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In applied mathematics generally and fluid dynamics in particular, the roleof complex variable methods is normally confined to two-dimensional motion andthe association of points with complex numbers via the assignment w = x+i y. Inthis framework 2D potential flow can be treated through the use of holomorphicfunctions and biharmonic flow through a simple, but superficiallynon-holomorphic extension. This paper explains how to elevate the use ofcomplex methods to three dimensions, using Penrose's theory of twistors asadapted to intrinsically 3D and non-relativistic problems by Hitchin. We firstsummarize the equations of 3D steady viscous fluid flow in their basicgeometric form. We then explain the theory of twistors for 3D, resulting incomplex holomorphic representations of solutions to harmonic and biharmonicproblems. It is shown how this intrinsically holomorphic 3D approach reducesnaturally to the well-known 2D situations when there is translational orrotational symmetry, and an example is given. We also show how the case ofsmall but finite Reynolds number can be integrated by complex variabletechniques in two dimensions, albeit under strong assumptions.

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