The orientations of prolate ellipsoids in linear shear flows

The dynamics of the orientations of prolate ellipsoids in general linear shear flow is considered. The motivation behind this work is to gain a better understanding of the motion and the orientation probability distribution of ice particles in clouds in order to improve the modelling of their collision. The evolution of the orientations is governed by the Jeffery equation. It is shown that the possible attractors of this equation are fixed points, limit cycles and an infinite set of periodic solutions, named Jeffery orbits, in the case of simple shear. Linear stability analysis shows that the existence and the stability of the attractors are determined by the eigenvalues of the linear part of the equation. If the eigenvalues possess a non-vanishing real part, then there always exists either a stable fixed point or a stable limit cycle. Pure imaginary eigenvalues lead to Jeffery orbits. The convergence to a stable fixed point or to a stable limit cycle may either be monotonic or may be retarded due to the occurrence of non-normal growth. If non-normal growth occurs the convergence rate may be much slower compared with the characteristic time scale of the shear. Expressions for the characteristic time scale of convergence to the stable solutions are derived. In the case of non-normal growth, expressions are derived for the delay in the convergence. The orientation probability distribution function (p.d.f.) is computed via the solution of the Fokkerâ€"Planck equation. The p.d.f. is either periodic, in the case of simple shear (pure imaginary eigenvalues), or it converges to singular points or strips in the orientation space (fixed points and limit cycles) on which it grows to infinity. Time-independent p.d.f.s exist only for imaginary eigenvalues. Unlike the case where Brownian diffusion is present, the steady solutions are not unique and depend on the initial conditions.