# Deformation of a macroscopic yield stress hydrogel during the free fall of a sphere

Nicolò R Sgreva, Anne Davaille

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Nicolò R Sgreva, Anne Davaille. Deformation of a macroscopic yield stress hydrogel during the free fall of a sphere. Physics of Fluids, 2024, 36 (12), 10.1063/5.0242825. hal-05408697

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# Deformation of a macroscopic yield-stress hydrogel during the free-fall of a sphere

Nicolò R. Sgreva*

Université de Lorraine, CNRS, LEMTA, 54000, Nancy, France

Anne Davaille

Université Paris-Saclay, CNRS, FAST, 91405, Orsay, France

## Abstract

We investigate the deformation of a macroscopic yield stress fluid consisting of a mixture of water and hydrogel (superabsorbent polymer) with high solid volume fractions and different grain size distributions during the settling of a sphere. The fluid's rheology combines viscous, elastic, and plastic behaviors and is described by a shear-thinning yield-stress model. Experiments were conducted in two motion regimes: the continuous fall, where the sphere reaches a constant terminal velocity, and the intermittent regime, with alternating periods of motion and no-motion. PIV and spatio-temporal images of laser-illuminated fluid cross-sections were used to determine the yield surface and analyze local grain motion dynamics.

In both regimes, a yield surface similar to an ovoid spheroid is observed. A scaling law is derived for lateral deformation (perpendicular to the fall direction), dependent on sphere size and Yield number (Y). This scaling law provides reliable estimations of maximum deformation widths, consistent with experiments in the literature using different yield-stress fluids. For all fluid samples used, the flow within the yielded region shows a strong fore-aft asymmetry, characterized by a negative wake at the back of the sphere related to the elastic component of the bulk deformation.

In the intermittent motion regime, we find that the transition from no-motion to motion is associated with reaching a maximum level of compression or triggering sufficient irreversible plastic events in the region of fluid under compression. This leads to a sudden avalanche-type of event where stresses are relaxed and the sphere starts moving downward again.

**Keywords:** Superabsorbent polymer, Falling sphere, yield-stress, elastic deformation, plastic events

<footnote>*nicolo.sgreva@univ-lorraine.fr</footnote>
1. Introduction

Yield stress fluids are a class of materials characterized by the yield stress ($\sigma_y$), a threshold that separates two main rheological behaviors. When the local stress exceeds the yield stress, the material flows and deforms indefinitely; otherwise, it behaves like a solid (Bonn et al., 2017). This stress-induced solid-liquid transition is a feature of many industrial processes (e.g. Wilson, 2019) and natural phenomena (e.g. Ancey, 2007; Sgreva et al., 2022).

Simple yield stress fluids (e.g., concentrated emulsions), can be described by rheological models such as the Bingham model or the Herschel-Bulkley model (Dinkgreve et al., 2018; Coussot, 2014). Once local stresses exceed the yield stress, a Bingham fluid exhibits Newtonian behavior through a linear flow curve and a constant value of viscosity. If a fluid shows a non-linear flow curve, it can very often be approximated by the Herschel-Bulkley model. The latter writes as:

$$ 
\begin{array}{l}
\sigma < \sigma_y \Rightarrow \dot{\gamma} = 0 \qquad (\text{solid-like regime}) \\
\sigma \geq \sigma_y \Rightarrow \sigma = \sigma_y + K_v \dot{\gamma}^n \quad (\text{liquid-like regime}),
\end{array}
\qquad (1) $$

where $\sigma$ is the stress, $\dot{\gamma}$ the shear rate, $K_v$ the consistency and $n$ the shear-thinning exponent. The Herschel-Bulkley model effectively describes the flow of many yield stress fluids, such as microgels (Di Giuseppe et al., 2015), soft particle glasses (Seth et al., 2011), emulsions (Bécu et al., 2006), foams (Gilbreth et al., 2006), and hydrogel mixtures (Sgreva et al., 2020).

In simple yield stress fluids, a denser spherical object falls when the drag force exerted by the fluid and the resistive force associated with the yield stress are both overcome. For small Reynolds numbers ($Re \ll 1$), two parameters characterize the free-fall of a sphere: the Bingham number ($Bi$), which compares the fluid's yield stress to viscous stress, and the Yield number ($Y$), which compares yield stress to buoyancy stress. For a Herschel-Bulkley fluid, where viscous stress is given by $K_v(v_s/d_s)^n$, the Bingham number becomes:

$$ Bi = \frac{\sigma_y}{K_v (v_s/d_s)^n}, \qquad (2) $$

with $v_s$ being the sphere's velocity and $d_s$ its diameter, while the Yield number is:

$$ Y = \frac{3\sigma_y}{g d_s \Delta\rho}, \qquad (3) $$

with $\Delta\rho = \rho_s - \rho_f$, i.e. the density difference between sphere and fluid. For a falling sphere, the relation between these two dimensionless numbers is described by a master curve (Tabuteau et al., 2007):

$$ \frac{1}{Y} = 7 + \frac{8.52}{Bi}. \qquad (4) $$

This equation defines the critical Yield number ($Y_c$) above which the sphere does not move. Experimental work with Carbopol microgel (Tabuteau et al., 2007) found this critical value to be $Y_c = 0.145$, consistent with previous numerical predictions (Beris et al., 1985; Blackery and Mitsoulis, 1997).
In addition to simple yield stress fluids, the analysis of the settling of objects is crucial in processes involving more complex systems. This includes studying the minimum size required for clasts to settle through seafloor sediments with a yield stress (Knappe et al., 2020), studying the mobility of gas bubbles in volcanic muds and crystal-rich magmas (Tran et al., 2015), and investigating the transport of small particles in drilling fluids (Rushd et al., 2019). These systems often involve fluids with heterogeneous structures, where solid particles form system-spanning networks that strongly affect flow dynamics and the settling sphere's behavior, leading, for instance, to oscillating or irregular falling velocities. Such phenomena have been observed in experiments with cornstarch suspensions (von Kann et al., 2011; Von Kann et al., 2013), Laponite (Fazilati et al., 2017), and wormlike micellar fluids (Jayaraman and Belmonte, 2003; Kumar et al., 2012). The influence of fluid structure on sphere dynamics becomes particularly significant when the fluid structure's length scale is relatively large compared to the sphere's size. For instance, in aqueous mixtures of hydrogels, the behavior of a falling sphere can be strongly influenced by the dimensions of the grains composing the mixture (Sgreva et al., 2020). This behavior is illustrated in the schematic regime diagram of Fig. 1, which shows the different motion regimes observed in hydrogels as a function of the Yield number and the ratio of the sphere's size to the fluid structure's length scale. While this figure is purely illustrative, with regime boundaries sketched based on experiments in Sgreva et al. (2020), it highlights how, beyond steady-state continuous fall and no-motion regimes, local interactions between the moving sphere and the macroscopic fluid structure result in an intermittent motion regime, where the sphere alternates between irregular falls and periods of no motion. Similar behavior is reported for a falling sphere in cellulose gels (Koochi et al., 2022), where the appearance of more chaotic motion regimes is related to the aging of the material.

In systems where interactions between the fluid structure and the falling sphere are expected, it is crucial to determine not only the sphere's falling dynamics but also the extent and shape of the region of fluid deformed by the sphere's passage, the nature of this deformation, and how different sizes of the fluid structure affect the deformation process. In the literature, these aspects have primarily been investigated for simple yield stress fluids, where local interactions between the intruder and the microscopic fluid structure have less impact compared to macroscopic yield stress fluids, where their structure can significantly affects the flow (Sgreva et al., 2020).

In experiments with Carbopol microgel, the extent of the yielded region is found to depend on the yield stress of the fluid sample (Holenberg et al., 2012). Within this region, the flow past the sphere remains confined nearby the sphere and can be asymmetric (Putz et al., 2008). A flow asymmetry, in the form of a negative wake behind the sphere, has also been reported in Laponite (Gueslin et al., 2009) and in aqueous viscoelastic solutions (Arigo and McKinley, 1998). This fore-aft asymmetry has been related to the fluid's elasticity (elastic recoil of shear stresses) and its occurrence with the relaxation of shear stresses generated near the sphere's side (Harlen, 2002; Putz et al., 2008).

In addition to experiments, recent theoretical and numerical studies have focused on the settling of differently shaped objects in simple yield-stress fluids. Fraggedakis et al. (2016) used transient finite element computations and an elasto-viscoplastic rheology to study the effects of elasticity on the falling sphere and the surrounding flow field. They found that $Y_c$ increases with the Deborah number when elastic effects are taken
<figure><img src="image_2.png" /><figcaption>Figure 1: Sketch of the different motion regimes of a sphere falling in aqueous mixtures of hydrogels, adapted from Sgreva et al. (2020). Here, $d_s$ is the sphere's diameter, $d_g$ the average particle size of the fluid, and $Y$ the Yield number. Aside from the regime boundary at $1/Y_c$, with $Y_c = 0.145$ (Tabuteau et al., 2007), the remaining boundaries are schematic, intended only to illustrate the general behavior observed in Sgreva et al. (2020).</figcaption></figure>

into account, and that the latter play a major role in achieving the for-aft asymmetry. Chaparian and Frigaard (2017b) studied how the 2D shape of settling particles affects the yield limit. Similarly, Chaparian and Frigaard (2017a) and Iglesias et al. (2020) investigated the impact of particle shapes on the yield surface's form.

The complexities in describing the fluid flow around the sphere highlight the importance of investigating the link between the macroscopic behavior of the already-yielded fluid and the behavior of the material at the particle scale. This includes studying the coupling between local elastic deformation of individual grains or particles and single or multiple irreversible plastic events (avalanches) that can suddenly un-jam the material (Balmforth et al., 2014). Another crucial aspect is understanding how the behavior of yield stress fluids with relatively large macroscopic structures deviates from that of simple yield stress fluids.

For these reasons, this work investigates how a macroscopic yield stress fluid deforms as a sphere passes through it. We aim to extend the study done in Sgreva et al. (2020) for a falling sphere in an aqueous mixture of hydrogels, highlighting the mechanisms of deformation (both reversible and irreversible) of the fluid and their relation to the different motion regimes of the sphere. For this purpose, we carried out new experiments to compare the yield surface obtained from Particle Image Velocimetry (PIV) measurements with the fluid deformation detected by analyzing spatio-temporal falling profiles.

The paper is organized as follows. In section 2, we introduce the fluid samples and describe the experimental setup and procedures used during the experiments. In section
3, we present and discuss the results of the experiments of the free fall of a sphere in the continuous fall motion regime, detailing the flow past the sphere and the extent of deformation in the surrounding fluid. Section 4 covers the same aspects for the irregular motion regime. We conclude with a summary of our findings in section 5.

## 2. Materials and methods

### 2.1. Fluids

Experiments were carried out using four fluid samples previously described and characterized in Sgreva et al. (2020) and Sgreva (2020b). They consist in different mixtures of water and swollen super-absorbent polymer (SAP) grains. Here, we used fluid samples n.1, n.2 and n.3, which correspond to “gel A”, “gel C”, and “gel E”, respectively, in Sgreva et al. (2020), and fluid sample n.4, corresponding to “gel F” in Sgreva (2020b). The procedures for preparing these samples and their descriptions are detailed in the supplementary material, while a rheological analysis is provided in Appendix A. The main fluid properties are summarized in Table 1.

According to the manufacturer (Omiya Green Service Co., Ltd), the original SAP powder is formed by the copolymerization of acrylic acid and acrylamide. When the dry SAP powder comes in contact with water, much of the latter is absorbed in the internal structure of SAP. This forms irregularly shaped swollen grains that can reach 100-200 times the size of the initial powder (Ramazani-Harandi et al., 2006). The final dimension of the swollen grains can be controlled by controlling the size of the initial raw powder. This allows the preparation of fluid samples that differ from each other in mean swollen grain (or particle) size, $d_g$ (Sgreva et al., 2020). The latter represents the mean equivalent diameter of a disk with the same area of the mean grains surface obtained from 2D images.

All the fluid samples were obtained from the same amount of dry SAP powder (17.5 ± 0.1 g) and distilled water (3.5 l). According to Ghannam and Esmail (1998), the low concentration of polymer in our final mixtures leads to negligible thixotropic behavior within single grains.

In similar heterogeneous fluids, such as pastes of soft particles, when the particle volume fraction $\phi$ reaches the random close packing limit $\phi_c$, the material undergo jamming transition (Seth et al., 2006). Hence, an yield-stress appears for $\phi > \phi_c$. The estimated solid volume fraction of our SAP-based fluids is around 0.63-0.70 (see Supplementary Material), meaning that grains are deformed and compressed together. Given the high solid volume fraction, the rheology of our fluids combines viscous, elastic and plastic aspects and can be associated with that of a Herschel-Bulkley fluid with a shear-thinning index $n=0.50$ (see Appendix A). Moreover, the size of the grains which constitute the fluid structure has a strong impact on the effective rheological properties and both yield stress ($\sigma_y$) and consistency ($K_v$) increase as $d_g$ increases (Sgreva et al., 2020). This occurs because an increment of grain size, for a constant distance between them, primarily causes an increase in their overlap, leading to a more solid network able to better withstand the flow (Seth et al., 2011).

