Experimental study of coherent structures of finite-size particles in thermocapillary liquid bridges

The motion of finite-size particles in thermocapillary flow is investigated experimentally. Particular attention is paid to the unique phenomenon known as a particle accumulation structure (PAS), where particles gather in a solidlike structure in a half-zone liquid bridge. We study the correlation between the PAS and the flow field due to the hydrothermal wave instability by measuring the surface temperature and dynamic surface deformation. An image processing method is developed in order to verify the physical model of the particle behavior. The structures of the particle depletion zone, the PAS, and the toroidal core are observed clearly using long-term averaged images in the rotating frame of reference. The results of precise experiments reveal that the correlation between the PAS and the temperature field over the free surface is strongly dependent on the test fluid. Reconstruction of the PASs is achieved by applying particle tracking velocimetry. Time-series analysis of the particle position indicate that the ratio of the hydrothermal wave frequency to the turnover frequency of the particle is independent of the intensity of the thermocapillary-driven convection.

Figure 1

Sketch of the cross-sectional view of the half-zone liquid bridge. Two horizontal disks of radius $R$ are separated by a distance $H$. The top disk temperature is higher than that of the bottom disk.

Figure 2

Snapshots of the particle accumulation structure: (a) $\mathrm{SL}1$ PAS $\left(\mathrm{Ma}=4.8\times {10}^4\right)$ and (b) $\mathrm{SL}2$ PAS $\left(\mathrm{Ma}=5.6\times {10}^4\right)$ with particles $15\mu \mathrm{m}$ in diameter in the liquid bridge of $\mathrm{\Gamma }=0.64$ and $R=2.5$ mm. Images are taken from above through the transparent top rod with a constant shutter speed at $1/60$ s.

Figure 3

Cross-sectional view of the experimental apparatus (not to scale). The liquid bridge is heated from above in this configuration. A narrow ZnSe window is placed at a portion of the external shield for the infrared camera to detect the surface temperature through the shield. The CMOS camera for the side view and the displacement sensor are omitted.

Figure 4

Typical coherent structures in the rotating frame of reference with respect to the HTW observed from above for a Marangoni number of (i) and (ii) $2.9\times {10}^4$ and (iii) $3.8\times {10}^4$. The diameter of suspended particles is (i) $15\mu \mathrm{m}$ and (ii) and (iii) $30\mu \mathrm{m}$. Each inverted image is obtained by averaging over 7 s (almost 10 times fundamental periods of HTW); black dots correspond to the particles and the white area corresponds to the region where no particles exist. Row (b) illustrates the major structures with multiple elements emerged in the liquid bridge schematically superimposed on the images shown in row (a). Note that the outer and inner edges of element $B$ are drawn in a circular shape for the sake of brevity, but the real edges obtained by the experiments exhibit wavy shapes. The arrow for element $A$ indicates the direction of the particles traveling in the bridge and its widths indicate the relative positions of the particles in the direction of the liquid-bridge height. The wider the arrows, the higher the position relative to the bottom rod. (b iii) illustrates three major coherent structures (the PAS, the core, and a structure wrapping the core) and corresponding KAM tori $(T_3^3,T_{\text{core}}$, and $T_3^9)$ following Mukin and Kuhlmann [64].

Figure 5

Variations of coherent structures by the particles as a function of $\mathrm{Ma}$ for (a) $d_{\text{p}}=15\mu \text{m}$ and (b) $d_{\text{p}}=30\mu \text{m}$. Each image is obtained by averaging over 7 s (almost 10 times fundamental periods of HTW) in the rotating frame of reference with respect to the HTW. The azimuthal direction of the HTW is counterclockwise for all images.

Figure 6

Definition of the minimum distance from the liquid-bridge center for the PAS $R_{\mathrm{PAS}}$ and for the PDZ $R_{\mathrm{PDZ}}$ through top-view images obtained by averaging over 7 s (10 times fundamental periods of HTW) in (i) the rotating and (ii) the absolute frames of reference with particles $15\mu \mathrm{m}$ in diameter. Results are illustrated for (a) $\mathrm{Ma}=3.13\times {10}^4$ [as shown in Fig. 5] and (b) $\mathrm{Ma}=4.30\times {10}^4$ [Fig. 5].

Figure 7

Minimum radii of the PAS $R_{\mathrm{PAS}}$ (circles) and the PDZ $R_{\mathrm{PDZ}}$ (triangles) against $\mathrm{Ma}$. The open and closed symbols correspond to the results with particles 15 and $30\mu \mathrm{m}$ in diameter, respectively.

Figure 8

Correlation between the PAS and HTW for (i) $n$-decane (at $\mathrm{Ma}=3.7\times {10}^4)$ and (ii) and (iii) 2-cS silicone oil (at $\mathrm{Ma}=4.3\times {10}^4)$ as test fluids from the (a) top view, (b) side view, and (c) surface temperature deviation superimposed on the side view (b). The particle diameters are (i) and (ii) $15\mu \mathrm{m}$ and (iii) $30\mu \mathrm{m}$. Red and blue indicate the hot and cold spots of the temperature deviation over the free surface, respectively. The black region on the right-hand side in each frame in (b) is the reflection of light and has nothing to do with the phenomenon concerned.

Figure 9

(a) Reconstructed $\mathrm{SL}1$ PAS configurations in the rotating frame of reference by tracking a single-particle forming the PAS, (b) time series of particle radial positions, and (c) its power spectral density (PSD) for $\mathrm{Ma}$ values of (i) $3.3\times {10}^4$, (ii) $3.7\times {10}^4$, (iii) $4.2\times {10}^4$, and (iv) $4.6\times {10}^4$, respectively. The particle diameter is $15\mu \mathrm{m}$. Arrows in row (c) indicate the fundamental frequency of the particle turnover motion $f_0$.

Figure 10

Dynamic surface deformation amplitudes for $\mathrm{Ma}=4.3\times {10}^4$ at 0.2 and 1.4 mm from the bottom-rod surface (or $z=0.2$ and 1.4 mm), measured based on enlarged side-view images. Dashed lines indicate the instants when the tip of the PAS blade approaches the measuring point in the azimuth.

Figure 11

Correlation between the DSD measured by the noncontact displacement sensor and temperature deviation of the thermal wave due to the HTW for $\mathrm{Ma}=4.3\times {10}^4$. The temperature deviation is indicated in grayscale and the DSD amplitude is indicated by the contour plot.

Figure 12

Map of typical patterns by suspended particles inside a liquid bridge of $\mathrm{\Gamma }=0.68$ against particle size $d_{\text{p}}$ and $\mathrm{Ma}$. Results are shown for (a) $d_{\text{p}}=5\mu \text{m}$, (b) $d_{\text{p}}=10\mu \text{m}$, (c) $d_{\text{p}}=15\mu \text{m}$, and (d) $d_{\text{p}}=30\mu \text{m}$. Each image is obtained by accumulating 625 inverted frames (for 5 s) in the rotating frame of reference against the fundamental frequency of the HTW under each condition. The direction of the HTW in the laboratory frame is counterclockwise for all images.