Paper detail

Complex Variable Methods for 3D Applied Mathematics: 3D Twistors and the biharmonic equation

In applied mathematics generally and fluid dynamics in particular, the role of complex variable methods is normally confined to two-dimensional motion and the association of points with complex numbers via the assignment w = x+i y. In this framework 2D potential flow can be treated through the use of holomorphic functions and biharmonic flow through a simple, but superficially non-holomorphic extension. This paper explains how to elevate the use of complex methods to three dimensions, using Penrose's theory of twistors as adapted to intrinsically 3D and non-relativistic problems by Hitchin. We first summarize the equations of 3D steady viscous fluid flow in their basic geometric form. We then explain the theory of twistors for 3D, resulting in complex holomorphic representations of solutions to harmonic and biharmonic problems. It is shown how this intrinsically holomorphic 3D approach reduces naturally to the well-known 2D situations when there is translational or rotational symmetry, and an example is given. We also show how the case of small but finite Reynolds number can be integrated by complex variable techniques in two dimensions, albeit under strong assumptions.