### 2.2. Experimental set-up and procedures

To study the fluid's deformation around a settling sphere we used the setup shown in Fig. 2. It consists in a Plexiglas cylinder of height $H=50$ cm and diameter $D=10$ cm
<table><caption>Table 1: Properties of the fluid samples used for experiments. $d_g$ is the mean grain diameter, $\sigma_y$ the effective yield-stress and $K_v$ the effective consistency. The shear-thinning index is $n=0.50$.</caption><thead><tr><th>Sample's name</th><th>d<sub>g</sub> (mm)</th><th>σ<sub>y</sub> (Pa)</th><th>K<sub>v</sub> (Pa.s<sup>n</sup>)</th><th>Source</th></tr></thead><tbody><tr><td>n.1</td><td>1.4</td><td>2.81</td><td>8.09</td><td>Sgreva et al. (2020)</td></tr><tr><td>n.2</td><td>3.3</td><td>6.04</td><td>10.23</td><td>Sgreva et al. (2020)</td></tr><tr><td>n.3</td><td>5.7</td><td>33.99</td><td>16.53</td><td>Sgreva et al. (2020)</td></tr><tr><td>n.4</td><td>6.2</td><td>34.39</td><td>21.31</td><td>Sgreva (2020b)</td></tr></tbody></table>

filled with approximately 3.5 liters of fluid sample, i.e. the mixture of water and touching swollen SAP grains. The cylinder is placed in a rectangular tank filled with water in order to reduce optical distortion when a 532-nm laser sheet is used to illuminate the central vertical plane of the cylinder. Individual spheres were gently released from the center of the top surface of the fluid. We used steel spheres of density $\rho_s=7975\pm5$ kg/m$^3$ and diameter $d_s$ between 5.0 and 20.0 mm. Spheres were painted with black spray paint to avoid strong reflections. We did not observe any variation in the falling velocity between painted spheres and unpainted ones.

The motion of the sphere and the fluid's deformation around it were captured by a 25 fps video camera (Canon Legria HF S21 1080p) placed in front of the cylinder and perpendicularly to the laser sheet. Videos were recorded in a cylinder portion of height $h$, as shown in Fig. 2. To obtain the fluid velocity field around the sphere and to identify the existence and extent of the fluidized region we analyzed fluid cross-sections, illuminated by a laser-sheet, using the PIV package from LaVision. We treated SAP grains as tracer particles in the classical PIV framework and we detected the variation in light intensity at their boundaries between consecutive frames (2 or 5 frames, depending on the sphere's velocity). The interrogation windows were of 32x32 pixels. The displacement field (in pixels) was calculated by cross-correlation and then converted to velocity field by knowing pixel length and elapsed time between frames. The presence of SAP grains within the interrogation windows can reduce the density of tracer spots ($N$) within them. This results in increased uncertainty in the displacement measurement (Raffel et al., 2018). For a low density of tracer particles (e.g., $N = 5$) and displacements greater than half a pixel, the systematic uncertainty in the velocity measurement in our case is about 0.1 mm/s.

Since the fluid column was illuminated by the laser sheet from the left side (from the camera's point of view), we show in the following figures only the left side of the velocity field because the right one is obscured by the shadow of the falling sphere.

To avoid possible preferential falling path of the sphere due to fluid hysteresis, the whole fluid column was gently stirred after each run. By stirring the fluid between two successive experiments, a reproducible falling velocity was obtained for identical spheres (Sgreva et al., 2020). When we run the same experiments several times with this protocol, we observe the same sphere velocity and velocity profiles around the sphere. This suggests that the fluid deformation characteristics reported in the next sections are not due to thixotropy of the assembly of elastic hydrogels.
<figure><img src="image_3.png" /><figcaption>Figure 2: Diagram of the experimental setup. Movies are recorded within the section *h* with a height of about 30 cm and centered at the mid-height of the cylinder.</figcaption></figure>

**3. Results and discussion for the continuous fall motion regime (d_s >> d_g)**

Here we present the deformation of the hydrogel mixture during the settling of a sphere in the continuous fall motion regime, characterized by a rapid, steady fall reaching a constant terminal velocity (Sgreva et al., 2020). The focus of this section is on fluid deformation around the falling sphere. Using PIV on vertical cross-sections, we captured grains motion and identified the solid-like and the liquid-like region of the fluid mixture. Additionally, spatio-temporal images of light intensity at the grain boundaries are used to assess fluid deformation. No filters were applied during image post-processing to preserve image quality. Terminal velocity measurements of falling spheres for the various experiments are provided in Supplementary Material (section 2).

