Finite-size Lagrangian coherent structures in thermocapillary liquid bridges

The rapid accumulation of small, finite-size rigid particles along closed rotating threads in high-Prandtl-number thermocapillary liquid bridges is investigated numerically when the flow arises as a traveling hydrothermal wave. For a dilute suspension, different motion models are investigated which could provide the dissipation mechanisms leading to the experimentally observed particle-motion attractors. Making use of a phenomenological particle-boundary interaction model, it is shown that the particle size effect, which becomes important when the particle moves near the thermocapillary free surface, provides the relevant source of dissipation for the remarkably fast particle accumulation. Furthermore, the accumulation phenomenon is tightly correlated with the Kolmogorov-Arnold-Moser (KAM) tori of the flow in the absence of particles. Therefore, KAM tori in the rotating frame of reference, in which the flow field is steady, can be considered templates for the accumulation structures. The numerical results obtained are compared with experimental data for the so-called spiral loop 1 and spiral loop 2 particle accumulation structures. In addition, other accumulation structures are numerically predicted which yet await experimental confirmation.

Figure 1

Geometry of the liquid bridge and coordinate system.

Figure 2

Sketch of the PSI model for the motion of a spherical particle (circles) with centroid indicated by a dot. The dotted curve represents the particle trajectory in the absence of the PSI model. The full line represents the trajectory when the PSI model is employed. The short-dashed line indicates the axis of the liquid bridge and the long arrow indicates the direction of the thermocapillary stresses along the interface (long-dashed line). The particle radius $a$, lubrication gap width $\delta$, and interaction parameter $\mathrm{\Delta }$ are indicated.

Figure 3

Distribution of finite volumes in the $(x,y)$ (a) and the $(x,z)$ plane (b). The grid employed for the simulations is five times finer than the one depicted, consisting of $\approx 20$ million cells. The double lines indicate the main blocks of the mesher software blockmesh.

Figure 4

Isotherms (left) and streamlines (right) of the unstable axisymmetric basic flow for $\mathrm{\Gamma }=0.68,\text{Re}=1600,\text{Pr}=28,\text{Bi}=0.3$, and $\text{Gr}=687$. The dashed lines and the dash-dotted line indicate the free surface and the axis of the liquid bridge, respectively.

Figure 5

Temperature field for $\mathrm{\Gamma }=0.68,\text{Re}=1600,\text{Pr}=28,\text{Bi}=0.3$, and $\text{Gr}=687$. (a) Isosurface $\theta =0.5$. (b) Isotherms in the midplane $z=0$. The direction of rotation is clockwise in panels (a) and (b). The gray scale indicates the velocity magnitude in panel (a) and the temperature field in panel (b).

Figure 6

Sketch of the topological elements of the supercritical flow. The three critical points $s_1,s_2$ and $s^{\prime }$ are indicated by dots. The three limit cycles $w_1,w_2$, and $w_c$ are represented as dotted and full lines. Arrows show the flow along the stable and unstable manifolds. The gray arrow indicates the broken connection between $s^{\prime }$ and $s_1$.

Figure 7

(a) The broken connection between $s_1$ and $s^{\prime }$ for $\text{Re}=1600$: The two unstable one-dimensional manifolds of $s^{\prime }$ obtained by integration forward in time are shown in gray. The stable manifold of $s_1$ (black) obtained by integrating backward in time from $s_1$ does not connect with $s^{\prime }$, but locally approximates the stable manifold of $s^{\prime }$. (b) Limit cycle $w_c$ (separation line) on the cold wall for $\text{Re}=1600$ (dots) and $\text{Re}=1850$ (squares). The arrows indicate the attraction to $w_c$ (repulsion from $w_1$) along the cold wall.

Figure 8

Poincaré section on the midplane $z=0$ of chaotic streamlines (light gray dots) and of streamlines on the largest reconstructible KAM tori (black dots) in the rotating frame for (a) $\text{Re}=1600$ with ${}^1{T}_3^3$, (b) $\text{Re}=1750$ with ${}^1{T}_3^3$, (c) $\text{Re}=1850$ with ${}^1{T}_3^3$ and ${}^1{T}_3^6$, and (d) $\text{Re}=1950$ with ${}^1{T}_3^6$ and ${}^0{T}_3^1$. Also shown are projections of the closed streamlines (lines) and of the largest reconstructible KAM tori (dark gray shading). The intersections of the closed streamlines with the Poincaré plane are marked by diamonds ($\diamond$). The arrow shows the direction of propagation of the hydrothermal wave in the laboratory frame. Fluid elements in the rotating frame move counterclockwise about the axis in the mean.

Figure 9

Three-dimensional view of the largest reconstructible KAM tori ${}^1{T}_3^3$ (dark gray) and ${}^1{T}_3^6$ (light gray) for different Reynolds numbers in the rotating frame of reference. A black line on a dark gray KAM torus represents part of the open streamline defining the torus. The arrow shows the direction of propagation of the hydrothermal wave.

Figure 10

Top view of finite-size Lagrangian coherent structures of SL-2 type for $\text{Re}=1950,a=0.002941$, and $\varrho =2.52$ (corresponding to $\mathrm{\Delta }=0.00422$), shown in the rotating frame of reference. (a) Experimental result (courtesy I. Ueno, see Ref. [46]). [(b)–(d)] Particle configuration (light gray dots) at $t=84$ resulting from numerical simulations according to model I (b), II (c), and III (d). The closed full line represents the closed streamline ${}^1{L}_3^6$ [see Fig. 8]. In panel (c), also an enlargement is shown into the region near the free surface indicated by the dark gray rectangle. The arrow indicates the direction of propagation of the pattern in the laboratory frame. The light gray full squares denote immobile particles having settled on the cold bottom wall.

Figure 11

Top view of finite-size Lagrangian coherent structures for $\text{Re}=1850$ shown in the rotating frame of reference. (a) Experimental result for $a=0.00424$, and $\varrho =1.77$ (courtesy I. Ueno, see Ref. [46]). [(b)–(d)] Particle configuration (light gray dots) at $t=50$ using the advection model II. Closed streamlines of the KAM tori ${}^1{T}_3^6$ (b) and ${}^1{T}_3^3$ [(c), (d)] from Fig. 8 are shown as full lines. The simulations differ by the interaction parameter: (b) $\mathrm{\Delta }={\mathrm{\Delta }}_{\text{exp}}=0.00497$, selected according to the experiment (line-like SL-2 PAS), (c) $\mathrm{\Delta }=0.011$ (tubular SL-1 PAS), and (d) $\mathrm{\Delta }={\mathrm{\Delta }}_{\text{fs}}{(}^1{L}_3^3)=0.01251$ (line-like SL-1 PAS). The dashed line refers to one of the three azimuths at which the closed streamline admits its minimum radial coordinate: $\phi ={\phi }_{r_{\text{min}}}=-0.9460$. The arrow indicates the direction of propagation in the laboratory frame. Using model I instead of model II yields visually indistinguishable particle attractors (not shown).

Figure 12

Poincaré section at $\phi ={\phi }_{r_{\text{min}}}=-0.9460$ of particle trajectories (gray dots) during the time interval $t\in [80,84]$ for $\text{Re}=1850$ and $\mathrm{\Delta }=0.011$ (tubular SL-1 PAS) corresponding to Fig. 11. The black dots delineate the largest reconstructible KAM torus ${}^1{T}_3^3$ and the diamond marks the closed streamline. The light gray strip at the right boundary indicates the prohibited region of width $\mathrm{\Delta }$ inaccessible by the particle centroid within the PSI model. The inset shows an enlargement.

Figure 13

(a) Experimental observation from the top view of the early phase of formation of SL-1 PAS for $\text{Re}=1600$; $a=0.00424,\varrho =1.77$ (courtesy of I. Ueno, see Ref. [46]). The top view of the early phase of a numerically simulated SL-1 type of particle accumulation for $\mathrm{\Delta }=0.00552$ at $t=25$ (1000 particles initially randomly distributed) is shown for model II (b) and model I (c). The closed streamline inside the ${}^1{T}_3^3$ torus is shown as a solid line. The particles are represented as light gray markers and the squares in panel (c) refer to settled particles.

Figure 14

Particle accumulation structure for $\text{Re}=1600$ and $\mathrm{\Delta }={\mathrm{\Delta }}_{\text{exp}}=0.00552$ at $t=84$ from an initially random distribution of 1000 particles using model II. (a) Three-dimensional view in the rotating frame showing the outermost KAM torus (dark gray), the temperature isosurface $\theta =0.5$ (light gray transparent surface), and particles (dark spherical markers). (b) Axial projection of panel (a) without the temperature field. The solid line indicates ${}^1{L}_3^3$ and the dashed line denotes ${}^5{L}_3^{15}$. The arrow indicates the direction of propagation of the pattern in the laboratory frame. (c) Poincaré section (gray dots) at $\phi ={\phi }_{r_{\text{min}}}=-1.7293$ during $t\in [80,84]$ of the closed streamline ${}^1{L}_3^3$ (diamond) and of ${}^1{T}_3^3$ (black dots). The thin gray strip of thickness $\mathrm{\Delta }$ adjacent to the free surface indicates the prohibited region for the particle centroids. The enlargement shows the Poincaré section of ${}^5{L}_3^{15}$ (diamond), ${}^5{T}_3^{15}$ (black dots), and PAS (gray dots).

Figure 15

Correlation between the free surface temperature $\theta$ (a) (grayscale), the free surface temperature deviation ${\theta }^{\prime }$ (b) (grayscale), and the closed streamline ${}^1{L}_3^3$ in radial projection (solid line) for $\text{Re}=1600$. The hydrothermal wave (HTW) wave travels from right to left (arrow).

Figure 16

PAS for $\text{Re}=1750$ and $\mathrm{\Delta }=0.01$ (a), 0.02 (b), and 0.03 [(c)–(e)] using model II and 1000 initially randomly distributed particles. Shown at time $t=84$ are tubular PAS for $\mathrm{\Delta }=0.01$ (a), fuzzy period-doubled PAS for $\mathrm{\Delta }=0.02$ (b), and the toroidal core for $\mathrm{\Delta }=0.03$ (c). Panel (d) shows a three-dimensional view of the particle configuration together with the attractor (line) at $t=280$ and panel (e) shows the radial projection of the attractor. The bold segments in panel (e) indicate the PSI.