poiseuille flow in a capillary with radius 1mm . . 384
18.8 Dimensionless accelerating and decelerating flow velocity profiles in rectangular
cross-sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
18.9 Accelerating and decelerating flow in a rectangular channel . . . . . . . . . . . . . . . . 392
18.10 Entrance flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
18.11 Normalized velocity profiles of the Hagen-Poiseuille entrance flow . . . . . . . . . . . . . 398
19.1 Static plug in microfluidic channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401
19.2 Moving plug in a microfluidic channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402
19.3 Example of Taylor-Aris dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
19.4 Balanced rectangular function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
19.5 Fourier series of the balanced rectangular pulse function f (x) . . . . . . . . . . . 413
bal.rect.
19.6 Effective diffusion D as a function of the pressure drop . . . . . . . . . . . . . . . . . 414
eff.
19.7 Taylor-Aris dispersion at a channel intersection with circular cross-sections . . . . . . . 415
20.1 Surface tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
20.2 Estimating the free surface energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
20.3 Forces originating from surface tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424
xxxvi List of Figures
20.4 Photograph of a gerridae walking on a water surface . . . . . . . . . . . . . . . . . . . . 425
20.5 Contact angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
20.6 Young-Laplace equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
20.7 Minimal surfaces demonstrated at a soap ring . . . . . . . . . . . . . . . . . . . . . . . . 429
20.8 Fluid flow in a tapered channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430
20.9 Advancing and receding contact angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431
20.10 Principle structure of a surfactant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431
20.11 Bilayer and micelle formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432
20.12 Langmuir-Blodgett films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433
20.13 Saponification reaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434
20.14 Surfactants based on carboxylic acid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434
20.15 Surfactants based on sulfonic acid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
20.16 Cationic surfactants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436
20.17 Zwitterionic surfactants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
20.18 Non-ionic surfactants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
20.19 Stabilization of suspensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440
20.20 Marangoni effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442
20.21 Demonstration of the Maragoni effect using a surfactant . . . . . . . . . . . . . . . . . . 443
21.1 Capillary pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
21.2 Capillary heights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
21.3 Meniscus formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
21.4 Calculated meniscus shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
22.1 Wilhelmy plate method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454
22.2 Drop-weight method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455
22.3 Determination of the surface tension using the d /d . . . . . . . . . . . . . . . . . . . . 456
e s
22.4 Maximum bubble pressure method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456
22.5 The spinning drop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
22.6 Geometry of the spinning drop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
22.7 Example of a Zisman extrapolation on a perfluorinated polyether acrylate surface . . . 463
23.1 Falling fluid jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468
23.2 Reduction of radius on a falling fluid jet . . . . . . . . . . . . . . . . . . . . . . . . . . . 470
23.3 Plateau-Rayleigh instability on fluid jets . . . . . . . . . . . . . . . . . . . . . . . . . . . 471
23.4 Dispersion relation for the Plateau-Rayleigh instability. . . . . . . . . . . . . . . . . . . 475
23.5 Stationary perturbation on a fluid jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475
23.6 Characteristic breakup time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476
23.7 Typical values for the Ohnesorge numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 477
24.1 Cut view through a sessile drop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479
24.2 Contour of a sitting water drop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484
24.3 Numerically calculated drop contour of mercury drop on a glass surface . . . . . . . . . 485
24.4 Height convergence of sessile drops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485
24.5 Difficulties using θ as independent variable . . . . . . . . . . . . . . . . . . . . . . . . . 487
24.6 Accommodated contact angles at a curved capillary wall . . . . . . . . . . . . . . . . . . 488
24.7 Pendant drop of water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490
24.8 Pendant drop of mercury . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491
24.9 Discontinuity of pendant drop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491
24.10 Comparison of the physical drop contour with the numerically derived drop contours . . 492
26.1 Example of a nonlinear system: the LORAN system . . . . . . . . . . . . . . . . . . . . 539
26.2 Intersection of the two equidistance lines . . . . . . . . . . . . . . . . . . . . . . . . . . 539
27.1 Visualization of the Euler method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550
27.2 Example of using Euler’s method to approximate a function . . . . . . . . . . . . . . . . 552
