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Copyright Brian J. Kirby. With questions, contact Prof. Kirby here. This material may not be distributed without the author's consent. When linking to these pages, please use the URL http://www.kirbyresearch.com/textbook.

This web posting is a draft, abridged version of the Cambridge University Press text. Follow the links to buy at Cambridge or Amazon or Powell's or Barnes and Noble. Contact Prof. Kirby here. Click here for the most recent version of the errata for the print version.

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Jump To: [Kinematics] [Couette/Poiseuille Flow] [Fluid Circuits] [Mixing] [Electrodynamics] [Electroosmosis] [Potential Flow] [Stokes Flow] [Debye Layer] [Zeta Potential] [Species Transport] [Separations] [Particle Electrophoresis] [DNA] [Nanofluidics] [Induced-Charge Effects] [DEP] [Solution Chemistry]

F.2 Stokes equations [multipolar solutions top]

The Stokes equations govern key viscous flows, as discussed in Chapter 8. As was the case for the Laplace equation, numerical approaches are the norm for solving the governing equations in complex geometries, but analytical solutions to the Stokes equation can be found by considering Green’s function solutions to the Stokes equations. These solutions have mathematical importance because they lead to a convenient series expansion (the multipolar expansion) that approximates the correct solutions. While the multipolar expansion for the Stokes equations is a bit more complicated than the multipolar expansion for the Laplace equations, we nonetheless are able to consider small viscous objects and their hydrodynamic interaction using the multipolar formulation. These multipolar formulations facilitate calculation of the forces on large numbers of interacting microparticles.

F.2.1 Green’s function for Stokes flow with a point source

The Stokes equationswith a point force at the origin microfluidics textbook nanofluidics textbook Brian Kirby Cornell can be written as:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(F.25)

where δ(microfluidics textbook nanofluidics textbook Brian Kirby Cornell) is the Dirac delta function. For an incompressible flow, we also have
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(F.26)

As discussed in Chapter 8, the solution to these equations is
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(F.27)

where
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(F.28)

and
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(F.29)

where
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(F.30)

microfluidics textbook nanofluidics textbook Brian Kirby Cornell and microfluidics textbook nanofluidics textbook Brian Kirby Cornell0 are the Green’sfunctions, respectively, for the velocity and pressure caused by a point force in Stokes flow. microfluidics textbook nanofluidics textbook Brian Kirby Cornellmicrofluidics textbook nanofluidics textbook Brian Kirby Cornell is referred to as aStokeslet, and constitutes the monopole of the multipolar expansion for Stokes flow. The multipole solutions for Stokes flow are also referred to as singular solutions or fundamental solutions. Figure F.4 shows a Stokeslet and a schematic of the resulting flow pattern.


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure F.4: Flow induced by a Stokeslet.


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Jump To: [Kinematics] [Couette/Poiseuille Flow] [Fluid Circuits] [Mixing] [Electrodynamics] [Electroosmosis] [Potential Flow] [Stokes Flow] [Debye Layer] [Zeta Potential] [Species Transport] [Separations] [Particle Electrophoresis] [DNA] [Nanofluidics] [Induced-Charge Effects] [DEP] [Solution Chemistry]

Copyright Brian J. Kirby. Please contact Prof. Kirby here with questions or corrections. This material may not be distributed without the author's consent. When linking to these pages, please use the URL http://www.kirbyresearch.com/textbook.

This web posting is a draft, abridged version of the Cambridge University Press text. Follow the links to buy at Cambridge or Amazon or Powell's or Barnes and Noble. Contact Prof. Kirby here. Click here for the most recent version of the errata for the print version.


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