Trajectory analysis for non-Brownian inertial suspensions in simple shear flow

Abstract

We analyse pair trajectories of equal-sized spherical particles in simple shear flow for small but finite Stokes numbers. The Stokes number, $\mbox{\textit{St}} \,{=}\, \dot{\gamma} \tau_p$, is a dimensionless measure of particle inertia; here, $\tau_p$ is the inertial relaxation time of an individual particle and $\dot{\gamma}$ is the shear rate. In the limit of weak particle inertia, a regular small-$\mbox{\textit{St}}$ expansion of the particle velocity is used in the equations of motion to obtain trajectory equations to the desired order in $\mbox{\textit{St}}$. The equations for relative trajectories are then solved, to $O(\mbox{\textit{St}})$, in the dilute limit, including only pairwise interactions. Particle inertia is found to destroy the fore–aft symmetry of the zero-Stokes trajectories, and finite-$\mbox{\textit{St}}$ open trajectories suffer net transverse displacements in the velocity gradient and vorticity directions. The vorticity displacement remains $O(\mbox{\textit{St}})$, while the scaling of the gradient displacement increases from $O(\mbox{\textit{St}})$ for far-field open trajectories, to $O(\mbox{\textit{St}}^{{1}/{2}})$ for open trajectories with $O(\mbox{\textit{St}}^{{1}/{2}})$ upstream gradient offsets. The gradient displacement also changes sign, being negative close to the plane of the reference sphere (the shearing plane) on account of dominant lubrication interactions, and then becoming positive at larger off-plane separations. The transverse displacements accompanying successive pair interactions lead to a diffusive behaviour for long times. The shear-induced diffusivity in the vorticity direction is $O(\mbox{\textit{St}}^2\phi \dot{\gamma} a^2)$, while that in the gradient direction scales as $O(\mbox{\textit{St}}^2 \ln \mbox{\textit{St}}\,\phi \dot{\gamma} a^2)$ and $O(\mbox{\textit{St}}^2 \phi \ln (1/\phi) \dot{\gamma} a^2)$ in the limits $\phi \,{\ll}\, \mbox{\textit{St}}^{{1}/{3}}$ and $\mbox{\textit{St}}^{{1}/{3}} \,{\ll}\, \phi \,{\ll}\, 1$, respectively. Further, the region of zero-Stokes closed trajectories is destroyed, and there exists a new attracting limit cycle whose location in the shearing plane is, at leading order, independent of $\mbox{\textit{St}}$. The extension of the present analysis to include a generic linear flow, and the implications of the finite-$\mbox{\textit{St}}$ trajectory modifications for coagulating systems are discussed.