The Effect of Mechanical Shaking on the Rising Velocity of Bubbles in High‐Viscosity Shear‐Thinning Fluids

The rising velocity of an air bubble in a non‐Newtonian shear‐thinning fluid at low Reynolds numbers is generally similar to the Newtonian case given by Stokes' law. However, when the shear‐thinning fluid is subject to mechanical oscillations, the rising velocity could significantly increase. Here, we present a series of experiments quantifying the rising velocity of single bubbles during shaking in very high‐viscosity (2,000–30,000 Pa·s) shear‐thinning silicone oils. Air bubbles (18–30 mm diameter) were injected in a tank mounted on a shaking table. The tank was horizontally oscillated, at accelerations between 0.4 and 2 g. We observed a small increase in the rising velocity of the shaking cases at our experimental conditions. The increase was larger when bubbles were large and accelerations were high. Larger accelerations experienced the largest observational errors and we emphasize the exploratory nature of our results. We also measured the change in bubble diameter during the oscillations and computed the shear rate at the bubble surface. Maximum shear rates were in the range of 0.04–0.08 s−1. At these shear rates, our analysis indicates that shear thinning behavior of our analog fluids is expected to be small and compete with elastic behavior. This transitional viscous/elastic regime helps explain the small and variable results of our experiments. Our results are relevant to the study of earthquake‐volcano interactions. Most crystal‐free silicate melts would exhibit a purely viscous, shear‐thinning behavior in a natural scenario. Seismically enhanced bubble rise could offer an explanation for the observed increased degassing and unrest following large earthquakes.

experimentally compared the rising velocity of bubbles at low Reynolds numbers, in Newtonian and shear-thinning fluids. Shear-thinning fluids (also known as pseudoplastic) are non-Newtonian fluids for which the viscosity decreases with increasing shear rates. When the fluids were kept quiescent, they found no difference in measured rising velocities between the Newtonian and the shear-thinning case in agreement with Equation 1. However, when the fluids were mechanically vibrated, the rising velocity increased by factors between 2 and 400 in the shear-thinning case, compared to the Newtonian case. Their findings were further analytically and numerically investigated and confirmed by De Corato et al. (2019). These results have never been applied to volcanological scenarios.
Understanding the dynamics of a bubble rising through a shear-thinning fluid is particularly relevant because silicate melts generally exhibit shear-thinning behavior and contain bubbles (e.g., Murase & McBirney, 1973;Shaw, 1969;Webb & Dingwell, 1990b). Additionally, magmas are very likely to undergo pressure oscillations both from internal sources (e.g., pulsatory flow and bubble resonance; Chouet, 1996;Wylie et al., 1999) and external sources (e.g., tectonic earthquakes and tides; Manga & Brodsky, 2006;Seropian et al., 2021). In particular, increased degassing rates are often observed at open-vent volcanoes following the passage of seismic waves (Avouris et al., 2017). Therefore, it is possible that seismic waves accelerate the rise of bubbles in magma, thus contributing to the observed increased degassing.
The zero-shear viscosity of the fluid used by Iwata et al. (2008) is 90 Pa⋅s. Silicate melt viscosity depends on composition, water content, and temperature, and generally lies within the range 10 0 − 10 9 Pa⋅s (e.g., Giordano et al., 2008). Here, we present a series of experiments to extend the results from Iwata et al. (2008) to higher viscosities (2,000-30,000 Pa⋅s). We measured the rising velocity of bubbles in shear-thinning fluids, subject to mechanical oscillations. We then discuss the application of our experimental results to natural volcanic scenarios.

