Techniques for generating centimetric drops in microgravity and application to cavitation studies

During each microgravity phase (20 s duration), one captured centimetric drop was created as follows. Distilled water was smoothly expelled through a custom-designed Aluminium injector tube (Fig. 1a) by a step-motor micropump (APT Instruments SP200 peristaltic pump). The injector tube was filled with hydrophilic porous foam for two purposes. First, to guarantee a homogeneous laminar flow and thereby control the actual drop volume during its formation. Second, to favor the attachment of the drop by water capillary cohesive forces through the tube foam interface. In addition, the top edge of the tube was trimmed at an angle of 45° in order to increase the effective contact surface with the drop. This inclined edge was coated with hydrophilic aluminium oxyde Al₂O₃ to ensure efficient contact, while the tube exterior was coated with hydrophobic silicon to prevent the water from flowing down. Four injector tubes with outer diameter D = 16, 12, 9 and 7 mm have been used (see Fig. 1a). They were interchanged according to the water drop volume to be produced, as larger contact surfaces allow to stabilize larger volumes (cf Sect. 3).

The micropump was programmed by an external computer, which triggered the pumping start, controlled its flux and its duration. The micropump was automatically started whenever the gravity level dropped below 0.05 g for more than 1 s (see Sect. 3.1). The flux was set to 0.6 ml/s and reduced to 0.3 ml/s during the 2–3 last seconds of the pumping process. This flux reduction aimed at slowing down the drop motion and avoiding subsequent drop oscillations. Varying the pumping time allowed to adjust the drop size. To precisely control the drop volume, the tube was filled with water up to its top foam interface prior to start the pumping. Thus, drop volumes of 2.4–8 ml (diameters of 16–25 mm) were formed in 5–15 s, respectively (the maximal pumping time was purposely restricted to 15 s according to Sect. 2.2). After each microgravity phase, the water drop fell down under hypergravity and was passively absorbed by sponges covering the vessel bottom.

During the microgravity phases provided by parabolic flights, the residual g-jitter can reach values of the order of 0.02 g at typical frequencies of 1–10 Hz. In 10% of the zero-g parabolas however, we recorded flight-based deviations from the microgravity level larger than 0.02 g for durations longer than 0.2 s. They mostly occurred approximately 3 s before the end of the zero-g phases. The injector tube being fixed to the aircraft reference, those fluctuations induced observable multi-pole oscillations of the water drop, visible in Fig. 2. Although the g-jitter spectrum contains eigenfrequencies of the drop (see Bauer and Chiba 2004, for a theoretical evaluation of the natural frequencies of captured drops), no resonance was observable and these oscillations always damped out rapidly within characteristic times of 0.5–1 s. We argue that the large area of support attaching the drop to the tube (1.1 cm² for an outer diameter of 12 mm) was responsible for this efficient attenuation, as the large porous foam ensures an inelastic rebound of the water drop. Due to the strong g-jitter sometimes seen in the final 3 s of the microgravity phase, we limited the pumping time to 15 s, thus creating the most undisturbed spherical drops for the cavitation studies described in Sect. 3.

We note that unwanted oscillations could also be partly avoided by increasing the viscosity of the fluid (e.g., using a solution of glycerol/water), or by using a second injector tube stabilizing the top of the drop, as realized by Treuner et al. (1995). On the other hand, the observed multipole oscillations qualitatively compare to the theoretical calculations of a harmonically excited captured drop by Bauer and Chiba (2004), indicating another potential application of this drop generation system.

Depending on the drop volume and the injector tube diameter, the drop could be unstable and sometimes even detached from the injector tube before the end of the microgravity phase. This leads to the following question: given a tube diameter, what is the maximum drop volume that is assured to remain attached on the tube under normal unevittable g-jitter?

We answer this question by formulating a simplified stability criterion for the drop, based on its mechanical equilibrium in the tube non-inertial frame. Only the force balance is considered, while the torques and shear-induced drop deformations are neglected. In the frame of the tube, the inertial force acting on the drop volume is opposed to its acceleration, F _V  = −ρVa, where V is the drop volume, ρ = 1,000 kg m⁻³ its mass density and a the residual acceleration (g-jitter) of the tube in microgravity. In reaction to this volume force, the cohesive forces at the tube interface develop a counterforce F _S  = σ _S S, where S is the tube surface and σ _S the surface force density along the direction considered. The drop will be stable as long as the tube surface force compensates the volume force, namely ρV a < σ _S,max S, where σ _S,max is the maximal value of the surface force density that can be exerted by the tube interface (considered as an intrinsic property of the tube). Although σ _S,max is in principle direction-dependent, we treat it as an average over all directions. During a standard parabola, the g-jitter reaches a maximum value a _max = 0.02 g and the stability criterion can be formulated as: Since the critical parameter c is a constant, the ratio V/S dictates whether the drop remains stable during the microgravity phase.

This theoretical result could be verified in our microgravity experiment, which also allowed a phenomenologial determination of c as follows. For all 155 parabolas (summing over both flight campaigns), we collected the drop volumes V and tube surface areas S and represented them as pairs (S,V) in the diagram of Fig. 3. In total, fourteen different (S,V) configurations were probed, corresponding to the fourteen data points. Each of these configurations was repeated in a variable number of parabolas. If during all of these parabolas, the drops remained stable on the tube (i.e., never detached and hanged on the side of the tube), the corresponding point was classified as stable (circles), otherwise as unstable (triangles).

In this representation, the critical parameter c corresponds to the slope of a straight line passing through the origin and separating the unstable from the stable regime; graphically, we find c ∼ 4.5 cm. This value has to be taken as a safe stability limit, as no configuration with V/S < 4.5 cm was unstable. For the 49 experimental cycles with V/S > 4.5 cm, the average rate of drop detachment was about 25%. Hence, configurations with V/S > 4.5 cm are not necessarily unstable, as rapid variations of the acceleration vector can prevent large displacements of the drop or even bring it back to a stable state. Therefore the relation (1) should be exclusively taken as a stability criterion, as its violation is only a necessary but not sufficient condition for unstability.

We like to end this analysis with a note for additional physical insight: given the observed value c ∼ 4.5 cm, we can estimate the maximal surface force density σ _S,max using (1). One finds σ _S,max ∼ 10 N m⁻², which compares well with the cohesive force of water observed in droplets suspended under their weights (see Young 1845, lecture L for further discussion on that matter).

Stimulated by this reflection, we can find a physical interpretation for the length c by noting that the ratio V/S corresponds to the height of a cylinder with base surface S and volume V. Thus c can be seen as the maximally allowed height of a hypothetical water cylinder, such that can be sustained by the tube interface in a weak gravitational field of 0.02 g.

Figure 4 shows the arrangement of the central components of the setup. The water drop, the injector tube as well as the electrodes for the bubble generation were contained in a sealed transparent cubic vessel (20 cm side length, 1 cm thickness) made of clear high-density polycarbonate (Lexan).

To record the fast bubble dynamics, we used a high-speed CCD camera (Photron Ultima APX, up to 120,000 frames/s) with a framing rate of 12,500–24,000 frames/s (depending on the temporal and spatial resolution desired), and an exposure time of 5 μs. The record sequence had a duration of 11 ms, during which the required illumination was provided by a perpendicularly placed high-power flashlight (Cordin Light Source Model 359). The slow water drop formation was filmed with a standard video camera (Sony Camcorder DCR-TRV900, 25 frames/s).

A diagram illustrating the links between the different functional units of the experiment as well as the order of their activation in time is given in Fig. 5. A control computer program triggered the micropump (drop creation), the electrodes discharge (bubble generation), the image recording and the flashlight release. The interface between the computer signals (USB) and the manual start button, micropump and electrodes was performed by standard input and output boards. The experimental sequence was automatically initiated at the beginning of each flight parabola, as soon as a stable level of microgravity (<0.05 g) was reached for more than 1 s. The gravity data was provided by a 100 Hz accelerometer (Memsic, 2125 Dual-axis Accelerometer) that continuously recorded the gravity level during the whole flight. Immediately after the water drop formation, the electrical discharge for bubble generation was released. Because of the low trigger time accuracy of the discharge generator, we used the electrodes discharge pulse current to trigger the recording system through an inductive response circuit. This precaution ensured that the recording system started at the instant of bubble creation, in order to capture the entire bubble dynamics and subsequent phenomena. For the exact synchronization of the high-speed camera and the flashlight, the induced signal from the electrodes was split by a TTL pulse generator sending a pulse to both units simultaneously.

The experiment was designed to be redundant in case of failure of any unit: As an alternative for the accelerometer, the experimental sequence could be started manually from an external button or directly from a computer key. If none of these devices were functional, the electrodes discharge generator could still be fired manually instead of being triggered by the computer. Finally, the recording system could be triggered directly from the electrodes inductive signal in case the TTL pulse generator were defective, but with a somewhat less accurate synchronization. However, none of these off-nominal procedures had to be used over the 155 parabolas.

The experimental setup was fixed on a single rack containing horizontal levels, as shown in Fig. 6. The lower level (base plate) was used to attach the experiment on the plane rails and to carry the heavy units (high-power generators for electrodes and flash light, camera control unit), while the upper level (working table) carried the devices requiring in-flight manipulation.

The cavitation bubble was generated using the standard spark-based technique, by which a spark released by electrodes allows the conversion of electrostatic energy into a point-like plasma of high temperature and pressure. The plasma immediately recombines and gives rise to a thermally growing bubble, which subsequently collapses under the surrounding liquid pressure (Chahine et al. 1995). This standard spark-based technique was chosen instead of a laser-based technique because of the difficulty to focus a laser beam across a (variable) spherical surface, and for aircraft security reasons.

To produce the discharge, a pair of electrodes was immersed in the water drop from above (Fig. 7b–e). The required electrostatic energy was stored into capacitors, whose fast discharge was enabled by a triggered spark gap (see Pereira et al. 1994 for more details). These devices were contained in the unit called electrodes discharge generator in Fig. 5. The capacitor’s voltage (4.7 kV) was high enough to induce the breakdown of water molecules and thereby create the plasma between the electrodes. To vary the bubble size, the discharge capacitance could be stepwisely altered between 30, 50, 100 and 200 nF. A minimum capacitance value of 30 nF was necessary to ensure water breakdown, probably due to the potential drop occuring during the finite impedance fall-time of the spark gap.

The electrodes had a thicker part which was mounted on an axis controllable by micro-positioners, and thinner ends to penetrate the drop (see Fig. 7a). The latter had a thickness of the order of 0.5 mm with a spacing between the tips of about 1 mm. Using the positioners, the electrode tips could be displaced in three dimensions, allowing their precise positioning inside the water drop.

In order to minimize the interaction of the electrodes with the water volume, their tips were coated with hydrophilic ferrous oxyde Fe₃O₄ and their upper part covered with hydrophobic silicon, as illustrated in Fig. 7a. Despite the careful preparation of the electrodes, we observed significant water drop distortions induced by electrode–water interaction. These distortions were strongly dependent on the specific coating of the electrodes, particularly on the thickness of the silicon layer covering their thick upper part. Without any silicon coating, the upper part of the electrodes was too hydrophilic, yielding an oblong distortion of the water volume (Fig. 7c). Conversely, additional silicon engendered water repulsion by the electrodes (Fig. 7d,e). Because of this effect, the electrodes position had to be readjusted during the flight to control the direction of liquid jets emission following bubble collapse (directed towards to and away from the closest free surface element, see Sect. 3.4), so that they did not escape from the high-speed camera field of view. Moreover, the drop deviations from a perfect sphere hinder the measurement of quantitative parameters such as the curvature ratio between drop and cavity, relative bubble eccentricity ε and relative bubble radius α (defined in Sect. 3.4). These parameters are nevertheless of importance for the analysis and modeling of the observations. The above thus reveals the necessity to optimize the electrodes surface coating and to render the tips finer in order to minimize such distortions.

To characterize the efficiency of our spark-based system, we determined the fraction of electrical discharge energy actually transformed into bubble energy. The electrical discharge energy E _e is given by the stored electrostatic energy of the capacitors. The hydrodynamic potential energy of the bubble E _b relates to its maximum radius R _b,max (as achieved at the end of its growth) via \(E_{\text{b}} = {\frac{4\pi}{3}} R^3_{\text{b},{\text{max}}},(p_\infty-p_v)\) , where p _∞ is the ambient static pressure and p _v the water vapor pressure at room temperature. In order to determine the relation E _b(E _e) experimentally, we therefore measured the maximum bubble radii R _b,max for the different values of the capacitance.

To complement the flight measurements within the drops, we also carried out on-ground measurements inside water flasks with additional values of the capacitance: 10 and 20 nF. For the measurements inside the drops, we corrected the apparent R _b,max using a model of optical refraction through a spherical surface (see Appendix). Care was taken to select bubbles sufficiently far from the drop surface to ensure the validity of the refraction model as well as to avoid bubble shape disturbances. For each measured value of R _b,max, the corresponding bubble energy E _b(R _b,max) was determined using the relation above, taking into account the different ambient pressures on ground and inside the aircraft cabin, 10⁵ and (8 ± 0.2)10⁴ Pa, respectively. Note that the cabin pressure contains an uncertainty owing to dynamic changes in the course of a parabola (e.g., during throttle-back maneuvers), but we did not correct for these changes (between different measurements) here and considered the pressure constant over the bubble lifetime. The mean values of E _b are plotted against E _e in Fig. 8. The errors on E _b are the standard deviations of the E _b(R _b,max) measurements, while the E _e errors account for the standard precision of capacitance values of about 5%. A weighted least-square linear regression (accounting for both errors) yielded the following discharge–cavity energy relation: E _b = 0.024(±0.005) E _e.

The obtained relation reveals that about 2–3% of the capacitors electrostatic energy was transformed into potential energy of the bubble, whereas the remaining energy was absorbed elsewhere. The latter fraction was presumably dissipated in electrical losses and radiated in the form of a shock wave caused by the initial plasma expansion (Chahine et al. 1995). To estimate the contribution of the shock wave proper, we used the method described in Obreschkow et al. (2006). Explicitely, we assumed that most of the shock energy was converted into submillimetric bubbles, arising from the excitation of impurities at the multiple passages of the shock confined within the drop. By integrating the potential energies of all these bubbles (using the energy-maximum radius relation given above), we found the fraction of shock wave energy to be roughly 4% (cf Obreschkow et al. 2006, where it was found that the shock wave energy is about 50–70% of the sum of the shock wave energy and bubble energy, confusingly referred to as the “initial electrical energy”). By taking into account additional sources of energy loss such as heat production and light emission (Brenner et al. 2002; Leighton 1994), the actual energy deposited in the water by the electrodes can be estimated to be of the order of 10%. The remaining 90% correspond to the initial energy fraction lost in electrical Joule heating, electro-magnetic radiation, and plausible incomplete discharge of the capacitors (as the electrodes voltage drops below the breakdown threshold of water). Although these values depend much on the specific devices used, they are roughly consistent with previous estimations (Chahine et al. 1995).

Next to a free surface, a cavitation bubble undergoes a toroidal collapse emitting a high-speed microjet directed away from the local surface element, while a counterjet called splash forms on the free-surface (Blake and Gibson 1981; Robinson et al. 2001; Pearson et al. 2004). In the present drop experiment, two liquid jets were then seen to escape the drop in opposite directions (Obreschkow et al. 2006). We observed that the importance of these liquid jets strongly depends on the bubble size and position within the drop. Furthermore, the collapsing bubble could induce instabilities on the drop surface or even cause a burst of the entire drop, thus rendering the jets investigation impossible. This motivated us to study the influence of the geometrical configuration of the bubble within the drop on these particular phenomena.

To characterize the jets emission and drop stability in function of the system’s geometry, we introduced three qualitative categories A, B, C, corresponding to the possible drop dynamics scenarios following bubble collapse shown in Fig. 9.

The geometrical quantities dominating the post-collapse dynamics are (1) the drop size, expressed by the minimum drop radius R _d,min before bubble generation, (2) the bubble size, given by the maximum bubble radius R _b,max, and (3) the bubble’s radial position within the drop, characterized by the distance d between drop center and bubble center. But these three parameters are redundant, inasmuch as a uniform stretching of all parameters reproduces a geometrically similar case. A natural choice of non-redundant dimensionless parameters was then obtained by defining: the relative bubble radius \(\alpha\equiv R_{{\text{b}},{\text{{max}}}}/R_{\text{d,min}}\) and the bubble eccentricity \(\varepsilon\equiv d/R_{\text{d,min}}\) .

For the 67 parabolas that qualified for this study (successful bubble generation, quasi-spherical drop, jets within the high-speed camera field of view), the geometrical parameters α and ε were measured as follows. Because of the drop deviation from spherical symmetry in presence of the electrodes (see Sect. 3.2), the geometrical drop center could not be defined accurately in most parabolas. We therefore adopted the natural convention to measure R _d,min and d along the axis of jets emission, being the relevant physical axis for the dynamics of the system. As this axis was determined only after the jets emission, it was exported to the first frames of the movies for the measurement of R _d,min and d. To obtain the true position (i.e., corrected for optical refraction through the drop’s surface) of the bubble center on this axis, the outer part of the electrode tips were prolongated inside the drop until its intersection with the jets axis. Finally, R _b,max was obtained from the linear regression between bubble energy and electrostatic energy E _b(E _e) (derived in Sect. 3), provided the corresponding value of the capacitance (and the electrostatic energy thereof) and the bubble energy-maximum radius relation E _b(R _b,max).

We then assigned each of these 67 parabolas to one of the three A-B-C categories, and represented them in the dimensionless parameter space {ε, α} shown in Fig. 10. In that space, we can roughly recognize three domains corresponding to the categories A, B and C, for which we have drawn straight boundaries to guide the eye (dashed lines). It can be observed that for 0.3 < α < 0.6, increasing the eccentricity leads to a transition from category A to category B. A similar transition occurs by increasing α for 0.25 < ε < 0.45. Hence, the splash strength qualitatively increases both with α and ε, as larger values of these parameters may produce a stronger pressure peak between the bubble boundary and the local free surface (as depicted in the numerical simulations of Robinson et al. 2001). Notably, for relative bubble radii above α ∼ 0.6, we assist to a burst of the entire water drop (cat. C). A closer look at the five cases belonging to this category revealed that the drops remain essentially stable during bubble growth and collapse, but explode immediately after, presumably under the effect of the strong shock wave radiated by the rebounding bubble (Shima 1997).

Consequently, as long as only the cavitation bubble life cycle is to be studied, the whole parameter space {ε, α} can be experimentally explored. However, to investigate the jets dynamics and surface effects, the parameter space should be restricted to α < 0.6.