Cornell University - Visit www.cornell.edu Kirby Research Group at Cornell: Microfluidics and Nanofluidics : - Home College of Engineering - visit www.engr.cornell.edu Cornell University - Visit www.cornell.edu
Cornell University, College of Engineering Search Cornell
News Contact Info Login

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] [Induced-Charge Effects] [DEP] [Solution Chemistry]

8.7 Exercises [Stokes flow top]

  1. Stokes flow is a low-Reynolds 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?

  2. Given a particle of radius a in an infinite domain and a freestream velocity in the x-direction 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 ur and uϑ, as is done in Equations 8.12 and 8.13.

    1. Write the continuity equation in spherical coordinates, delete terms involving the azimuthal velocity uφ or derivatives with respect to φ, and show that the continuity equation can be written as
      microfluidics textbook nanofluidics textbook Brian Kirby Cornell
      (8.52)

    2. Write the radial momentum equation in spherical coordinates (Equation D.16), eliminate the right-hand side owing to the Stokes approximation, delete terms involving the azimuthal velocity uφ or derivatives with respect to φ, and show that the radial momentum equation for Stokes flow can be written as
      microfluidics textbook nanofluidics textbook Brian Kirby Cornell
      (8.53)

    3. Write the colatitudinal momentum equation in spherical coordinates (Equation D.16), eliminate the right-hand side owing to the Stokes approximation, delete terms involving the azimuthal velocity uφ or derivatives with respect to φ, and show that the colatitudinal momentum equation for Stokes flow can be written as
      microfluidics textbook nanofluidics textbook Brian Kirby Cornell
      (8.54)

    4. Assume that the solutions for the radial and colatitudinal velocities have the following forms:
      microfluidics textbook nanofluidics textbook Brian Kirby Cornell
      (8.55)

      microfluidics textbook nanofluidics textbook Brian Kirby Cornell
      (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 Navier-Stokes 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.

  3. 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] [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.


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: