On the motion of flexible threads in a stokes shear field

TLDR: In this article, the motion of a flexible thread-like filament suspended in an unbounded Stokes shear flow is considered, and the model is solved in closed form, for inextensible filaments and general time-dependent shear flows.

Abstract: We consider the motion of a flexible threadlike filament suspended in an unbounded Stokes shear flow. As the extension of a well-established slender-body theory, we treat the case of a highly anisotropic drag relation, as reflected by a small ratio ϵ of transverse to longitudinal filament mobility. This provides one potentially useful model of hindered filament motion in highly concetrated filamentary suspensions. In the limit ϵ → 0, which corresponds to the kinematics associated with a recent model (of Doi and Edwards) for polymer chains, the filament motion is governed by a non-linear kinematic-wave equation. We show that this equation can be solved in closed form, for inextensible filaments and general time-dependent shear flows. We investigate the permanence and stability of threads having uniform axial mobility, showing that only piecewise straight threads have permanent shape. Also, we investigate the stability of fully-extended treads by means of both finite and infinitesimal stability analyses, and we provide equations for the growth rate of arbitrary intial disturbances. These are applied to simple shear and simple-extensional flows. Finally, we discuss the failure of our model for small but non-zero ϵ, near points of extreme curvature of “entaglement”, and the possible relevance to the rheology of fibrous composites and suspensions.