Book contents
- Frontmatter
- Contents
- Preface
- Nomenclature
- Introduction
- 1 Kinematics, Conservation Equations, and Boundary Conditions for Incompressible Flow
- 2 Unidirectional Flow
- 3 Hydraulic Circuit Analysis
- 4 Passive Scalar Transport: Dispersion, Patterning, and Mixing
- 5 Electrostatics and Electrodynamics
- 6 Electroosmosis
- 7 Potential Fluid Flow
- 8 Stokes Flow
- 9 The Diffuse Structure of the Electrical Double Layer
- 10 Zeta Potential in Microchannels
- 11 Species and Charge Transport
- 12 Microchip Chemical Separations
- 13 Particle Electrophoresis
- 14 DNA Transport and Analysis
- 15 Nanofluidics: Fluid and Current Flow in Molecular-Scale and Thick-EDL Systems
- 16 AC Electrokinetics and the Dynamics of Diffuse Charge
- 17 Particle and Droplet Actuation: Dielectrophoresis, Magnetophoresis, and Digital Microfluidics
- APPENDIX A Units and Fundamental Constants
- APPENDIX B Properties of Electrolyte Solutions
- APPENDIX C Coordinate Systems and Vector Calculus
- APPENDIX D Governing Equation Reference
- APPENDIX E Nondimensionalization and Characteristic Parameters
- APPENDIX F Multipolar Solutions to the Laplace and Stokes Equations
- APPENDIX G Complex Functions
- APPENDIX H Interaction Potentials: Atomistic Modeling of Solvents and Solutes
- Bibliography
- Index
8 - Stokes Flow
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- Nomenclature
- Introduction
- 1 Kinematics, Conservation Equations, and Boundary Conditions for Incompressible Flow
- 2 Unidirectional Flow
- 3 Hydraulic Circuit Analysis
- 4 Passive Scalar Transport: Dispersion, Patterning, and Mixing
- 5 Electrostatics and Electrodynamics
- 6 Electroosmosis
- 7 Potential Fluid Flow
- 8 Stokes Flow
- 9 The Diffuse Structure of the Electrical Double Layer
- 10 Zeta Potential in Microchannels
- 11 Species and Charge Transport
- 12 Microchip Chemical Separations
- 13 Particle Electrophoresis
- 14 DNA Transport and Analysis
- 15 Nanofluidics: Fluid and Current Flow in Molecular-Scale and Thick-EDL Systems
- 16 AC Electrokinetics and the Dynamics of Diffuse Charge
- 17 Particle and Droplet Actuation: Dielectrophoresis, Magnetophoresis, and Digital Microfluidics
- APPENDIX A Units and Fundamental Constants
- APPENDIX B Properties of Electrolyte Solutions
- APPENDIX C Coordinate Systems and Vector Calculus
- APPENDIX D Governing Equation Reference
- APPENDIX E Nondimensionalization and Characteristic Parameters
- APPENDIX F Multipolar Solutions to the Laplace and Stokes Equations
- APPENDIX G Complex Functions
- APPENDIX H Interaction Potentials: Atomistic Modeling of Solvents and Solutes
- Bibliography
- Index
Summary
The Navier–Stokes equations have not been solved analytically in the general case, and the only available analytical solutions arise from simple geometries (for example, the 1D flow geometries discussed in Chapter 2). Because of this, our analytical approach for solving fluid flow problems is often to solve a simpler equation that applies in a specific limit. Some examples of these simplified equations include the Stokes equations (applicable when the Reynolds number is low, as is usually the case in microfluidic devices) and the Laplace equation (applicable when the flow has no vorticity, as is the case for purely electrokinetic flows in certain limits). These simplified equations guide engineering analysis of fluid systems.
In this chapter, we discuss Stokes flow (equivalently termed creeping flow), in which case the Reynolds number is so low that viscous forces dominate over inertial forces. The approximation that leads from the Navier–Stokes equations to the Stokes equations is shown, and analytical results are discussed. The Stokes flow equations provide useful solutions to describe the fluid forces on small particles in micro- and nanofluidic systems, because these particles are often well approximated by simple geometries (for example, spheres) for which the Stokes flow equations can be solved analytically. The Stokes flow equations also lead to simple solutions (Hele-Shaw flows) for wide, shallow microchannels of uniform depths.
- Type
- Chapter
- Information
- Micro- and Nanoscale Fluid MechanicsTransport in Microfluidic Devices, pp. 178 - 198Publisher: Cambridge University PressPrint publication year: 2010