Orientation dynamics of a spheroid in the simple shear flow of a weakly elastic fluid

TL;DR

This paper studies how weak viscoelasticity affects the orientation and rotation of spheroids in shear flow, revealing transitions in their tumbling and spinning behaviors depending on fluid elasticity and particle shape.

Contribution

It provides a novel analytical framework for understanding spheroid orientation dynamics in weakly viscoelastic fluids using reciprocal theorem and multiple-scales analysis.

Findings

Weak elasticity induces a spiraling orientation trajectory.

Prolate spheroids tend to spin mode in polymeric fluids.

Oblate spheroids switch between tumbling and kayaking modes depending on aspect ratio.

Abstract

We investigate the orientation dynamics of a neutrally buoyant spheroid, of an arbitrary aspect ratio ($\kappa$), freely rotating in a weakly viscoelastic fluid undergoing simple shear flow. Weak elasticity is characterized by a small but finite Deborah number ($De$), and the suspending fluid rheology is therefore modeled as a second-order fluid, with the constitutive equation involving a material parameter $\epsilon$ related to the ratio of the first and second normal stress differences; polymer solutions correspond to $\epsilon\in[-0.7,-0.5]$. Employing a reciprocal theorem formulation, along with expressions for the relevant disturbance fields in terms of vector spheroidal harmonics, we obtain the spheroid angular velocity to $O(De)$. In the Newtonian limit, a spheroid rotates along Jeffery orbits parametrized by an orbit constant $C$, although this closed-trajectory topology is structurally unstable, being susceptible to weak perturbations. For $De$ well below a threshold, $De_c(\kappa)$, weak viscoelasticity transforms the closed-trajectory topology into a tightly spiralling one. A multiple-scales analysis is used to interpret the resulting orientation dynamics in terms of an $O(De)$ orbital drift. The drift in orbit constant over a Jeffery period $\Delta C$, when plotted as a function of $C$, identifies four different orientation dynamics regimes on the $\kappa-\epsilon$ plane. For $\epsilon$ in the polymeric range, prolate spheroids always drift towards the spinning mode. Oblate spheroids drift towards the tumbling mode for $\kappa > \kappa_c(\epsilon)$, but towards an intermediate kayaking mode for $\kappa < \kappa_c(\epsilon)$. The rotation of spheroids of extreme aspect ratios, either slender prolate spheroids ($\kappa \gg 1$) or thin oblate ones ($\kappa \ll 1$), about the vorticity axis, is arrested for $De \geq De_c(\kappa)$