Fluids
We used three different transparent silicone oils, from the KORASILON Fluids M series produced by Obermeier GmbH & Co. KG (Obermeier, 2016). All three fluids exhibit a shear-thinning behavior, with respective zero shear viscosities of 2,000, 10,000, and 30,000 Pa⋅s, referred to as M2, M10, and M30, respectively. All three fluids have a density of 970 kg⋅m −3 at 25°C.
The documentation from the manufacturer does not specify the minimum shear rate at which shear-thinning effects become significant (Obermeier, 2016). Rudolf et al. (2016) published a detailed rheological study of silicone oils, which includes fluids similar to our M2 and M30 fluids (respectively KOR2000 and G30M in their study), but no fluid like M10. They report transitional shear rates (i.e., when shear-thinning effects become dominant) of 3.8 and 0.26 s −1 for the M2 and M30 fluids, respectively.
We measured the viscosity of the three fluids at different shear rates, using a Brookfield DVIII+ rheometry measuring head and a wide-gap concentric cylinder geometry. The rheometer could not be calibrated for such high viscosity values, and thus did not provide absolute viscosity values. However, we were able to obtain the relative viscosity values at different shear rates, and thus determine whether the viscosity decreased with increasing shear rates. For each fluid, we first measured viscosity at the smallest possible shear rate (η 1 ). We then measured viscosity at increasing shear rates (η n ), and computed the ratio η n /η 1 , to determine when shear-thinning effects become significant (Table 1). We were able to measure viscosity at four different shear rates for the M2 fluid. For the M10 and M30 fluids, however, we could only measure viscosity at two different shear rates because the torque quickly exceeded the maximum limit of the instrument (0.7187 mN⋅m). We found that the viscosity of the M2 fluid remains constant up to a shear rate of 1.45 s −1 . In contrast, the M10 and M30 fluids already exhibit shear-thinning behavior at shear rates of 0.21 and 0.10 s −1 , respectively. Our measurements are thus in agreement with and extend the results of Rudolf et al. (2016).

Experimental Procedure
We used a 240 × 85 × 235 mm 3 rectangular tank, mounted on a GeoSIG GSK-166 Uniaxial Linear shaking table at GFZ Potsdam, Germany to impose horizontal oscillations on the fluids (Figure 1). This is the same apparatus used by Namiki et al. (2016). The tank was filled with fluid to a depth of 220 mm, sealed with a lid 10.1029/2022JB025741 3 of 13 to prevent hydration (and thus viscosity reduction; Rudolf et al., 2016) and left to rest for a day in order for air bubbles to escape. This was sufficient to remove air bubbles in the M2 and M10 fluids. For the M30, we additionally injected large bubbles (>50 mm diameter)) at the bottom of the tank to clear a path wide enough through the remaining small bubbles. For the bubble rise experiments, air bubbles were injected through a sealable port at the bottom of the tank using a graduated syringe affixed with a 200-mm long needle. After injection of the bubble, the syringe was removed. Needle removal often created a slight elongated cusp at the bottom of the bubble (see Section 3.1). Pressure is monitored in the fluid throughout experiments using a Keller 33X/80794 pressure transmitter located on the side, at the bottom of the tank.
An experiment starts 1-5 min after a bubble is injected. We considered three bubble diameters, namely 18, 24, and 30 mm. The bubble is first left to rise in still fluid for 1.5-4 min, depending upon the fluid viscosity. We then started shaking, for a duration of 30 s (see shaking parameters in Section 2.2.2). After the shaking phase ended, the bubble was left to rise in still fluid again for 1.5-4 min. For the M2 fluid, the bubble was then allowed to rise and escape the tank before another experiment took place with a new bubble injection. For the M10 and M30 experiments, for expediency purposes, we waited a minimum of 2 min and then used the same bubble to run the next experiment.
Experiments were recorded with a Ricoh GR camera with a resolution of 1,920 × 1,080 at 24 frames per second (fps), giving at least six frames per oscillation. The camera was mounted on an arm attached to the tank on the shaking table, and thus followed the tank's movements during the oscillations. A 10 × 10 mm 2 grid was placed on the back of the tank to serve as a scale, and detect any relative motion between the camera and the tank. The grid only covered half of the back of the tank, leaving a clear background where the bubble was injected, to facilitate bubble detection during the post-experimental analysis.

Shaking Parameters
The shaking table produces a horizontal, sinusoidal displacement according to: where A is the oscillation amplitude and f is the oscillation frequency, both of which can be controlled independently. Oscillation amplitude and frequency can be used to compute the maximum acceleration according to: Accelerations are normalized by the gravitational acceleration g = 9.81 m.s −1 . We considered five different combinations of amplitude and frequency (i.e., five different accelerations), referred to as P1-5 and summarized in Table 2. We did not detect any tank deformation during the oscillations. Note. For each fluid, the first line represents η 1 , that is, the viscosity measured at the smallest possible shear rate.

Data Analysis
For each experiment, the video recording was converted into a series of still frames (at 24 fps) which were then processed using the scikit-image library in Python (van der Walt et al., 2014). RGB images were first converted to grayscale, then thresholded using Otsu's method (Otsu, 1979). A Canny edge detector was then applied to detect the shape of the bubble (Canny, 1986). We then computed the area  , width w and height h of the bubble shape, as well as the position of its centroid. Finally, using the computed area, we calculated the equivalent diameter (i.e., the diameter of a disk with the same area) as: To determine the rising velocity of a bubble, we plot the height of its centroid as a function of time and fit a line through these data. The slope of the line is then the rising velocity. For each experiment, we obtain three different rising velocities: V pre , V shake , and V post , the rising velocities of the bubble before, during, and after shaking, respectively. For V shake , we generally exclude the first 2-4 s of shaking from the fitting, due to large initial wobble of the camera.

Results
We present the results of 82 experiments. A summary of the experimental parameters and corresponding results is provided in Appendix A. The time series from a typical experiment are shown in Figure 2. These include a subset of the original video frames (cropped around the bubble) as grayscale images, with the corresponding segmented image following the detection algorithm, and a graph of the bubble centroid height as a function of time, along with the three best-fitted lines whose slope give V pre , V shake , and V post .

Shape and Size of the Bubbles
For each frame, we measured the bubble's width w and height h, and computed the equivalent diameter d eq (Figure 3a). At such low Reynolds numbers (Re < 10 −6 ), bubbles are expected to have a spherical shape (Clift et al., 1978). In practice, however, bubbles are generally not perfect spheres, but retain a cusp at the bottom, due  to the removal of the needle after injection. Therefore, most bubbles have aspect ratio ℎ < 1 (average of 0.92, Figure 3b). A few bubbles have aspect ratios >1; these occurred when a small remnant bubble was present behind the main bubble, and incorporated as part of the main bubble by the detection algorithm.

Comparison to Newtonian Theory
We compare the pre-shaking and post-shaking still rising velocity V pre and V post to the theoretical rising velocity V St predicted by Equation 1 (Figure 4). Air density is negligible compared to the density of our fluids; hence, we used Δρ ≈ 970 kg⋅m −3 . For the bubble diameter, we used the average of the measured equivalent diameters d eq (Equation 4) over the pre-shaking period. In general, there is good agreement between our measured still rising velocities V pre and V post and the theoretical values V St , with most measurements falling within 10% of the expected value.

Rising Velocities With Shaking
We now consider the effect of shaking on bubble rising velocity. The ratio of the rising velocity during and before shaking V shake /V pre is plotted as a function of acceleration a and equivalent bubble diameter d eq in Figures 5 and 6, respectively. Larger accelerations led to larger uncertainties in the ratio V shake / V pre , especially for the M10 and M30 fluids. We also compute the average ratio V shake /V pre for each fluid and each acceleration investigated (Table 3).
The ratio V shake /V pre exhibits a weak apparent positive relationship with both acceleration and bubble diameter (except in Figure 6b). Overall, the bubble rising velocities measured during shaking are slightly higher compared to the pre-shaking ones, although averages given in Table 3 are within 2 standard deviations of 1, implying that V shake does not significantly differ from V pre in general. We note however that, for some cases, the sample size is very small, and hinders any further statistical analysis.

Discussion
We experimentally tested whether mechanical shaking would change bubble rising velocity in three different high-viscosity shear-thinning fluids. We found only weak evidence supporting our hypothesis at the explored conditions. This differs from the significant results of Iwata et al. (2008) andDe Corato et al. (2019). We now explore different reasons for this weak relationship.

Shear Rates in Our Experiments
First, we estimate the shear rate around the bubbles during shaking. The local shear rate at the surface of a bubble during shaking can be computed using  the change in equivalent diameter of the bubble (Iwata et al., 2008). This was difficult for our experiments due to the high signal-to-noise ratio in the equivalent diameter time series data. Instead, we fitted our diameter measurements with a function of the form where d 0 , δ, ω, and φ are the four fitting parameters. An example of an equivalent diameter time series along with the fitted function is shown in Figure 7. Assuming vertical axial symmetry, the change in equivalent diameter suggests that the volume of the bubbles successively expand and contract in response to bulk pressure changes, as opposed to an isochoric pure shear mechanism (as depicted in Figure 7). This is supported by the anticorrelation  between pressure and equivalent diameter in our experiments: d eq increases as pressure decreases, and vice versa (Figure 7). We can then compute the local shear rate at the bubble's surface as (De Corato et al., 2019): where ̇ is the time derivative of d(t), given by: Finally, we consider the maximum shear rate, which can be computed as: Note. The number of experiments for each case is also indicated (n). For the largest accelerations achieved in our experiments (1.93 g, parameter P5), we obtain maximum shear rate values in the range 0.04-0.08 s −1 . These shear rate values are well below the transitional shear rate of the M2 fluid (3.8 s −1 ; Rudolf et al., 2016) but only slightly below the transitional values for M10 (<0.21 s −1 , see Section 2.1) and M30 (between 0.10 s −1 , Section 2.1, and 0.26 s −1 , Rudolf et al., 2016). Therefore, it is likely that the shaking in our experiments was not sufficient to reach the shear-thinning regimes of our fluids. For comparison, Iwata et al. (2008)

Limitations
We saw in Section 4.1 that the shaking in our experiments did not produce shear rates large enough to reach the shear-thinning regime of our fluids. Stronger accelerations could be explored in the future, by increasing either the displacement amplitude A or the shaking frequency f. However, it should be noted from Figure 5 that the uncertainty in rising velocity increases with acceleration. There are two main reasons for this. First, the camera is mounted on a vertical arm attached to the shaking table, which wobbles for strong shaking. This leads to a decrease in the accuracy of the bubble height measurement. Second, the light source is stable (i.e., not mounted on the shaking table); hence, the reflection of light on the bubble varies during shaking, which influences the edge detection. An increased uncertainty in the edge detection yields larger uncertainty in both the bubble height measurement (and hence rising velocity) and the bubble area (and hence into the equivalent diameter and shear rate calculations). The increased uncertainty in determining the bubble's position during shaking also hindered a detailed study of any potential lateral displacement of the bubble. Such lateral displacement could be the source of additional strain on the bubble and will require further attention in future studies.

Geological Implications
Our study was motivated by the observation that degassing increases at open vent volcanoes following the passage of seismic waves from tectonic earthquakes (e.g., Avouris et al., 2017;Walter et al., 2009). We hypothesized that seismic waves could increase the rising velocity of bubbles contained in magma, and thus enhance degassing processes. We now consider the natural conditions where the results of our experiments could be applied.
10.1029/2022JB025741 9 of 13 Typical peak ground accelerations (PGAs) for moderate earthquakes with M W < 7 are <1 g (e.g., Douglas, 2003). For larger events (M W > 7) however, local accelerations can reach values >1 g, with values up to 2.7 g recorded after the M W 9.0 Tohoku and M W 7.8 Kaikōura events (e.g., Bradley et al., 2017;Goto & Morikawa, 2012). Overall, the accelerations considered in our experiments (0.48-1.93 g) are representative of the PGAs expected from moderate to large earthquakes. Similarly, the frequencies considered in our experiments (1-4 Hz) are representative of typical seismic waves frequency (0.1-10 Hz).
The viscosity of silicate melts covers an extremely wide range of values depending primarily on melt composition, temperature, and water content, from 10 0 Pa⋅s for hot, water-rich basalt to 10 9 Pa⋅s for dry, cold rhyolite (Giordano et al., 2008). Our fluids have viscosities in the range 2,000-30,000 Pa⋅s and thus correspond to low to intermediate melt viscosities. Overall the similar accelerations, frequencies, and viscosities between our experiments and natural systems remove the need for scaling and allow more direct applications of our results.
The shear rates at the surface of the bubbles in our experiments were of order 0.04-0.08 s −1 . For our fluids, the onset of non-Newtonian shear-thinning effects occurs at shear rates between 0.2 and 3.8 s −1 , that is, above the range of shear rates explored experimentally. In contrast, shear-thinning effects are observed in silicate melts at much lower shear rates, generally in the range 10 −4 − 10 −2 s −1 (e.g., Sonder et al., 2006;Webb & Dingwell, 1990a). Therefore, we surmise that, shear-thinning effects could play a more important role, if natural magma was subject to our experimental shaking, and hence bubble rising velocity could be increased in this case.
The shear relaxation time of magma may be estimated as: where G ≈ 10 10 Pa is the elastic shear modulus (Dingwell & Webb, 1989). Thus, for a range of viscosity 10 0 − 10 9 Pa⋅s, we obtain relaxation times of 10 −10 − 10 −1 s. For a maximum seismic frequency of 10 Hz, this yields Deborah numbers in the range 10 −9 − 10 0 . Assuming that elastic effects may be neglected for De < 0.01, we conclude that natural melts with viscosities η ≤ 10 7 Pa⋅s will satisfy the condition and behave viscously during shaking.
The bubbles in our experiments had diameters 18-30 mm. Natural magmas contain bubbles spanning a very wide range of diameters, from a few microns (e.g., Polacci et al., 2003) to over a decameter (e.g., Bouche et al., 2010). Therefore, centimeter-size bubbles are likely to occur in natural magmas, in particular in the lower viscosity cases.
Importantly, our analysis is for pure melt only, and neglects any effect from the presence of crystals. Crystals may drastically change magma rheology (e.g., Caricchi et al., 2007). Additionally, as the crystal fraction increases, the physics of rising bubbles transitions from bubbly flow (i.e., individual spherical bubbles rising, examined here) to channel flow and capillary fracturing (e.g., Degruyter et al., 2019). Therefore, our experiments are only applicable for low crystal fractions, where the crystals do not interfere with the bubbles. Similarly, our experiments only feature single bubbles. Magmas may contain many bubbles closely packed together, leading to increased interactions between bubbles (e.g., bubble coalescence, hindered settling, Davis & Acrivos, 1985).
Air is insoluble in our experimental fluids, as opposed to volatiles in natural magmas. As a result, our experiments do not exhibit any phenomenon related to the exsolution or dissolution of volatiles occurring during shaking (e.g., viscosity increase as water exsolves), though such processes may be important. In particular, shaking a pack of bubbles composed of soluble volatiles may lead to phenomena such as bubble growth reduction or Ostwald ripening (e.g., Masotta & Keppler, 2017;Seropian et al., 2022).

Conclusions
In this contribution, we report the results of a series of experiments where we measured the rising velocity of an air bubble in a shear-thinning fluid subject to mechanical oscillations. This is an important problem to volcanology, where magmas are often non-Newtonian and contain bubbles, but also to many industrial processes where the removal of gas bubbles from a fluid is key. We performed experiments with high-viscosity fluids and found that the shaking had only minor effects on the bubble rising velocity, at the explored conditions. We quantified the shear rates at the bubble's surface, which were below although close to the minimum values at which shear-thinning effects become significant in our fluids. Similarly, we calculated Deborah numbers and showed that our experiments were within the transitional zone where elastic and viscous effects may be competing. We finally discussed the applicability of our experiments to volcanic scenario. In particular, we note that, for silicate melts, shear-thinning behavior generally occurs at much lower shear rates than for our fluids. We thus postulate that rising bubbles in magma may accelerate when subject to shaking, for instance, from seismic waves.

Appendix A: List of Experiments
Full list of experiments performed and their parameters (Table A1).

Table A1 Full List of Experiments Performed and Their Parameters
Exp Note. See Table 2 for code definitions. η is the fluid viscosity, eq is the average equivalent diameter, A is the shaking displacement amplitude, f is the shaking frequency, a is the acceleration, V St is the theoretical Newtonian rising velocity, V pre , V shake , and V post are the rising velocities before, during, and after shaking, respectively, and shake pre is the ratio of the rising velocity during shaking to pre-shaking.