A model for alignment between microscopic rods and vorticity

Numerical simulations show that microscopic rod-like bodies suspended in a turbulent flow tend to align with the vorticity vector, rather than with the dominant eignevector of the strain-rate tensor. This paper investigates an analytically solvable limit of a model for alignment in a random velocity field with isotropic statistics. The vorticity varies very slowly and the isotropic random flow is equivalent to a pure strain with statistics which are axisymmetric about the direction of the vorticity. We analyse the alignment in a weakly fluctuating uniaxial strain field, as a function of the product of the strain relaxation time $\tau_{\rm s}$ and the angular velocity $\omega$ about the vorticity axis. We find that when $\omega\tau_{\rm s}\gg 1$, the rods are predominantly either perpendicular or parallel to the vorticity.