Minimal Surfaces and Harmonic Vector Fields
Osserman in his book on A Survey on Minimal Surfaces demonstrates that a minimal surface can be defined by a non-constant map into E^N, where the "position vector" defined by the map (call it V) is harmonic. In 3-D this means that the vector Laplacian of V vanishes.

Now in hydrodynamics based upon the Navier Stokes equations the viscous dissipation term is assumed to be proportional to the Laplacian of the Velocity vector field (times a coefficient of viscosity). Suppose the velocity field was harmonic. Then there would be no viscous dissipation! IF such a flow could be generated on the tips of aircraft wings, drag would be reduced dramatically. The little winglets you see on the big jets are attempts to do something about the non-harmonic vortices generated from the wingtips.

If a velocity field started out with both harmonic and non-harmonic components, then after the passage of time the flow should reduce (by viscous dissipation) to the Harmonic components alone. Such is the theory of wakes.

Another remarkable theorem is that of Sophus Lie: a holomorphic function generates a minimal surface. That is, the position vector or 4 space, [u,v,phi(u,v),chi(u,v)] where phi and chi are the real and imaginary parts of the holomorphic function of the complex variable, u+iv, generates a minimal surface (actually a pair of conjugate minimal surfaces). But in turn these can be related to Harmonic vector fields, which are "dissipation free".


Copyright © 1995-2009, CSDC Inc. All rights reserved.
Last update 01/23/2009
to HomePage