**3.1. Flow past the sphere from PIV measurements**

The fluid flow around a falling sphere of diameter *d*<sub>s</sub>=20 mm is shown in Fig. 3 in terms of magnitude of the instant velocity field (|*v*|). In this case the fluid is sample n.3 (*d*<sub>g</sub>=5.7 mm) and Y=7.46×10<sup>-2</sup>, Bi=1.13 and Re=1.45×10<sup>-1</sup>.

In Fig. 3a, the fluid velocity field is shown superimposed on the fluid's internal structure to display the relationship between swollen SAP grains and the moving sphere. Since this experiment was carried out in the continuous fall regime, the fluid structure does not significantly perturb the motion of the sphere, which falls with a constant terminal velocity of v_s=66.0±8.8 (mm/s). In Fig. 3b, |v| has been normalized by the sphere's velocity v_s. In this way, it is recognizable a region of fluid surrounding the sphere that move under the sphere passage, while more peripheral areas remain motionless. Although the velocity of the fluid at the sphere-fluid interface cannot be obtained with our setup due to light reflection from the sphere during PIV measurements, Fig. 3b shows that the velocity near the side surface of the sphere is about 20% of the falling velocity. Ahonguio et al. (2014) showed that for slip along the sphere-fluid interface, the lateral flow near the side of the sphere remains around 80% of the sphere's velocity. Moreover,
<figure><img src="image_4.png" /></figure>

Figure 3: Sphere settling through fluid sample n.3 in the continuous fall motion regime ($d_s=20$ mm, $d_g=5.7$ mm). The direction of movement is indicated by the white arrow. The snapshot is taken 6.2 s after the release of the sphere. (a) Magnitude of the instant velocity $|v|$ superimposed on the illuminated fluid structure. (b) $|v|$ normalized by the sphere's velocity $v_s$. Contour lines show intervals of 0.05. Dashed lines indicate sections in which spatio-temporal profiles are shown in the following images.
slip conditions lead to a lower drag coefficient and higher critical conditions of motion (Deglo de Besses et al., 2004; Ahonguio et al., 2014). In our case, we did not investigate the drag coefficient; however, the other two conditions correspond to what is expected for rough surfaces, suggesting no-slip at the sphere-fluid interface for our experiments.

In addition to the instant velocity field, the average velocity field, divided into its vertical and horizontal components, is shown in Fig. 4a-b for the same experiment of Fig. 3. These figures are obtained by averaging the velocity field at the sphere's reference point over 50 frames. The horizontal velocity field (Fig. 4a) averages to zero at the sphere's mid-height, highlighting two regions of opposite horizontal direction generated by fluid recirculation. From the vertical component of the velocity (Fig. 4b), it is recognizable a large fluid region ahead of the sphere that moves downward in the sphere's direction, while a smaller region behind the sphere moves upward. This asymmetry is a typical feature of flow around a settling sphere in simple yield-stress fluids, where the flow field ahead of the sphere differs from that behind it, generating the so-called fore-aft asymmetry (e.g. Putz et al., 2008; Arigo and McKinley, 1998).

In our case, we always observed an upward return flow at the rear of the sphere, with a strong fore-aft asymmetry in all fluid samples. As shown in Fig. 5a, the fluidized front zone remains close to the sphere. By plotting the vertical component of fluid velocity along the central axis of fall and normalizing it by the sphere velocity, $v_s$, we see that the fluidized front vanishes when $v_y/v_s=0$. To compare with the case of a Newtonian fluid, we define an effective viscosity $\eta_{eff} = \sigma/\dot{\gamma}$ with a characteristic shear rate of $\dot{\gamma} = v_s/d_s$. For a fluid without yield stress, stress writes $\sigma = K_v \dot{\gamma}^n$, leading to $\eta_{eff} = K_v(v_s/d_s)^{-0.5}$ for $n=0.5$. The fluid velocity profile along the vertical y-axis for a Newtonian fluid is (Batchelor, 1967):

$$v_y = v_{\infty} \left( -\frac{1}{2} \frac{R_s^3}{y^3} + \frac{3}{2} \frac{R_s}{y} \right), \qquad (5)$$

for a radial angle equal to zero, where $v_{\infty}$ is the Stokes' velocity for a fluid with viscosity $\eta = \eta_{eff}$.

