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]
For systems solved by the Laplace equation, certain mapping functions make it easy for us to transform problems
spatially in a manner that leads to simple solution of the governing equations.
G.5.1 Joukowski transform
One common mapping function for potential flow or electric field solutions is theJoukowski transform,
 (G.41) 
This transform maps a distance to a new distance J (). In particular, a line of length 4b centered on = 0 and
rotated an angle α with respect to the xaxis is transformed into a circle of radius b centered on = 0. This
transform, for example, maps a uniform flow into a flow over a cylinder. If the complex potential for a uniform
flow is given by = Uexp[jα], then the transformed flow is given by = U J (exp[jα],b),
or
 (G.42) 
which is identical to Equation 7.75 for a = b. Because any function of J () only is also a function of only, these
transformed solutions also satisfy the Laplace equations and thus are solutions of the flow equations, but with a
spatially transformed boundary.
The inverse Joukowski transform, then, maps a circle of radius b onto a line. The inverse Joukowski transform,
though generally more useful, is harder to implement, because the transform is dualvalued in a manner that is not
immediately obvious. The inverse Joukowski transform J ^{1} is given by
 (G.43) 
where sgn(x) = x∕x is the sign function of x. The sign functions make this relation not so obvious–basically, what
the sign functions do is account for the fact that the inverse is dual valued since it involves a square root. If the point
in question is either (a) outside the circle and in the right half of the complex plane or (b) inside the circle and in the
left half of the complex plane, we use the principal square root, and the inverse Joukowski transform is given
by
 (G.44) 
and otherwise, we use the negative square root, and the inverse transform is given by
 (G.45) 
If we define shapes as loci of points defined by distances from some reference point, we can describe how
the Joukowski transform and its inverse maps these shapes. The inverse Joukowski transform maps a
circle of radius b centered on = 0 onto a line. It simultaneously maps circles centered on = 0 with
radius greater than a to ellipses. This becomes useful if we want to analytically calculate the flow
around an ellipse. For example, the complex potential for flow around a cylinder of radius a is given
by
 (G.46) 
and, if transformed by the inverse Joukowski transform:
 (G.47) 
the result is flow around an ellipse with semimajor axis a+b^{2}∕a and semiminor axis ab^{2}∕a.
G.5.2 SchwarzChristoffel transform
Given locations in 2D planes denoted using complex variables = x+jy, the SchwarzChristoffel transform maps
an arbitrary polygon onto a halfspace. This is useful primarily for approximating electric fields in polygonal
microchannels, since many microfabrication techniques lead to polygonal microchannels (e.g., rectangular or
trapezoidal). The SchwarzChristoffel formula states that, given specific constraints that are met for physically
wellposed problems, there exists a conformal mapping between a halfspace in a complex plane and a polygon in an
alternate complex plane. Given a location in the upper half plane (i.e., for Im() > 0), S() is a complex number
that gives the location of that point inside a specified polygon. The function that gives this mapping, S(), is given
by
 (G.48) 
where the polygon consists of n vertices with locations , each of which has an exterior angle α_{i} defined in the
range π < α_{i} < π.
[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:
