Coherent structures of $m=1$ by low-Stokes-number particles suspended in a half-zone liquid bridge of high aspect ratio: Microgravity and terrestrial experiments

We investigate the coherent structures by low-Stokes-number particles suspended in half-zone liquid bridges of high Prandtl number via microgravity and terrestrial experiments. We especially focus on the structure by the suspended particle accumulation, or, the particle accumulation structure (PAS), of $m=1$ in azimuthal wave number emerged in the traveling-wave-type time-dependent “oscillatory” convection. The particles form the closed path with a spiral structure in tall or high-aspect-ratio liquid bridges under both micro- and normal gravity conditions. Their formation process is evaluated by applying the accumulation measure, and it is indicated that the formation time of the structure is of the order of a thermal diffusion time regardless of the difference of Prandtl number as well as the gravitational acceleration. The spatial correlation between the coherent structure and the thermal wave over the free surface is illustrated in the rotating frame of reference. By the terrestrial experiments, we indicate how the PAS is conformed by the individual particle motions in the laboratory frame and the rotating frame of reference via two-dimensional particle tracking. We also indicate their occurring condition as a function of the slenderness of the liquid bridges in terms of the volume ratio, and we evaluate the morphological characteristics of the spiral structure.

Figure 1

Apparatus for microgravity experiment installed in the Japanese Experiment Module “Kibo” on the International Space Station (ISS) [47]. “$\mathrm{Ch}i$” ($i=1,2,3$) indicates the CCD camera of $i\mathrm{th}$ channel, and “MIDM” indicates the micro-imaging displacement meter.

Figure 2

Apparatus for terrestrial experiment.

Figure 3

(1) Variations of top view of the particle images, and (2) corresponding $K$ measure under $\mathrm{Ma}=1.30\times {10}^4$ ($\mathrm{\Delta }T=8.2\mathrm{K}$ and $\epsilon =0.8$) in liquid bridge of $V/V_0=0.95$ and $\mathrm{\Gamma }=1.0$ under microgravity condition. In the first frame in panel (1), the periphery of the top disk (circle) and the azimuthal propagation direction of the traveling-wave-type oscillation (arrow) are illustrated. Time in GMT (and the elapsed time in second) is indicated in panel (1). “LOS” in panel (2) is the abbreviation for “loss of signal” from the International Space Station. More detailed time series of the particle images is illustrated in Fig. 11 in Appendix pp1.

Figure 4

Temporal variations of (a) top view and (b) side view of the particle images, and (c) surface temperature deviation obtained by microgravity experiment: $\mathrm{Ma}=1.30\times {10}^4$ ($\mathrm{\Delta }T=8.2\mathrm{K}$ and $\epsilon =0.18$), and $V/V_0=0.95$ for $\mathrm{\Gamma }=1.0$. Schematic of the trajectory of the particles forming the coherent structure is illustrated in the first panel superimposed on the same images indicated in (i). The point “A” indicates the position of the trajectory where the coherent structure locating closest to the free surface near the heated disk. The particles travel along the positions on the trajectory in the alphabetical order.

Figure 5

(1) Variations of top view of the particle images, and (2) corresponding $K$ measure by terrestrial experiment; $\mathrm{Ma}=1.9\times {10}^4$ ($\mathrm{\Delta }T=20\mathrm{K}$) for $V/V_0=0.8$ and $\mathrm{\Gamma }=$ $2.0$. At $t=-1.1\mathrm{s}$ a disturbance to the free surface is put by blowing a tiny amount of the air through the external shield, and at $t=0$ the disturbance to the liquid bridge is stopped. Note that the azimuthal direction of rotation before adding the disturbance is counterclockwise as indicated by an arrow in the first frame in (1), and that after adding the disturbance becomes clockwise. More detailed time series is illustrated in Fig. 11 in Appendix pp1.

Figure 6

Temporal variations of (a) top view and (b) side view of particle images, and (c) surface temperature deviation obtained by terrestrial experiment: $\mathrm{Ma}=1.9\times {10}^4$ ($\mathrm{\Delta }T=20\mathrm{K}$) for $V/V_0=0.8$ and $\mathrm{\Gamma }=2.0$. Schematic of the trajectory of the particles forming the coherent structure is illustrated in the first panel superimposed on the same images indicated in (i). The point “A” indicates the position of the trajectory where the coherent structure locating closest to the free surface near the heated disk, and the particles travel along the positions on the trajectory in the alphabetical order. Note that this result was obtained in the different experimental run shown in Fig. 5.

Figure 7

(a) Coherent structure in rotating frame of reference observed through hot disk, (b) its schematics, and (c) temperature-deviation distribution over free surface for (1) microgravity- and (2) terrestrial experiments. The coherent-structure images are obtained by integrating the particle images for (1) four fundamental periods or $131.4\mathrm{s}$ and for (2) ten fundamental periods or $3.8\mathrm{s}$. Arrows in row (a) indicate the direction of the traveling wave. Solid and dashed arrows in row (b) indicate the upward ($\partial z/\partial t\ge 0$) and downward ($\partial z/\partial t\le 0$) motions of the particles forming the PAS, respectively. Circle in row (b) illustrates the position “A” where the coherent structure seems locating closest to the free surface, as shown in Figs. 4 and 6.

Figure 8

Typical motion by two different particles forming the PAS in (I) the laboratory frame and (II) the rotating frame of reference observed from above for $3.992\mathrm{s}$ with a constant interval of $1/125\mathrm{s}$, and (III) their temporal variations of radial positions. Two large filled marks (square and circle) in frames (I) and (II) indicate the initial positions of the particles in the observation period. Panels (a) and (b) in frame (III) indicate the temporal variations of the particle whose initial position is illustrated by the filled square (in blue) and (b) the particle by the filled circle (in red), respectively. The circles by solid line and dashed line in frames (I) and (II) show the periphery of the top rod of $0.75\mathrm{mm}$ in radius and the averaged position of the free surface of minimum radius, respectively. An arrow in frame (I) indicates the azimuthal direction of the thermal wave over the free surface or the azimuthal direction of the PAS rotation.

Figure 9

Variation of particle distribution in top view for $\mathrm{Ma}=2.3\times {10}^4$ ($\mathrm{\Delta }T=24\mathrm{K}$); in (a) rotating frame of reference and in (b) laboratory frame as a function of $V/V_0$ in liquid bridge of $\mathrm{\Gamma }=2.0$ under normal gravity. Each frame for top views is obtained by accumulating the particle images for five fundamental cycles. Dashed circle in each frame indicates the minimum position of the concave free surface. The azimuthal direction of the PAS for each case is counterclockwise. Row (c) illustrates the side-view snapshots of the liquid bridge.

Figure 10

Maximum and minimum distances of spiral structure in $r$ direction, $R_{\mathrm{s}}^{\left(\max \right)}$ and $R_{\mathrm{s}}^{\left(\min \right)}$, respectively, in liquid bridge of $\mathrm{\Gamma }=$ $2.0$ under normal gravity. (1) Definition of $R_{\mathrm{s}}^{\left(\min \right)}$ and $R_{\mathrm{s}}^{\left(\max \right)}$ with the particle distribution (a) in the rotating frame of reference and (b) in the laboratory frame for the liquid bridge of $V/V_0=$ $0.68$ for $\mathrm{Ma}=$ $1.9\times {10}^4$ ($\mathrm{\Delta }T=$ $20\mathrm{K}$), and (2) their variation against $V/V_0$ under $\mathrm{Ma}=$ $1.9\times {10}^4$ ($\mathrm{\Delta }T=$ $20\mathrm{K}$) and $\mathrm{Ma}=$ $2.3\times {10}^4$ ($\mathrm{\Delta }T=$ $24\mathrm{K}$). Dashed circle in each frame indicates the minimum position of the concave free surface.

Figure 11

Time series of the particle image observed from above via (a) microgravity- and (b) terrestrial experiment: (a) $\mathrm{Ma}=$ $1.30\times {10}^4$ ($\mathrm{\Delta }T=$ $8.2\mathrm{K}$ and $\epsilon =$ $0.8$) in the liquid bridge of $V/V_0=$ $0.95$ and $\mathrm{\Gamma }=$ $1.0$, and (b) $\mathrm{Ma}=$ $1.9\times {10}^4$ ($\mathrm{\Delta }T=$ $20\mathrm{K}$) in the liquid bridge of $V/V_0=$ $0.8$ for $\mathrm{\Gamma }=$ $2.0$. The azimuthal direction of the PAS and the thermal wave over the free surface due to the hydrothermal wave instability is counterclockwise. Note for (b) the terrestrial experiment that the azimuthal direction of the PAS rotation changes from the counterclockwise to the clockwise after the disturbances added in the period of $-1.1\mathrm{s}$ $\le t\le$ $0\mathrm{s}$.

Figure 12

Predictions by linear stability analysis (LSA) for the straight liquid bridge of $\mathrm{Pr}=$ $28.6$ [68]: (1) neutral curves under $\mathrm{Bd}=0.34$ (in red) and 0 (in black), (2) typical examples of distributions of isosurfaces of $\widehat{T}=\pm 0.2$ inside liquid bridge (top) and of temperature deviation $\widehat{T}$ over free surface for different types of HTW. The eigenfunctions of the induced thermal flow fields are named as the HTW(b) for small $\mathrm{Bi}$ (left), the HTW(${\mathrm{a}}^{\prime }$) for moderate $\mathrm{Bi}$ (middle), and the HTW(a) for large $\mathrm{Bi}$ (right). The azimuthal traveling direction of the thermal wave over the free surface is counterclockwise for all cases in this figure.