In our fluid samples, we find a more limited velocity profile (red dashed line in Fig. 5b) compared to that obtained from eq. (5) (gray dashed line in Fig. 5b). This is expected because, unlike in viscous sedimentation, in yield-stress fluids the flow is localized near the sphere as the stress applied by the sphere on the surrounding fluid decays with distance (Beris et al., 1985). In our case, the velocity decay in the front region is exponential and can be fitted to:

$$\frac{v_y}{v_s} = (2.51 \pm 0.28) \exp(-(1.27 \pm 0.10)(r/R_s)), \qquad (6)$$

with a R-square of 0.98.

At the back of the sphere the velocity field is significantly different since the fluid moves also upward creating a negative wake. The presence of such feature in viscoelastic and elasto-viscoplastic fluids have already been reported both in numerical studies (Harlen, 2002; Frank and Li, 2006; Kim et al., 2005; Kemiha et al., 2006) and in experiments (Kemiha et al., 2006; Putz et al., 2008; Holenberg et al., 2012; Arigo and McKinley, 1998).

In our experiments, $Re$ is always small ($3.3 \times 10^{-4} < Re < 1.5 \times 10^{-1}$), making the influence of inertia on the formation of the negative wake irrelevant. In simple yield-
<figure><img src="image_5.png" /><figcaption>Figure 4: Horizontal (a) and vertical (b) components of the averaged velocity field for a sphere settling through fluid sample n.3 in the continuous fall motion regime ($d_s$=20 mm, $d_g$=5.7 mm). The figures are obtained by averaging the velocity field at the sphere's reference point over 50 frames. The coordinate axes are centered at the center of the sphere. The white arrow indicates the direction of motion.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Figure 5: Flow past a sphere in the continuous fall motion regime ($d_s=19$ mm, fluid sample n.4, $Y=8.1 \times 10^{-2}$, $Bi=1.52$, $Re=2.0 \times 10^{-2}$). The direction of motion is indicated by the red arrow. (a) Vertical component of the instant velocity field ($v_y$), with solid lines as streamlines. (b) $v_y$ normalized by the sphere's velocity $v_s$ as a function of the distance from the sphere's center $r$ normalized by the sphere radius $R_s$. $v_y$ here represents the mean velocity in a 2-mm-wide band along the vertical axis of the sphere's fall. Black solid lines delimit the region overlaid by the sphere. The red dotted line fits eq. (6). The gray dashed line shows the velocity profile from eq. (5) for a fluid with viscosity $\eta = \eta_{\text{eff}} = K_v \dot{\gamma}^{n-1}$.</figcaption></figure>

stress fluids, the nature of the negative wake is assumed to be linked to the elasticity of the fluid and its appearance to the relaxation of shear stresses generated close to the side of the sphere (Harlen, 2002; Putz et al., 2008). In our hydrogel mixture, the rise of the fore-aft asymmetry also seems linked to the elastic component of the bulk deformation. The presence of elastic deformation here directly relates to the forces generated by the interaction between two neighboring SAP grains. When in a suspension of soft particles the volume fraction exceeds the jamming limit, particles are compressed together. The interaction between two neighboring particles consists of two forces: an elastic force, perpendicular to the contact facet formed between the two particles, and a drag force, parallel to the facet, causing the particles to glide against one another. During flow, the rheology of these fluids is dominated by the elastic component of the stress (Seth et al., 2006, 2011), which is linked to the elastic deformation of particles during rearrangement and to the elastic force perpendicular to the contact facet. Both forces depend on the elastic properties of the particles, the contact radius (representing particle overlap), and the overlap distance (how much they are compressed).

Unfortunately, for swollen SAP grains, reliable measurements of the elastic properties of individual grains were not possible due to the material's extreme softness, which limits us in further characterization of the overall elastic deformation. Future work will focus on accurately measuring the shear modulus associated with the rheological transition
and quantifying the role of individual grain elasticity in bulk deformation.

