Rotation of a fibre in simple shear flow of a dilute polymer solution

Abstract

The motion of a freely rotating prolate spheroid in a simple shear flow of a dilute polymeric solution is examined in the limit of large particle aspect ratio, $\kappa$ . A regular perturbation expansion in the polymer concentration, $c$ , a generalized reciprocal theorem, and slender body theory to represent the velocity field of a Newtonian fluid around the spheroid are used to obtain the $O(c)$ correction to the particle's orientational dynamics. The resulting dynamical system predicts a range of orientational behaviours qualitatively dependent upon $c\, De$ ( $De$ is the imposed shear rate times the polymer relaxation time) and $\kappa$ and quantitatively on $c$ . At a small but finite $c\, De$ , the particle spirals towards a limit cycle near the vorticity axis for all initial conditions. Upon increasing $\kappa$ , the limit cycle becomes smaller. Thus, ultimately the particle undergoes a periodic motion around and at a small angle from the vorticity axis. At moderate $c\, De$ , a particle starting near the flow–gradient plane departs it monotonically instead of spirally, as this plane (a limit cycle at smaller $c\, De$ ) obtains a saddle and an unstable node. The former is close to the flow direction. Upon further increasing $c\, De$ , the saddle node changes to a stable node. Therefore, depending upon the initial condition, a particle may either approach a periodic orbit near the vorticity axis or obtain a stable orientation near the flow direction. Upon further increasing $c\, De$ , the limit cycle near the vorticity axis vanishes, and the particle aligns with the flow direction for all starting orientations.