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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]

G.5 Conformal mapping [complex algebra top]

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,
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(G.41)

This transform maps a distance microfluidics textbook nanofluidics textbook Brian Kirby Cornell to a new distance J (microfluidics textbook nanofluidics textbook Brian Kirby Cornell). In particular, a line of length 4b centered on microfluidics textbook nanofluidics textbook Brian Kirby Cornell= 0 and rotated an angle α with respect to the x-axis is transformed into a circle of radius b centered on microfluidics textbook nanofluidics textbook Brian Kirby Cornell= 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 microfluidics textbook nanofluidics textbook Brian Kirby Cornell = Umicrofluidics textbook nanofluidics textbook Brian Kirby Cornellexp[-], then the transformed flow is given by microfluidics textbook nanofluidics textbook Brian Kirby Cornell = U J (microfluidics textbook nanofluidics textbook Brian Kirby Cornellexp[-],b), or
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(G.42)

which is identical to Equation 7.75 for a = b. Because any function of J (microfluidics textbook nanofluidics textbook Brian Kirby Cornell) only is also a function of microfluidics textbook nanofluidics textbook Brian Kirby Cornell 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 dual-valued in a manner that is not immediately obvious. The inverse Joukowski transform J -1 is given by
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(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
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(G.44)

and otherwise, we use the negative square root, and the inverse transform is given by
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(G.45)

If we define shapes as loci of points defined by distances microfluidics textbook nanofluidics textbook Brian Kirby Cornell 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 microfluidics textbook nanofluidics textbook Brian Kirby Cornell= 0 onto a line. It simultaneously maps circles centered on microfluidics textbook nanofluidics textbook Brian Kirby Cornell= 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
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(G.46)

and, if transformed by the inverse Joukowski transform:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(G.47)

the result is flow around an ellipse with semimajor axis a+b2∕a and semiminor axis a-b2∕a.

G.5.2 Schwarz-Christoffel transform

Given locations in 2D planes denoted using complex variables microfluidics textbook nanofluidics textbook Brian Kirby Cornell= x+jy, the Schwarz-Christoffel transform maps an arbitrary polygon onto a half-space. 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 Schwarz-Christoffel formula states that, given specific constraints that are met for physically well-posed problems, there exists a conformal mapping between a half-space in a complex plane and a polygon in an alternate complex plane. Given a location microfluidics textbook nanofluidics textbook Brian Kirby Cornell in the upper half plane (i.e., for Im(microfluidics textbook nanofluidics textbook Brian Kirby Cornell) > 0), S(microfluidics textbook nanofluidics textbook Brian Kirby Cornell) is a complex number that gives the location of that point inside a specified polygon. The function that gives this mapping, S(microfluidics textbook nanofluidics textbook Brian Kirby Cornell), is given by
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(G.48)

where the polygon consists of n vertices with locations microfluidics textbook nanofluidics textbook Brian Kirby Cornell, each of which has an exterior angle αi defined in the range -π < αi < π.

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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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