The Laplace and Stokes equations are both linear in the fluid velocity and are both amenable to solution by superposition of Green's functions; further, each equation has a set of solutions whose superposition is termed the multipolar expansion. This appendix details these solutions.

LAPLACE EQUATION

The Laplace equation governs key electromagnetic solutions and, in the case of purely electroosmotic flow far from walls, the Laplace equation can govern fluid flow as well. Although numerical approaches are the norm for solving the governing equations in complex geometries, a great wealth of analytical solutions to the Laplace equation can be found by solving by separation of variables, especially for systems with symmetry. These solutions have mathematical importance because they lead to a convenient series expansion (the multipolar expansion) that approximates the correct solutions, and these solutions have physical importance because the building blocks of electrical systems (point charges), magnetic systems (point magnetic dipoles), and fluid systems (sources, sinks, vortexes) correspond to individual terms in this expansion.

This appendix describes solutions to the Laplace equation with both axial and plane symmetry. The solutions for plane-symmetric flows forecast the potential flows discussed in Chapter 7, and the solutions for axial symmetry forecast the use of the multipolar expansion for modeling the dielectrophoretic response of particles (Chapter 17).