Mean transport of inertial particles in viscous streaming flows

Viscous streaming has emerged as an effective method to transport, trap, and cluster inertial particles in a fluid. Previous work has shown that this transport is well described by the Maxey-Riley equation augmented with a term representing Saffman lift. However, in its straightforward application to viscous streaming flows, the equation suffers from severe numerical stiffness due to the wide disparity between the timescales of viscous response, oscillation period, and slow mean transport, posing a severe challenge for drawing physical insight on mean particle trajectories. In this work, we develop equations that directly govern the mean transport of particles in oscillatory viscous flows. The derivation of these equations relies on a combination of three key techniques. In the first, we develop an inertial particle velocity field via a small Stokes number expansion of the particle's deviation from that of the fluid. This expansion clearly reveals the primary importance of Faxén correction and Saffman lift in effecting the trapping of particles in streaming cells. Then, we apply the generalized Lagrangian mean theory to unambiguously decompose the transport into fast and slow scales, and ultimately, develop the Lagrangian mean velocity field to govern mean transport. Finally, we carry out an expansion in small oscillation amplitude to simplify the governing equations and to clarify the hierarchy of first- and second-order influences, and particularly, the crucial role of Stokes drift in the mean transport. We demonstrate the final set of equations on the transport of both fluid and inertial particles in configurations involving one cylinder in weak oscillation and two cylinders undergoing such oscillations in sequential intervals. Notably, these equations allow numerical time steps that are $O\left({10}^3\right)$ larger than the existing approach with little sacrifice in accuracy, allowing more efficient predictions of transport.

Figure 1

Schematic of flow maps used in this work. ${\mathcal{X}}_t$ and ${\mathcal{Y}}_t$ represent slices of the configuration spaces ${\mathcal{X}}_t\times [0,\infty )$ and ${\mathcal{Y}}_t\times [0,\infty )$ at some instant $t$. Illustrations of particle trajectories are shown as colored curves (though strictly speaking, these trajectories would proceed along the time axis of the respective space).

Figure 2

Mean (green) and actual (blue) particle trajectories with initial position ${\mathit{x}}_0$. The position ${\mathit{x}}^{\xi }$ is the actual position whose mean is $\mathit{x}$. Adapted from Bühler and McIntyre [40].

Figure 3

Overall algorithm for the fast Lagrangian averaged transport of particles.

Figure 4

Diagram of the oscillating cylinder (right) and a time sequence illustrating the repeated oscillation cycles.

Figure 5

Configuration and oscillation sequence with two cylinders adapted from the work of Chong et al. [14].

Figure 6

Time-resolved inertial particle trajectories from the Maxey-Riley equation (black) and the asymptotic expansion in small Stokes number (green) for particles initially located at ${\mathit{x}}_0=(2,2)$ (left column) and ${\mathit{x}}_0=(1,3)$ (right column). The top row (a), (d) shows the trajectories over the first 20 oscillation periods, and the middle (b), (e) and lower (c), (f) rows depict the time histories of the $x$ and $y$ components, respectively, sampled once per period.

Figure 7

Trajectory of an inertial particle initially located at ${\mathit{x}}_0=(-2,3)$ over 65 000 periods of oscillation. Trajectories from the Maxey-Riley equation (black) and small Stokes number expansion (green) are both sampled once per period.

Figure 8

Lagrangian mean trajectories (magenta circles, spaced by 30 periods) generated for fluid particles started at ${\mathit{x}}_0=(2,2)$ and ${\mathit{x}}_0=(1,3)$ (denoted by the larger gray circle), compared with the Lagrangian streamlines, depicted as black lines. (Other Lagrangian streamlines are shown in light gray.) The black region shows the mean position of the cylinder, and the lighter shaded region in the vicinity of the cylinder shows the range of displacement of the cylinder over one oscillation cycle.

Figure 9

(a) Lagrangian mean trajectory (magenta) for fluid particle started at ${\mathit{x}}_0=(2,2)$ compared with the full time-resolved trajectory (blue) for the same particle. (b) Magnified view of fluid particle trajectory. (c), (d) Comparison of $x$ and $y$ components, respectively, of the Lagrangian mean trajectory and the full time-resolved trajectory (sampled once per cycle).

Figure 10

Left: Time-resolved fluid particle trajectory, sampled once per cycle (blue), and Lagrangian mean trajectory (magenta circles, spaced by 30 periods), and Lagrangian mean streamlines (in light gray) for ${\mathit{x}}_0=(-1,3)$, with mean cylinder configurations depicted in black. Right: Time histories of the $x$ and $y$ components of both trajectories shown in the left panel.

Figure 11

Lagrangian mean trajectories (magenta) generated for inertial particles started at (a) ${\mathit{x}}_0=(2,2)$ and (b) ${\mathit{x}}_0=(1,3)$ (denoted by the small circle), compared with the full time-resolved trajectories (blue) of the same particles, sampled once per cycle. The Lagrangian streamlines of the fluid are shown for reference as light gray lines. The black region shows the mean position of the cylinder, and the lighter shaded region in the vicinity of the cylinder shows the range of displacement of the cylinder over one oscillation cycle.

Figure 12

Left column: (a), (b) Comparison of $x$ and $y$ components, respectively, of the Lagrangian mean trajectory and the full time-resolved trajectory (sampled once per cycle) for particle ${\mathit{x}}_0=(2,2)$. (c) Lagrangian mean trajectory (magenta) for inertial particle ${\mathit{x}}_0=(2,2)$, compared with the full time-resolved trajectory (blue) for the same particle. Right column: (d), (e) Comparison of $x$ and $y$ components, respectively, of the Lagrangian mean trajectory and the full time-resolved trajectory (sampled once per cycle) for particle ${\mathit{x}}_0=(1,3)$. (f) Lagrangian mean trajectory (magenta) for inertial particle started at ${\mathit{x}}_0=(1,3)$, compared with the full time-resolved trajectory (blue) for the same particle.

Figure 13

Lagrangian mean trajectories generated for inertial particles with ${\rho }_p/{\rho }_f=0.95$ (magenta) and 0.05 (blue) started at ${\mathit{x}}_0=(2,2)$.

Figure 14

(a) Full time resolved inertial particle trajectory, sampled once per cycle (blue), and Lagrangian mean trajectory (magenta) for a particle released from ${\mathit{x}}_0=(-2,3)$ in the sequential oscillator configuration. (b), (c) Time histories of the $x$ and $y$ trajectory components, respectively.

Figure 15

Time-resolved inertial particles trajectories (blue) and mean transport algorithm (yellow) for ${\mathit{x}}_0=(-2.0,3.0)$ over the transient regime of the left cylinder (left) and ${\mathit{x}}_0=(-1.9285,1.0266)$ over the transient regime of the right cylinder (right).