Cross-streamline migration of slender Brownian fibres in plane poiseuille flow

We consider fibre migration across streamlines in a suspension under plane Poiseuille flow. The flow investigated lies between two infinite, parallel plates separated by a distance comparable to the length of a suspended fibre. We consider the weak flow limit such that Brownian motion strongly affects the fibre position and orientation. Under these conditions, the fibre distribution, fibre mobility and fluid velocity field all vary on scales comparable to the fibre's length thus complicating a traditional volume-averaging approach to solving this problem. Therefore, we use a non-local derivation of the stress. The resulting fully coupled problem for the fluid velocity, fibre stress contribution and fibre distribution function is solved self-consistently in the limit of strong Brownian motion. When calculated in this manner, we show that at steady state the fibres' centre-of-mass distribution function shows a net migration of fibres away from the centre of the channel and towards the channel walls. The fibre migration occurs for all gap widths (0 less than or equal to lambda less than or equal to 35) and fibre concentrations (0 less than or equal to c less than or equal to 1.0) investigated. Additionally, the fibre concentration reaches a maximum value around one fibre half-length from the channel walls. However, we find that the net fibre migration is a relatively small change over the fibre's uniform bulk distribution, and typically the centre-of-mass migration changes the uniform concentration profile by only a few percent.