## 3.2. Extent of deformation

The limit between the yielded and unyielded region of the fluid cannot be directly defined from the velocity field of Fig. 3b, as contour lines become increasingly irregular farther from the sphere. To better define the deformation limit, Fig. 6 displays the map of the second invariant of the shear rate tensor, $\dot{\gamma}_2 = \sqrt{\frac{1}{2}\dot{\gamma}_{ij}^2}$, calculated from the averaged velocity field. The figure shows the overall shape of the yielded region and its boundary, defined as the outline between the undeformed zone (set to $\dot{\gamma}_2=0.2 \text{ s}^{-1}$ due to the noise level in the PIV measurements) and the deformed zone ($\dot{\gamma}_2 > 0.2 \text{ s}^{-1}$), highlighting the strong shear localization near the sphere as expected for yield stress fluids (Gabard and Hulin, 2003). The small ring of low shear rates immediately around the sphere's outline is an artifact caused by light reflection, which slightly widens the sphere's section during PIV analysis. Shear localization, and hence shear banding, in yield stress fluids can be of three types (Fall et al., 2010): near the walls due to wall slip, due to aging effects within the fluid, and as a result of a heterogeneous stress field. In our experiments, the first two types should not be present since we did not observe slip at the walls, and no polymer chains are present between SAP grains to form system-spanning structures that can be affected by the flow. Thus, the shear localization in Fig. 6 can be attributed to the heterogeneous stress field induced by the falling sphere, resulting in a portion of the fluid sample being yielded (the region around the sphere) and another (more peripheral) remaining unyielded.

The 2D shape of the yield surface obtained from Fig. 6 is similar to those reported for experiments in Carbopol by Putz et al. (2008) and Holenberg et al. (2012). They found a yield surface approximating an ovoid spheroid, a geometrical shape that also describes our yielded area. In our case, the dimension of the yielded region ahead of the sphere is about twice the sphere's radius (Fig. 6), leading to a semi-major axis of the spheroid ($R_f$) being more than three times the sphere radius ($R_s$), i.e. $R_f \approx 3.1 R_s$. A similar $R_f$ is measured for the yielded region boundary obtained by Holenberg et al. (2012) for a smooth and light sphere ($R_f \approx 3.4 R_s$). However, the yielded envelop ahead the sphere reported by Putz et al. (2008) is more pronounced ($R_f \approx 5 R_s$). These discrepancies can be attributed to the different sensitivities of the experimental techniques used to derive the yield surface and the value of $\dot{\gamma}_2$ taken as the limit to distinguish between yielded and unyielded regions (here $\dot{\gamma}_2 = 0.2 \text{ s}^{-1}$, whereas in Putz et al. (2008) $\dot{\gamma}_2 = 0.1 \text{ s}^{-1}$).

As detailed in Appendix B, a yielded surface based on the second invariant of the shear rate should be interpreted only as a first approximation of the real surface subjected to solid-liquid transition. For this reason, a more effective approach to estimate the portion of fluid deformed by the sphere's fall is by directly analyzing the 2D spatio-temporal falling profile. In our case, this is possible due to the large size of SAP grains, enabling direct visualization of their behavior as the sphere approaches. We conducted analyses for both pixel lines of the light intensity of the illuminated fluid structure and for the velocity components obtained from PIV. In the first case (Fig. 7a), grain motion in the x-direction is detected from the spatio-temporal profile derived directly from the video of the fluid column illuminated by the laser sheet. In the second case (Fig. 7b), the spatio-temporal profile is obtained from the corresponding velocity field obtained from PIV. In both cases, one can observe a deformation front preceding and following
<figure><img src="image_7.png" /><figcaption>Figure 6: Second invariant of the rate-of-shear tensor obtained from the averaged velocity field (same experiments of Fig. 4). Contour lines show intervals of 0.2 s⁻¹. R_f and L_f indicate the extents of the deformation from the center of the sphere.</figcaption></figure>
the sphere, triggering motion of SAP grains. This can be observed, for example, by examining how bright and dark bands in the black and white image of the internal fluid structure evolve over time (Fig. 7a). The horizontal spatio-temporal profile also highlights the lateral extension (perpendicular to the falling direction) of the deformed area: bright bands on the side of the sphere bend upon its appearance and return to their original positions once the sphere has passed, indicating lateral motion of SAP grains alongside the sphere. This allows estimation of the maximum extent of lateral deformation from the sphere's outline, $l_d$, as shown in Fig. 7a.

