Particle accumulation structures in noncylindrical liquid bridges under microgravity conditions

The emergence of particle accumulation structures (PAS) in noncylindrical liquid bridges (LBs) is studied numerically for a high Prandtl number liquid considering microgravity conditions. Simulations are conducted in the framework of a finite-volume (Eulerian) approach with nonisodense particles tracked using a Lagrangian, one-way coupling scheme. First, the threshold of the Marangoni-flow instability is determined as a function of the aspect ratio and the volume of liquid held between the supporting disks, thereafter, PAS formation is investigated for supercritical conditions. The overall approach is specifically conceived to provide details about the morphological evolution of these structures as the main control parameters are varied. For this reason a set of dedicated notions and definitions (such as the linear extension of the PAS, its inner core radius, and the area of the “petals” or “blades”) are introduced to allow a precise quantification of a series of purely geometrical effects. Though the analysis is deliberately limited to illustrating the macroscopic patterning behavior and its relationship with the overarching factors, a model is proposed to interpret the increased ability of slender (concave) LBs to support the formation of PAS over extended ranges of values of the particle Stokes number. This model yet relies on essentially geometrical arguments, that is, the triadic relationship among the curvature of the free surface, the topology of fluid streamlines, and particle mass effects.

Figure 1

Sketch of LBs with different volume ratios, $S$, and the coordinates system adopted to represent the interface.

Figure 2

Schematics of the LB interface showing the unit normal and tangent vectors, and the interface temperature gradient ${\mathbf{\nabla }}_{s}T$.

Figure 3

Computational cell next to a boundary showing the particle velocity before “time frame $h$”), and after “time frame $h+1$”) the particle-boundary interaction treatment.

Figure 4

Computational domain (mesh $M_1$) for the configuration $\text{AR}=0.34$, $S=1.1.$

Figure 5

Top view of PAS obtained obtained numerically: (a) present simulation, (b) after [42]. Flow conditions are summarized in Table 1. In panel (a), the larger circle represents the periphery of the closest disk, while the smaller circle is periphery of the opposite disk; the shaded region corresponds to the axial projection of the interface.

Figure 6

Critical Marangoni number and respective critical frequency versus volume ratio for the cases with aspect ratio $\text{AR}=0.34.$

Figure 7

Critical Marangoni number and respective critical frequency versus volume ratio for the cases with aspect ratio $\text{AR}=0.5.$

Figure 8

Critical Marangoni number and respective critical frequency versus volume ratio for the cases with aspect ratio $\text{AR}=0.9.$

Figure 9

Meridian plane (top), and axial view (below) of the temperature field in the presence of hydrothermal waves for three different aspect ratios: $\text{AR}=0.34$, $S=0.92$ (left), $\text{AR}=0.5$, $S=0.82$ (center), and $\text{AR}=0.9$, $S=0.8$ (right). The periphery of the circular cross sections corresponds to the free surface of the LB.

Figure 10

Lateral and top views of PAS obtained numerically for the case $\text{AR}=0.34$ and three different volume ratios: $S=0.92$ ($t^{*}\approx 0.375$) (left), $S=1$ ($t^{*}\approx 2.5$) (center), and $S=1.1$ ($t^{*}\approx 0.88$) (right). Flow conditions are summarized in Table 4. (For the axial views, the larger circle represents the periphery of the closest disk, while the smaller circle is the periphery of the opposite disk; the shaded region corresponds to the axial projection of the interface). The same perspective mode has also been used for other similar three-dimensional figures included in the present work.

Figure 11

Lateral and top views of PAS obtained numerically for the case $\text{AR}=0.5$ and three different volume ratios: $S=0.82$ ($t^{*}\approx 0.16$) (left), $S=1$ ($t^{*}\approx 0.45$) (center), and $S=1.3$ ($t^{*}\approx 0.24$) (right). Flow conditions are summarized in Table 6.

Figure 12

Streamlines for an axis-symmetric two-dimensional simulation for $\text{AR}=0.5$ and $S=0.82$. Flow conditions are $\text{Ma}=14000$ and $\text{Pr}=8.$

Figure 13

Lateral (above) and axial (below) views of the PAS found for the case $\text{AR}=0.9$ and $S=0.6$ ($t^{*}\approx 0.28$).

Figure 14

Dimensionless PAS length, ${\mathit{l}}_{PAS}$, as a function of $S$ for $\text{AR}=0.34,0.5$ ($\text{Ma}/{\text{Ma}}_c\sim 2$). The dashed lines represent quadratic polynomial fits.

Figure 15

View resulting from the superposition of the different images obtained through rotation of a single time-frame (i.e., a single image of the PAS) by constant angular increments. The black line indicates the pinning area in correspondence of the cold disk (conditions are $\text{AR}=0.5$, $S=0.82$).

Figure 16

Top view of the structure for the case $\text{AR}=0.34$, $S=1$ with a schematic depiction of the projected area of a single petal, $A_p$ and it maximum radial extension, $\overline{r}$. Note that $r_m=d_m/2$ (cf. $d_m$ indicated in Fig. 15).

Figure 17

Dimensionless extension of the cavity, $d_m^{*}$ as a function of $S$ for $\text{AR}=0.34,0.5$ ($\text{Ma}/{\text{Ma}}_c\sim 2$; the dashed lines represent quadratic polynomial fits).

Figure 18

Dimensionless extension of the cavity, $d_m^{*}$ as a function of the Marangoni number for the case $S=1$ and $\text{AR}=0.34$).

Figure 19

Location of the main toroidal vortex in a shallow LB with convex interface; the deformation of the free surface has been exaggerated for illustration purposes.

Figure 20

Dimensionless width of a single PAS petal as a function of the volume ratio ($\text{Ma}/{\text{Ma}}_c\sim 2$). The dashed lines represent quadratic polynomial fits.

Figure 21

Module of the dimensionless velocity field (left) and velocity distribution on its interface (right) for a two-dimensional, concave LB. The flow conditions adopted were $\mathrm{Pr}=8$, $\mathrm{AR}=0.5$, $S=0.82$, and $\text{Ma}=18000.$

Figure 22

Schematics of the trajectories described by a particle next to the interface moving toward the “cold” corner of the LB for two opposite configurations: concave interface (top), and convex (below). The dashed line indicates the free surface.

Figure 23

Deviation of the trajectory of a single particle (evaluated with respect to the reference position obtained for the case $\xi =1$) as a function of the density ratio for two different LB configurations: concave bridge $\text{AR}=0.5$ and $S=0.82$ (cc), and $\text{AR}=0.5$ and $S=1.3$ (cv). The positions are taken in correspondence of two different cross sections at $z/L=1/2$ and $z/L=1/4$.