Donations keep this resource free! Give here:
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.
[Return to Table of Contents]
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]
[InducedCharge Effects]
[DEP]
[Solution Chemistry]
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 can be written as:
 (F.25) 
where δ() is the Dirac delta function. For an incompressible flow, we also have
 (F.26) 
As discussed in Chapter 8, the solution to these equations is
 (F.27) 
where
 (F.28) 
and
 (F.29) 
where
 (F.30) 
and _{0} are the Green’sfunctions, respectively, for the velocity and pressure caused by a point force in
Stokes flow. ⋅ 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.
[Return to Table of Contents]
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]
[InducedCharge 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.
Ad revenue from these pages is used to support student research. The presence of an advertisement on these pages does not
constitute an endorsement by the Kirby Research Group or Cornell University.
Donations keep this resource free! Give here:
