On the Arnold cat map and periodic boundary conditions for planar elongational flow

In this paper we show that the periodic boundary conditions used to simulate planar elongational flow are closely related to the Arnold cat map. In particular the relationship between the Arnold cat map and the periodic boundary conditions devised by Kraynik and Reinelt [1992, Int. J. multiphase Flow, 18, 1045], the so-called K-R map, is demonstrated. It is shown that the family of lattices found by Kraynik and Reinelt corresponds to a subset of hyperbolic toral automorphisms. These lattices were previously found to be sufficient to enable molecular dynamics simulations of steady-state planar elongational flow of unrestricted duration. Within the frame of the cat map we provide a re-derivation for the set of eigenvalues, eigenvectors and orientation angles of the K-R map and find it to be considerably simpler than the original derivation provided by Kraynik and Reinelt.