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]

G.1 Complex numbers and basic operations [complex algebra top]

Acomplex number microfluidics textbook nanofluidics textbook Brian Kirby Cornellis written in the form microfluidics textbook nanofluidics textbook Brian Kirby Cornell = a+jb, where a and b are real numbers and j = microfluidics textbook nanofluidics textbook Brian Kirby Cornell. The real part a of the complex number is denoted by Re[microfluidics textbook nanofluidics textbook Brian Kirby Cornell], while the imaginary part b of the complex number is denoted by the real value Im[microfluidics textbook nanofluidics textbook Brian Kirby Cornell]. Complex numbers are denoted with undertildes throughout this text.

The complex conjugate ofmicrofluidics textbook nanofluidics textbook Brian Kirby Cornell is denoted by microfluidics textbook nanofluidics textbook Brian Kirby Cornell* and is found by reversing the sign of the imaginary part:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(G.1)

The product of a number and its complex conjugate is purely real.

A complex number microfluidics textbook nanofluidics textbook Brian Kirby Cornell can be thought of as a position vector in a two dimensional coordinate system (microfluidics textbook nanofluidics textbook Brian Kirby Cornell = x+jy; cf. Equation 7.10). Alternately, cylindrical coordinates (magnitude and angle) can specify position in this plane. The magnitude or absolute value of a complex number is equivalent to the radial distance in the complex plane and is denoted by |microfluidics textbook nanofluidics textbook Brian Kirby Cornell| and given by
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(G.2)

while the angle of a complex number is equivalent to the azimuthal coordinate of the position in the complex plane:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(G.3)

where we use tan-1 to denote the two-argument inverse tangent, whose value (in the range 0-2π) depends on the sign of y and x, as well as the value of y∕x:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

Given Euler’s formula (exp() = cosθ+jsinθ), we can thus equivalently write
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(G.6)

a notation termed thepolar form of the complex number.

G.1.1 Arithmetic operations

Complex numbers can be added and subtracted by adding and subtracting the real and imaginary parts separately, so
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(G.7)

and
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(G.8)

Complex numbers can be multiplied term by term, noting that j2 = -1:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(G.9)

Complex numbers are divided by first multiplying the numerator and denominator by the complex conjugate of the denominator. The resulting real denominator divides both the real and imaginary parts of the numerator:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(G.10)

G.1.2 Calculus operations

Differentiation of an analytic functionwith respect to a complex number can be evaluated in a number of ways. If the desired result as a function of microfluidics textbook nanofluidics textbook Brian Kirby Cornell, the derivative is often taken symbolically, as is the case throughout most of Chapter 7. When evaluating the derivative for plotting in the xy-plane, though, we often want to write the derivative with respect to x or y. We do this by evaluating the limit in the definition of the derivative along either the real axis or the imaginary axis. This leads to

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

There is a critical distinction here between differentiation of complex numbers as compared to real numbers. The definition of the derivative of a real function:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(G.12)

must be the same for Δx approaching zero from the positive direction or the negative direction. If these two are not the same, we say the function is not differentiable at x. For example, |x| is not differentiable at x = 0, since the derivative (Equation G.12) evaluated with positive Δx is 1, while the derivative evaluated with negative Δx is -1. For a derivative of an analytic function with respect to a complex variable:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(G.13)

For a complex function two requirements must be satisfied for differentiability. First, the derivative must be the same regardless of whether microfluidics textbook nanofluidics textbook Brian Kirby Cornell is positive or negative. Second, the derivative must be the same regardless of whether microfluidics textbook nanofluidics textbook Brian Kirby Cornell is real or imaginary. The second condition is satisfied if Equation G.11 holds:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(G.14)

This means that differentiability for analytic functions has a much grander meaning than differentiability for real functions. For real functions, we think of a differentiable function as one that is smooth. For complex functions, differentiability implies not just smoothness, but that there is a specific relation (Equation G.14) between the derivatives of the function in the real and imaginary directions. If we use a complex variable to represent spatial distances in a 2D system, as we do in Chapter 7, then the differentiability of the analytic function (a relation between derivatives along the real axis versus the imaginary axis) implies a specific spatial relation between the derivatives in the x- and y-directions. To show this, consider the complex function microfluidics textbook nanofluidics textbook Brian Kirby Cornell = ϕv +. By taking the derivatives of microfluidics textbook nanofluidics textbook Brian Kirby Cornell along the real and complex directions, we can show that
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(G.15)

and
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(G.16)

These two relations are theCauchy-Riemann equations for analytic functions. Combining these two relations (by taking partial derivatives with respect to x or y), we obtain two relations:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

and

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

So, analytic functions are differentiable only if their real and imaginary parts each satisfy the Laplace equation. In Chapter 7, we take advantage of this by defining microfluidics textbook nanofluidics textbook Brian Kirby Cornell in terms of microfluidics textbook nanofluidics textbook Brian Kirby Cornell only. These functions are differentiable, and thus also satisfy the Laplace equation.


microfluidics textbook nanofluidics textbook Brian Kirby Cornell


[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: