Vibroequilibria in microgravity: Comparison of experiments and theory

Experiments on vibrated fluids confined in cylindrical and cuboidal containers were performed under the reduced gravity conditions of a parabolic flight. The results constitute a systematic quantitative investigation of the vibroequilibria effect, which refers to the reorientation of vibrated fluids in response to the inhomogeneous oscillatory velocity field and the accompanying dynamic pressure. This effect is amplified in microgravity where the restoring force of gravity is small or absent. Here the vibrations are transmitted via a pair of piezoelectric ceramics and a cantilever beam, excited in a resonant mode. The first and second resonances exhibit different types of motion and lead to different types of vibroequilibria surfaces, one with a dip or crater in the interior and the other flattened compared to the unforced reference experiment. The general tendency for interfaces to orient more perpendicular, on average, to the vibrational axis is confirmed. In the case of water in a cuboidal container, a quantitative comparison is made with vibroequilibria theory and with direct simulations of the Navier-Stokes equations. The good agreement confirms the predictions of vibroequilibria theory and suggests the capacity of this phenomenon to manipulate and position fluids in space environments through the choice of frequency and resonant mode.

Figure 1

Pictures of the experimental cells and the bender beam: (a) cuboid and (b) cylinder. The cells are held closed by four bolts, which compress the PMMA body and cover against an O-ring. The aluminum bender beam has two piezoelectric ceramics attached to opposite horizontal faces. Cables associated with electrical excitation of the ceramics and the clamping bolt can also be seen.

Figure 2

Sketch of experimental setup with (a) cuboid and (b) cylinder. Shown are the (partially filled) experimental cell, bender beam, piezoelectric ceramics, mirror, and CMOS camera. Selected optical paths indicate the direct and reflected images referred to in later sections. The inset in (b) shows a lateral view of the setup for a cylindrical cell. Shaded curves illustrate the excited motion of the beam for the first (b, inset) and second (a) resonant modes.

Figure 3

(a) Midplane cut of the experimental setup for a cuboidal container showing the bender beam, PZTs, fluid volume $\mathrm{\Omega }$, free surface $S$, and solid boundaries $W$. The container and laboratory reference frames are illustrated; since the displacement is very small compared to the container size, we assume hereafter that $(x,y,z)\simeq (x_{\mathrm{lab}},y_{\mathrm{lab}},z_{\mathrm{lab}})$. This geometry is used for the numerical results shown in Sec. 5c. The motion associated with the first two resonant modes is illustrated in (b) and (c), respectively.

Figure 4

Snapshots (front and lateral views) showing the evolution of the surface of silicone oil in a cylindrical cell. The upper row of panels (a)–(c) shows the fluid behavior in the reference cell (experiencing residual acceleration of the airplane) at the given times, paired with corresponding snapshots of a vibrated experiment. The vibrated experiments are excited at $f=59.3\mathrm{Hz}$ with vibrational velocities [(a)–(c)] $v_z=73.9$, [(d)–(f)] 84.6, and [(g)–(i)] 94.8 mm/s, which correspond to applied (peak) voltages of 80, 90, and 100 V, respectively. To help visualize the free surface deformation, contact lines at “front” and “rear” halves of the lateral wall are highlighted with solid and dashed lines.

Figure 5

Evolution in time of the highest visible point of the contact line (solid dots) and the minimum visible point of the fluid surface (diamonds) in vibrated (black) and reference (gray) cells. Vibrations are delivered at $f=59.3\mathrm{Hz}$ and different applied vibrational velocities: (a) $v_z=73.9$, (b) 84.6, and (c) 94.8 mm/s, which correspond to applied (peak) voltages of 80, 90, and 100 V, respectively. Measurements of the residual gravity level are shown below each panel, with the labeled components measured in the airplane reference frame. Vertical lines mark the times when the snapshots in Fig. 4 are taken [labeled as 4(b), 4(c), 4(e), 4(f), 4(h), 4(i)].

Figure 6

Snapshots (top and front views) showing the evolution in time of experiments with silicone oil in a cuboidal container. The upper two rows show the reference cell, which is subjected to the airplane's residual gravity, while the lower two rows show the cell excited at $f=83.07\mathrm{Hz}$ and $v_{\eta }=161.6\mathrm{mm}/\mathrm{s}$, corresponding to a peak voltage $V=100\mathrm{V}$. When feasible, the contact lines along the front and rear walls (front view) are highlighted using solid and dashed lines, respectively, as is the contact line in the upper views.

Figure 7

[(a)–(c)] Snapshots of vibroequilibria in a cuboidal container vibrated at 79.8 Hz in the first resonant mode. Vibrational velocities are as follows: (a) 70.7, (b) 122.6, and (c) 161.6 mm/s, corresponding to peak voltages of 50, 80, and 100 V, respectively. The location of the “front” and “rear” contact lines is outlined with solid and dashed curves, respectively. Images of the reference cell at identical times are included for comparison. (d) Illustration of the measurement points used for quantitative comparison with the theory of Sec. 5c.

Figure 8

Snapshots (front views) showing the time evolution of experiments with water a cuboidal container when vibrated in the second resonant mode. The upper rows show the reference cell, while the lower rows show the results of vibrating at $f=215.4\mathrm{Hz}$ and $v_{\eta }=49.19\mathrm{mm}/\mathrm{s}$, corresponding to an applied voltage of 100 V.

Figure 9

Comparison of experimental results (solid black dots) with direct numerical simulation of Eqs. (2, 3, 4, 5) (open gray circles with solid fitting curve), with analogous simulations that ignore the rotational motion (open gray squares with dashed fitting curve), and with the predictions of vibroequilibria theory based on the minimization of Eq. (10) (solid black curve). The contact angle used in these calculations is ${50}^{\circ }$, while the residual gravity is set to a representative value of $g_z=0.05g_0$.

Figure 10

Influence of viscosity and gravity level for a fixed voltage $V=50\mathrm{V}, \eta =0.57$ on the predicted contact point separation used to quantify the vibroequilibria effect. (a) Simulations of the Navier-Stokes equations with $g_z=0.01g_0$ and $\nu =10\mathrm{cSt}$ (light gray curves), $50\mathrm{cSt}$ (medium gray curves), and $200\mathrm{cSt}$ (black curves). (b) Simulations using $\nu =10\mathrm{cSt}$ with $g_z=0$ (light gray curves) and $g_z=0.05g_0$ (black curves). The remaining parameters are as in Fig. 9.

Figure 11

Measurements (performed on ground) of the maximum linear and angular velocities at the center of the side wall for the first resonant mode. (a) Horizontal velocity $|v_x|$ (solid dots) and vertical velocity $|v_z|$ (solid squares). (b) Angular velocity ${\varpi }_0$.