Kinematics of vorticity: Vorticity-strain conjugation in incompressible fluid flows

An exact kinematic analysis is made of the three-dimensional incompressible Euler flows. It is found that the vorticity and rate-of-strain tensors are connected with each other through an identical singular integral transform. Some formal properties of this transform are derived. In particular, there exist harmonic functions in (3+1)-dimensional space so that the boundary values (toward our three-dimensional physical space) of a pair of conjugates are simply the vorticity and rate-of-strain tensors. The generalized Cauchy-Riemann equations are explicitly written. As an application, three of Siggia's invariants are related by some integrals.