Numerical study of two disks settling in an Oldroyd-B fluid: From periodic interaction to chaining

In this article, we present a numerical study of the dynamics of two disks sedimenting in a narrow vertical channel filled with an Oldroyd-B fluid. Two kinds of particle dynamics are observed: (i) a periodic interaction between the two disks, and (ii) the formation of a two-disk chain. For the periodic interaction of the two disks, two different motions are observed: (a) the two disks stay far apart and interact periodically, and (b) the two disks interact closely and then far apart in a periodic way, like the drafting, kissing, and tumbling of two disks sedimenting in a Newtonian fluid, due to a weak elastic force. Concerning the formation of a two-disk chain occurring at higher values of the elasticity number, either a tilted chain or a vertical chain is observed. Our simulations show that, as expected, the values of the elasticity and Mach numbers are the determining factors concerning the particle chain formation and its orientation.

Figure 1

An example of a two-dimensional flow region with four circular particles.

Figure 2

Histories of the two-disk horizontal velocity (left) and vertical velocity (right) obtained by different mesh sizes and time steps for ${\rho }_s=1.0025$ and $E=0.8$.

Figure 3

Positions of the two disks (left three) and trajectories of their centers (right) for ${\rho }_s=1.0025$ and $E=0.8$.

Figure 4

Positions of the two disks interacting apart (left four) and the trajectories of their centers (right) for ${\rho }_s=1.0025$ and $E=0.16$.

Figure 5

Positions of the two disks forming a tilted chain (left four) and the trajectories of their centers (right) for ${\rho }_s=1.0025$ and $E=0.256$.

Figure 6

Positions of the two disks forming a vertical chain (left four) and the trajectories of their centers (right) for ${\rho }_s=1.0025$ and $E=0.32$.

Figure 7

The limit cycles for ${\rho }_s=1.0025$ in phase space: The periods of the two disks interacting apart are 58.55, 56.80, 55.485, 54.205, 52.067, 49, and 47.9 for $E=0.016$, 0.080, 0.144, 0.192, 0.224, 0.2272, and 0.2288, respectively.

Figure 8

Trajectories of the two-disk centers for ${\rho }_s=1.0015$ at $E=0.1393$, 0.2424, 0.4848, and 0.12606 (from left to right).

Figure 9

Phase diagram (left) and associated values of the Mach number (right) for two disks interacting in a narrow vertical channel for ${\rho }_s=1.0015$, 1.0025, 1.0035, and 1.005.

Figure 10

Positions of the two disks drafting, kissing, and nonchaining for ${\rho }_s=1.005$ and $E=0.4$.

Figure 11

The limit cycles for ${\rho }_s=1.005$ in phase space: The periods of the two disks interacting apart are 21.65, 18.05, and 16.28 for $E=0.16$, 0.24, and 0.32, respectively (left) and the periods of two disks drafting, kissing, and nonchaining are 34.2, 38.45, 46.45, and 64 for $E=0.36$, 0.4, 0.44, and 0.48, respectively (right).

Figure 12

Positions of two disks forming an almost horizontal chain for ${\rho }_s=1.005$ at $E=0.56$.

Figure 13

Trajectories of the two-disk centers forming vertical chains for $E=1.6$ and the density ${\rho }_s=1.0015$, 1.0025, 1.0035, and 1.0045 (from left to right).

Figure 14

Positions of the two disks forming a tilted chain (left three) and trajectories (right) of the two disks for ${\rho }_s=1.01$ and $E=1.6$ and $M=3.0784$.

Figure 15

Phase diagram (left) and associated values of the Reynolds number (right) for two disks interacting in a vertical channel with different confined ratios.

Figure 16

Left: Histories of the two-disk horizontal velocity (left top), vertical velocity (left middle), and angular velocity (left bottom) for ${\rho }_s=1.0015$, $d=0.1$, $K=10$, and $E=0.2$. Right: The limit cycles for the angular velocity and the horizontal position of the disk mass center for $K=8$ (solid line), 10 (dashed-dotted line), and 12 (dashed line).

Figure 17

Phase diagram for two different sized disks interacting in a vertical channel: One disk has a fixed diameter of $d=0.125$ and the other one has $0.9d$, $0.95d$, $d$, $1.05d$, and $1.1d$ (from bottom to top in the plot).

Figure 18

Positions and orientations of the two rigidly connected disks (left four) and the trajectory of the long body mass center (right) for ${\rho }_s=1.0025$ and $E=0.02$.