Inertial effects on the orientation of nearly spherical particles in simple shear flow

Abstract

We investigate theoretically the first effects of inertia on the orientation dynamics of a torque-free spheroidal particle in simple shear flow when the deviation from sphericity is small. The inertialess motion of any axisymmetric particle in simple shear represents a degenerate limit, the spheroidal geometry being a special case; as originally found by Jeffery (Proc. R. Soc. Lond. A, vol. 102, 1922, p. 161), the orientation vector moves indefinitely along any one of a single-parameter family of closed orbits centred around the vorticity axis, the distribution across orbits being determined by initial conditions. We consider both the inertia of the particle and that of the suspending fluid, characterized by the Stokes ($\hbox{\it St}$) and Reynolds numbers ($\hbox{\it Re}\,{=}\, \rho_f/\rho_p\hbox{\it St}$, $\rho_p$ and $\rho_f$ being the particle and fluid densities), respectively, as mechanisms for breaking the aforementioned degeneracy. The former is defined as $\hbox{\it St} \,{=}\, a^2\dot{\gamma}\rho_p/\mu$, where $\dot{\gamma}$ is the shear rate, $a$ is the radius of the unperturbed sphere and $\mu$ is the fluid viscosity. When the particles are much denser than the suspending fluid, as is the case for aerosols, $\hbox{\it St} \,{\gg}\, \hbox{\it Re}$ (both parameters being much less than unity), inertial forces in the fluid may be neglected. It is then found, in the absence of gravity, that a slightly prolate spheroid drifts toward the shearing plane, while the axis of a slightly oblate spheroid tends toward the vorticity axis, both on a time scale of $O(|\epsilon | \hbox{\it St} \dot{\gamma})^{-1}$, where $\epsilon ({\ll}\,1)$ is the deviation from sphericity. For the case of neutrally buoyant particles ($\hbox{\it St} \,{=}\,\hbox{\it Re}$), inertia of both the particle and fluid come into play. In contrast to the small but finite $\hbox{\it St}$ zero $\hbox{\it Re}$ case, the orientation vector of a neutrally buoyant prolate spheroid now migrates toward the direction of vorticity, while that of an oblate spheroid drifts towards the shearing plane. The time scale of drift towards the asymptotic state is $O(| \epsilon | \hbox{\it Re} \dot{\gamma})^{-1}$ in both cases. Thereafter, we also examine the rotations of prolate and oblate spheroids in the presence of both gravity and shear, the analysis again being restricted to weak inertial effects. A wide range of interesting orientational behaviour arises, and the long-time orientation dynamics of the spheroids are determined as a function of both the density ratio $\rho_p/\rho_f$ and a shear parameter $N$, defined as $N\,{=}\, 2a \rho_f g/(9\mu \dot{\gamma})$.