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Copyright Brian J. Kirby. With questions, contact Prof. Kirby here.
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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]

Stokes flow is a lowReynolds number limit, and the resulting solution (Equations 8.12 and 8.13) is
independent of the Reynolds number. Why, then, in Equation 8.26, is the drag coefficient dependent
on the Reynolds number? How could the drag coefficient be defined differently such that the drag
coefficient is not a function of the Reynolds number?

Given a particle of radius a in an infinite domain and a freestream velocity in the xdirection with magnitude
U, solve for the velocity around the particle. Assume that the solution is axisymmetric and write the
solution in terms of axisymmetric velocity components u_{r} and u_{ϑ}, as is done in Equations 8.12
and 8.13.
 Write the continuity equation in spherical coordinates, delete terms involving the azimuthal velocity
u_{φ} or derivatives with respect to φ, and show that the continuity equation can be written
as
 (8.52) 
 Write the radial momentum equation in spherical coordinates (Equation D.16), eliminate the righthand
side owing to the Stokes approximation, delete terms involving the azimuthal velocity u_{φ} or derivatives
with respect to φ, and show that the radial momentum equation for Stokes flow can be written
as
 (8.53) 
 Write the colatitudinal momentum equation in spherical coordinates (Equation D.16), eliminate the
righthand side owing to the Stokes approximation, delete terms involving the azimuthal velocity u_{φ} or
derivatives with respect to φ, and show that the colatitudinal momentum equation for Stokes flow can be
written as
 (8.54) 
 Assume that the solutions for the radial and colatitudinal velocities have the following
forms:
 (8.55) 
 (8.56) 
These six constants can be fixed by examining (a) the boundary conditions at r = a, which gives two
constraints (one for each component of velocity at the wall); (b) the continuity equation, which gives
three constraints (one for each term in the assumed form for velocity); and (c) the NavierStokes
equation, which gives one new constraint (two total, but one is redundant with one from the continuity).
With these constants fixed, the solution is complete.

Most exercises are excluded from this web posting. Follow the links to buy the text at
Cambridge or
Amazon or Powell's or
Barnes and Noble.
[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.
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