Invisible Anchors Trap Particles in Branching Junctions

We combine numerical simulations and an analytic approach to show that the capture of finite, inertial particles during flow in branching junctions is due to invisible, anchor-shaped three-dimensional flow structures. These Reynolds-number-dependent anchors define trapping regions that confine particles to the junction. For a wide range of Stokes numbers, these structures occupy a large part of the flow domain. For flow in a $\mathsf{V}$-shaped junction, at a critical Stokes number, we observe a topological transition due to the merger of two anchors into one. From a stability analysis, we identify the parameter region of particle sizes and densities where capture due to anchors occurs.

Figure 1

Analysis of a $\mathsf{T}$-junction experiment with hollow glass beads in a steady flow of water ($\mathrm{Re}=277$; cf. the video in the Supplemental Material [5]). The density of the beads is ${\rho }_p=0.15{\rho }_f$, where ${\rho }_f$ is the density of water. (a) Time-lapse image, with arrows marking the inlet (top) and two outlets (left, right). Beads are released at the inlet and exit through one of the outlets. The spiraling of some bead paths (dark gray curves) is due to vortex breakdown in the junction arms. The black box marks the subregion shown in (c)–(f). (b) Illustration of the sequences $S1–S4$ shown in (c)–(f). (c) A particle is at rest at a fixed point, while other particles pass by quickly. (d) The particle is displaced by another particle and begins to move downstream. (e) After spiraling downstream, the particle stops its downstream motion and shows damped oscillations perpendicular to the downstream direction. (f) The particle slowly creeps back to its starting position along a line (orange).

Figure 2

Primary trapping regions (TRs, colored yellow) in the $\mathsf{T}$- and $\mathsf{V}$-junction flows for $ϵ=0.05$. This value of $ϵ$ corresponds to, e.g., hollow glass beads ($\rho =0.15$) with $\beta =2.3078$, $\tau =0.0382$, and $\mathrm{St}=0.0588$ used in experiments [4], or gas bubbles ($\rho ={10}^{-3}$) with $\beta =2.994$, $\tau =0.0251$, and $\mathrm{St}=0.0501$. Blue crosses mark the fixed points $P_{1,2,3,4}$ and $Q_{1,2,3,4}$. (a) $\mathsf{T}$ junction at $\mathrm{Re}=320$ (beads with $a=2.9\times {10}^{-2}$, bubbles with $a=2.7\times {10}^{-2}$). (b) $\mathsf{T}$ junction viewed from the top. (c) $\mathsf{V}$ junction at $\mathrm{Re}=230$, with representative particle trajectories (beads with $a=3.4\times {10}^{-2}$, bubbles with $a=3.1\times {10}^{-2}$). The boundary of the TR separates trajectories that exit through the outlet (red) and trajectories that remain inside the TR and spiral onto the extended vortex axis (green).

Figure 3

Trapping regions in the $\mathsf{V}$-junction flow at $\mathrm{Re}=230$ for different Stokes numbers St representing the range of experimentally observed bubble sizes (between $a=5.0\times {10}^{-4}$ and $a=2.5\times {10}^{-2}$ [4]). (a) $\mathrm{St}\approx ϵ=1.9\times {10}^{-3}$ ($a=6.0\times {10}^{-3}$). (b) $\mathrm{St}\approx ϵ=6.6\times {10}^{-3}$ ($a=1.1\times {10}^{-2}$). (c) Critical case, merger of TRs: $\mathrm{St}\approx ϵ={ϵ}_M=0.0161$ ($a=1.8\times {10}^{-2}$). (d) $\mathrm{St}\approx ϵ=0.0319$ ($a=2.5\times {10}^{-2}$). (Bottom-left panels) $z-y$ views showing that, below the critical value $\mathrm{St}\approx {ϵ}_M=0.0161$, the TRs are well separated. (Bottom-right panels) Intersections of the inlet cross section ($y=3$) with the TRs.

Figure 4

Parameter regions of St and $\rho$ where capture occurs. Vertical lines and squares mark St and $\rho$ values of particles from the experiments in Ref. [4]. Gas bubbles, blue; hollow glass beads, magenta; and polystyrene beads, gray. Thin black curves are level sets of $ϵ$ [see Eq. (2)], indicating parameter regions where we expect that Eq. (1) ceases to be valid. (a) $\mathsf{T}$ junction at $\mathrm{Re}=320$. (b) $\mathsf{V}$ junction at $\mathrm{Re}=230$. Capture occurs either in two TRs, or, for $ϵ>{ϵ}_M=0.016$, in a single TR.