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

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.

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Cambridge University Press
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Journal of Fluid Mechanics
Medium: X
Sponsoring Org:
National Science Foundation
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