Complex Trkalian Fields and Solutions to Euler's Equations for the Ideal Fluid

Abstract

We consider solutions to the complex Trkalian equation,~$ \vec{\nabla} \times \vc = \vc ,$ where~$\vc$ is a 3 component vector function with each component in the complex field, and may be expressed in the form~$ \vc = e^{ig} \vec{\nabla} F, $ with~$g$ real and~$F$ complex. We find, there are precisely two classes of solutions; one where~$g$ is a Cartesian variable and one where~$g$ is the spherical radial coordinate. We consider these flows to be the simplest of all exact 3-d solutions to the Euler's equation for the ideal incompressible fluid. The novel approach we use in solving for these classes of solutions to these 3-dimensional vector pdes involves differential geometric techniques: one may employ the method to generate solutions to other classes of vector pdes.

30 pages, no figures; to appear-March PRE

References in corpus (1)

• Continuous topological evolution

Cited by in corpus (3)

• Beltrami vector fields with an icosahedral symmetry
• Topologically Massive Abelian Gauge Theory
• Scale Invariant Streamline Equations and Strings of Singular Vorticity