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Theoretical Microfluidics
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Contents List of symbols xiii 1 Basic concepts in microfluidics 1 1.1 Lab-on-a-chip technology 1 1.2 Scaling laws in microfluidics 2 1.3 Fluids and fields 3 1.4 SI units and mathematical notation 7 1.5 Perturbation theory 11 1.6 Eigenfunction expansion 13 1.7 Further reading 15 1.8 Exercises 15 1.9 Solutions 17 2 Governing equations 19 2.1 Mass flux, conservation of mass, and the continuity equation 19 2.2 Momentum flux, force densities, and the equation of motion 21 2.3 Energy flux and the heat-transfer equation 28 2.4 Further reading 31 2.5 Exercises 32 2.6 Solutions 33 3 Basic flow solutions 37 3.1 Fluids in mechanical equilibrium 37 3.2 Liquid film flow on an inclined plane 39 3.3 Couette flow 40 3.4 Poiseuille flow 41 3.5 Poiseuille flow in shape-perturbed channels 51 3.6 Poiseuille flow for weakly compressible fluids 55 3.7 Stokes drag on a sphere moving in steady state 60 3.8 Exercises 63 3.9 Solutions 65 vii viii Contents 4 Hydraulic resistance and compliance 71 4.1 Viscous dissipation of energy for incompressible fluids 71 4.2 Hydraulic resistance of some straight channels 74 4.3 Shape dependence of hydraulic resistance 75 4.4 Reynolds number for systems with two length scales 79 4.5 Hydraulic resistance, two connected straight channels 81 4.6 Compliance 83 4.7 Equivalent circuit theory and Kirchhoff’s laws 84 4.8 Exercises 86 4.9 Solutions 88 5 Diffusion 91 5.1 A random-walk model of diffusion 91 5.2 The convection-diffusion equation for solutions 93 5.3 The diffusion equation 95 5.4 The H-filter: separating solutes by diffusion 98 5.5 Taylor dispersion; a convection-diffusion example 100 5.6 Exercises 105 5.7 Solutions 106 6 Time-dependent flow 109 6.1 Starting a Couette flow 109 6.2 Stopping a Poiseuille flow by viscous forces 111 6.3 Flow induced by slowly oscillating boundaries 113 6.4 Accelerated motion of a spherical body in a liquid 116 6.5 Other time-dependent flows 117 6.6 Exercises 118 6.7 Solutions 118 7 Capillary effects 123 7.1 Surface tension 123 7.2 Contact angle 127 7.3 Capillary length and capillary rise 128 7.4 Capillary pumps 131 7.5 Marangoni effect; surface-tension gradients 134 7.6 Exercises 134 7.7 Solutions 136 8 Electrohydrodynamics 141 8.1 Polarization and dipole moments 141 8.2 Electrokinetic effects 143 8.3 The Debye layer near charged surfaces 145 8.4 Further reading 152 8.5 Exercises 152 8.6 Solutions 154 Contents ix 9 Electroosmosis 157 9.1 Electrohydrodynamic transport theory 157 9.2 Ideal electro-osmotic flow 157 9.3 Debye-layer overlap 161 9.4 Ideal EO flow with backpressure 162 9.5 The many-channel EO pump 165 9.6 The cascade EO pump 166 9.7 Further reading 169 9.8 Exercises 169 9.9 Solutions 170 10 Dielectrophoresis 173 10.1 Induced polarization and dielectric forces; heuristically 173 10.2 A point dipole in a dielectric fluid 174 10.3 A dielectric sphere in a dielectric fluid; induced dipole 175 10.4 The dielectrophoretic force on a dielectric sphere 177 10.5 Dielectrophoretic particle trapping in microfluidics 178 10.6 The AC dielectrophoretic force on a dielectric sphere 180 10.7 Exercises 182 10.8 Solutions 184 11 Magnetophoresis 187 11.1 Magnetophoresis and bioanalysis 187 11.2 Magnetostatics 188 11.3 Basic equations for magnetophoresis 190 11.4 Calculation of magnetic-bead motion 191 11.5 Magnetophoretic lab-on-a-chip systems 193 11.6 Further reading 194 11.7 Exercises 194 11.8 Solutions 195 12 Thermal transfer 197 12.1 Thermal effects in hydrostatics 198 12.2 Poiseuille flow in a transverse temperature gradient 201 12.3 Equivalent circuit model for heat transfer 205 12.4 The PCR biochip 208 12.5 Exercises 210 12.6 Solutions 211 13 Two-phase flow 213 13.1 Two-phase Poiseuille flow 213 13.2 Capillary and gravity waves 215 13.3 Gas bubbles in microfluidic channels 220 13.4 Droplets in microfluidic junctions and digital fluidics 224 13.5 Further reading 226 13.6 Exercises 226 13.7 Solutions 228 x Contents 14 Complex flow patterns 231 14.1 Pressure-driven flow in shape-perturbed microchannels 231 14.2 Streamlines in a shape-perturbed channel 235 14.3 Lubrication theory 237 14.4 The staggered herring-bone mixer 238 14.5 Induced-charge electrolytic flow 240 14.6 Exercises 248 14.7 Solutions 249 15 Acoustofluidics 255 15.1 The acoustic-wave equation for zero viscosity 256 15.2 Acoustic waves in first-order perturbation theory 258 15.3 Viscous damping of first-order acoustic waves 260 15.4 Acoustic resonances 262 15.5 Acoustic waves in multilayer systems 264 15.6 Second-order acoustic fields 267 15.7 Further reading 270 15.8 Exercises 271 15.9 Solutions 272 16 Optofluidics 275 16.1 The optical wave equation in electrolytes 276 16.2 Molecular absorption and Beer–Lambert’s law 278 16.3 Molecular fluorescence and phosphorescence 281 16.4 Onchip waveguides 282 16.5 Onchip laser sources 283 16.6 Photonic bandgap structures in optofluidics 286 16.7 Further reading 288 16.8 Exercises 288 16.9 Solutions 289 17 Nanofluidics 291 17.1 Investigation of the no-slip boundary condition 291 17.2 Capillary filling of nanochannels 294 17.3 Squeeze flow in nanoimprint lithography 298 17.4 Nanofluidics and molecular dynamics 302 17.5 Exercises 304 17.6 Solutions 305 Appendix A Physical constants 309 A.1 Water 309 A.2 Viscosity 309 A.3 Diffusivity 310 A.4 Surface tension and contact angle 310 Contents xi Appendix B Dimensionless numbers 311 Appendix C Curvilinear co-ordinates 313 C.1 Cartesian co-ordinates 313 C.2 Cylindrical polar co-ordinates 314 C.3 Spherical polar co-ordinates 316 Appendix D The chemical potential 319 D.1 The partition function and the free energy 319 D.2 The chemical potential of a solution 320 Appendix E The wave equation 321 Appendix F Numerical simulations 325 F.1 The finite-element method (FEM) 325 F.2 The level set method and motion of interfaces 329 Bibliography 333 Index 33 |
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