Tumbling of a rigid rod in a shear flow

TL;DR

This paper analyzes the orientation and tumbling behavior of a rigid rod in shear flow, deriving a probability distribution and examining the effects of shear rate and thermal diffusion across different flow regimes.

Contribution

It introduces a Master Equation approach for the orientation distribution of a rod in shear flow, including the derivation of a crossover function for tumbling dynamics.

Findings

Stationary orientation distribution derived for all Weissenberg numbers.

Tumbling time and frequency depend on shear rate and diffusion, with a proposed crossover function.

Flow pattern shows the balance between shear-induced alignment and thermal diffusion.

Abstract

The tumbling of a rigid rod in a shear flow is analyzed in the high viscosity limit. Following Burgers, the Master Equation is derived for the probability distribution of the orientation of the rod. The equation contains one dimensionless number, the Weissenberg number, which is the ratio of the shear rate and the orientational diffusion constant. The equation is solved for the stationary state distribution for arbitrary Weissenberg numbers, in particular for the limit of high Weissenberg numbers. The stationary state gives an interesting flow pattern for the orientation of the rod, showing the interplay between flow due to the driving shear force and diffusion due to the random thermal forces of the fluid. The average tumbling time and tumbling frequency are calculated as a function of the Weissenberg number. A simple cross-over function is proposed which covers the whole regime from small to large Weissenberg numbers.