The inertial orientation dynamics of anisotropic particles in planar linear flows

Abstract

In the Stokes limit, the trajectories of neutrally buoyant torque-free non-Brownian spheroids in ambient planar linear flows are well known. These flows form a one-parameter family, with the velocity gradient tensor given by $\unicode[STIX]{x1D735}\boldsymbol{u}^{\infty \dagger }=\dot{\unicode[STIX]{x1D6FE}}(\mathbf{1}_{x}^{\prime }\mathbf{1}_{y}^{\prime }+\unicode[STIX]{x1D706}\mathbf{1}_{y}^{\prime }\mathbf{1}_{x}^{\prime })$. The parameter $\unicode[STIX]{x1D706}$ is related to the ratio of the vorticity to the extension (given by $(1-\unicode[STIX]{x1D706})/(1+\unicode[STIX]{x1D706})$), and ranges from $-1$ to 1, with $\unicode[STIX]{x1D706}=1\,,0$ and $-1$ being planar extensional flow, simple shear flow and solid-body rotation respectively. The unit vectors $\mathbf{1}_{x}^{\prime }$ and $\mathbf{1}_{y}^{\prime }$ are unit vectors along the flow and gradient axes of the simple shear flow ($\unicode[STIX]{x1D706}=0$). The trajectories, as described by a unit vector along the spheroid symmetry axis, are closed orbits for $\unicode[STIX]{x1D706}<\unicode[STIX]{x1D706}_{crit}$, where $\unicode[STIX]{x1D706}_{crit}=\unicode[STIX]{x1D705}^{2}(1/\unicode[STIX]{x1D705}^{2})$ for an oblate (a prolate) spheroid of aspect ratio $\unicode[STIX]{x1D705}$. We investigate analytically the orientation dynamics of such a spheroid in the presence of weak inertial effects. The inertial corrections to the angular velocities at $O(Re)$ and $O(St)$, where $Re$ and $St$ are the Reynolds ($Re=\unicode[STIX]{x1D70C}_{f}\dot{\unicode[STIX]{x1D6FE}}L^{2}/\unicode[STIX]{x1D707}$) and Stokes numbers ($St=\unicode[STIX]{x1D70C}_{p}\dot{\unicode[STIX]{x1D6FE}}L^{2}/\unicode[STIX]{x1D707}$) respectively, are derived using a reciprocal theorem formulation. Here, $L$ is the semimajor axis of the spheroid, $\unicode[STIX]{x1D707}$ is the viscosity of the suspending fluid, $\dot{\unicode[STIX]{x1D6FE}}$ is the shear rate, and $\unicode[STIX]{x1D70C}_{p}$ and $\unicode[STIX]{x1D70C}_{f}$ are the particle and fluid densities respectively. A spheroidal harmonics formalism is then used to evaluate the reciprocal theorem integrals and obtain closed-form expressions for the inertial corrections. The detailed examination of these corrections is restricted to the aforementioned Stokesian closed-orbit regime ($\unicode[STIX]{x1D706}<\unicode[STIX]{x1D706}_{crit}$). Here, even weak inertia, for asymptotically long times, of $O(1/(\dot{\unicode[STIX]{x1D6FE}}Re))$ or $O(1/(\dot{\unicode[STIX]{x1D6FE}}St))$, will affect the leading-order orientation distribution on account of the indeterminate nature of the distribution across orbits in the Stokes limit. For $\unicode[STIX]{x1D706}<\unicode[STIX]{x1D706}_{crit}$, inertia results in a drift across the closed orbits in Stokes flow, and this orbital drift is characterized using a multiple time scale analysis. The orbits stabilized by the inertial drift, at $O(Re)$ and $O(St)$, are identified in the $\unicode[STIX]{x1D706}{-}\unicode[STIX]{x1D705}$ plane. For the majority of ($\unicode[STIX]{x1D706},\unicode[STIX]{x1D705}$) combinations, the stabilized orbit is either one confined to the plane of symmetry (the flow-gradient plane) of the ambient flow (the tumbling orbit) or one where the spheroid is aligned with the ambient vorticity vector (the spinning orbit). However, for some ($\unicode[STIX]{x1D706},\unicode[STIX]{x1D705}$) combinations, depending on the initial orientation, the orbit stabilized can be either the spinning or the tumbling orbit, since both orbits have non-trivial basins of attraction, separated by a pair of unstable (repelling) limit cycles, on the unit sphere of orientations. A stochastic orientation decorrelation mechanism in the form of rotary Brownian motion, characterized by a Péclet number, $Pe_{r}$ ($Pe_{r}=\dot{\unicode[STIX]{x1D6FE}}/D_{r}$, where $D_{r}$ is the rotary Brownian diffusivity), is included to eliminate the aforementioned dependence on the initial orientation distribution for certain ($\unicode[STIX]{x1D706}$, $\unicode[STIX]{x1D705}$) combinations. The unique steady-state orientation distribution determined by the combined effect of Brownian motion and inertia is obtained by solving a closed-orbit-averaged drift–diffusion equation. The steady-state orientation dynamics of an inertial spheroid in a planar linear flow, in the presence of weak thermal orientation fluctuations, has similarities to the thermodynamic description of a one-component system. Thus, we identify a tumbling–spinning transition in a $C{-}\unicode[STIX]{x1D705}{-}Re\,Pe_{r}$ space. Here, $C$ is the orbital coordinate that acts as a label for the closed orbits in the Stokes limit. This transition implies hysteretic orientation dynamics in certain regions in the $C$–$\unicode[STIX]{x1D705}$–$Re\,Pe_{r}$ space, although the hysteretic volume shrinks rapidly on either side of simple shear flow. In the hysteretic region, one requires exceedingly large times to achieve the unique steady-state distribution (underlying the thermodynamic interpretation), and for durations relevant to experiments, the system may instead attain an initial-condition-dependent metastable distribution.