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
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
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 10 µm height 328
16.5 Velocity and shear force profile in a channel of 50 µm 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 
for elliptical cross-sections 364
17.6 Correction factor α over compactness factor for planar infinitesimally extended cross-sections 365
17.7 Correction factor α over compactness factor 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 1 mm 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 fbal.rect. (x) 413
19.6 Effective diffusion Deff. as a function of the pressure drop 414
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
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 de/d s 456
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
27.3 Numerical solution using Euler’s method 554
27.4 Numerical instability of the Euler method 556
27.5 Numerical stability of the backward Euler method 558
27.6 Example of nonlinear ODE solved by the backward Euler method 559
27.7 Numerical stability and precision of the Crank-Nicolson method 562
27.8 Comparison of the forward Euler and the second-order Adams-Bashforth method 567
27.9 Comparison of forward Euler, second-order Adams-Bashforth and second-order Adams-Moulton methods 571
27.10 Comparison of forward Euler, second-order Adams-Bashforth, second-order Adams-Moulton and fourth-order Runge-Kutta methods 575
27.11 Example of damped harmonic oscillation 576
27.12 Over-, under-, and critically-damped oscillation 579
27.13 Numerical output of the forward Euler and the fourth-order Runge-Kutta method 583
27.14 Example of the method of shooting applied to the ODE for the circular channel cross-section 588
27.15 Numerical output of the shooting method applied to the pendant drop ODE 590
28.1 4 × 4 mesh for numerical calculation of the flow profile in rectangular channels 596
28.2 Numerical output of the solution to the Poisson equation 600
28.3 Numerical output of the solution to the Poisson equation using SOR 601
28.4 Numerical output of the SOR solver in C 605
28.5 Application of the numerical SOR solver in C 606
29.1 Example of the Galerkin method used to solve the differential equation for the Hagen-Poiseuille flow 615
29.2 Example of the Galerkin method used to solve the two-dimensional Poisson equation for the rectangular channel flow 618
29.3 The first four Chebyshev polynomials 620
30.1 Spreadsheet for solving the Poisson equation for pressure-driven flow (part 1) 625
30.2 Spreadsheet for solving the Poisson equation for pressure-driven flow (part 2) 627
30.3 Spreadsheet for solving the Poisson equation for pressure-driven flow (part 3) 628
30.4 Use of the spreadsheet to calculate nonzero boundary conditions 628
30.5 Use of the spreadsheet for fluid mechanical problems with Neumann boundary conditions using the Couette flow as example 629
30.6 Use of the spreadsheet to derive the flow profiles in arbitrarily shaped microfluidic channels 630
30.7 Use of the spreadsheet to solve the electric field distribution 631
31.1 Cells used in FVM 637
31.2 Function reconstruction schemes commonly used in FVM 638
31.3 Example of a finite volume method (FVM) using the one-dimensional heat equation 645
31.4 Microsoft Excel spreadsheet used to solve the one-dimensional heat equation using FVM 647
31.5 Numerical output of the Microsoft Excel spreadsheet used to solve the one-dimensional heat equation using FVM 648
31.6 Two-dimensional mesh for FVM 649
32.1 Mesh discretization used in FEM 655
32.2 Lagrangian coordinate system 656
32.3 Common approximation functions in FEM 659
32.4 Numerical solution of the flow profile in infinitesimally extended channels using finite element method (FEM) 666
32.5 Hat functions plotted using the values obtained from listing 32.1 667
32.6 Mesh for solving the Poisson equation for a circular cross-section 669
32.7 Pyramid function supporting the value ĝ(1) in the two-dimensional mesh 670
32.8 Two-dimensional FEM applied to solving the Poisson equation on a circular cross-section 677
33.1 Numerical results for the accelerating flow in rectangular channel profiles 686
33.2 Numerical results for the accelerating flow in circular channel profiles 688
33.3 Numerical results for the space-transient flow in circular channel profiles during entry flow 693
33.4 Numerical results for the space-transient flow in circular channel profiles during entry flow taking convection into account 696
33.5 Comparison of numerical results for the space-transient flow in circular channel profiles during entry flow studying the effects of convection (part 1) 697
33.6 Comparison of numerical results for the space-transient flow in circular channel profiles during entry flow studying the effects of convection (part 2) 698
34.1 Usage of the three-dimensional solver for case 1: channel by-flow 728
34.2 Three-dimensional flow field for case 1: channel by-flow 730
34.3 Usage of the three-dimensional solver for case 2: channel through-flow 731
34.4 Three-dimensional flow field for case 2: channel through-flow 732
34.5 Usage of the three-dimensional solver for case 3: step expansion 733
34.6 Three-dimensional flow field for case 3: step expansion 734
34.7 Usage of the three-dimensional solver for case 4: flow around objects 736
34.8 Three-dimensional flow field for case 4: flow around objects 736
34.9 Usage of the three-dimensional solver for case 5: rectangular channel flow 738
34.10 Three-dimensional flow field for case 5: rectangular channel flow 738
34.11 Comparison of the flow profiles in a rectangular channel cross-section 739
34.12 Derived velocity and pressure profile along the x-axis for case 5 739
34.13 Usage of the three-dimensional solver for case 6: double-fin channel flow 740
34.14 Numerical output for case 6 in the xy-plane 741
34.15 Numerical output for case 6 in the xz-plane 741
34.16 Three-dimensional flow field for case 6: double-fin channel flow 742