A rough estimation of $l_d$ can also be inferred from wall effects. The influence of the falling sphere affects only a portion of fluid close to the sphere itself and does not extend into the unyielded region of the fluid. This means that beyond a critical tube size $D_c$, wall effects are not expected. The reduction of sphere velocity due to wall effects occurs in yield-stress fluids when $d_s/D > d_s/D_c$, where the critical ratio between sphere and container diameter $d_s/D_c$ is given by Atapattu et al. (1990):

$$d_s/D_c = 0.055 + 1.114Y \quad \text{for } Y \le Y_c. \qquad (7)$$

Since wall effects arise when the sheared fluid envelope surrounding the sphere reaches the wall, $d_s/D_c$ can be interpreted as the critical diameter ratio at which the lateral fluidized portion $l_d$ extends exactly to the distance between the edge of the sphere and the wall. In Fig. 8, we show the length of the experimental fluidized horizontal section from the center of the sphere, $L_f$ (i.e $L_f = l_d + d_s/2$), obtained from the spatio-temporal profiles as a function of the corresponding $D_c/2$ estimated by wall effects according to Atapattu et al. (1990):

$$\frac{D_c}{2} = d_s \left( 0.11 + 2.228 \frac{3 \sigma_y}{gd_s \Delta \rho} \right)^{-1}. \qquad (8)$$

From the linear fit of the data in Fig. 8, we find that the fluidized lateral section $L_f$ scales as:

$$L_f \sim d_s [(1.65 \pm 0.16) (0.11 + 2.228 Y)]^{-1}. \qquad (9)$$

Using eq. (9) to estimate $L_f$ for the experiments in Carbopol of Putz et al. (2008) and Holenberg et al. (2012), we obtain a good estimation of the maximum width of their yield surfaces, as summarized in Table 2. For the experiments of Holenberg et al. (2012), where a smooth and light sphere was used, the deviation is approximately 18% between the directly measured $L_f$ and the value obtained from eq. (9). A larger deviation is expected for the experiments of Putz et al. (2008), as their yield surface does not precisely correspond to the zero contour of shear rate, leading to an underestimation of the actual extent of the fluidized region. The scaling provided by eq. (9) can thus be effectively employed for rapid estimations of the maximum extent of deformation around a settling sphere, highlighting its relevance for a wide range of practical applications.

The value of $L_f$ obtained from eq. (9) can be compared with the maximal lateral extension of the yielded envelope obtained from PIV measurements. For the case reported in Table 2 and shown in Fig. 6 ($d_s = 20$ mm and fluid sample n.3), $L_f$ calculated from eq. (9) is slightly larger than that measured from the $\dot{\gamma}_2$-field. This discrepancy is consistent with the fact that the latter comes from a yielded envelope arbitrary defined at $\dot{\gamma}_2 = 0.2$ s$^{-1}$ instead of at zero, due to noise during PIV measurements.
<figure><img src="image_8.png" /><figcaption>Figure 7: Sphere settling through fluid sample n.3 in the continuous fall motion regime ($d_s=20$ mm, $d_g=5.7$ mm). (a) Horizontal pixel line of light intensity plotted over time (time elapsed since the sphere was released). The position of the pixel line is indicated by the green line in Fig. 3(b). $l_d$ indicates the extent of the deformed area at the sphere side. (b) Temporal evolution of the vertical ($v_y$) and horizontal ($v_x$) components of the velocity field along the same horizontal pixel line shown in (a).</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Figure 8: Measured extension of the yield surface $L_f$ from the center of the sphere as a function of the critical tube radius $D_c/2$ at which wall effects are expected for a given $Y$. Filled symbols represent experiments in the continuous fall regime of motion, while the empty symbol corresponds to the intermittent regime. The solid line is a linear fit of equation $y(x) = 1.65x$ with $R^2 = 0.89$.</figcaption></figure>

<table><caption>Table 2: Dimensions of the yielded envelope around a falling sphere from different experiments.</caption><thead><tr><th>Source</th><th>Y</th><th>Bi</th><th>Re</th><th>Measured R<sub>f</sub> (mm)</th><th>Measured L<sub>f</sub> (mm)</th><th>L<sub>f</sub> (mm) from eq. (9)</th></tr></thead><tbody><tr><td>Putz et al. (2008)</td><td>5.1×10<sup>-2</sup></td><td>3.1×10<sup>-1</sup></td><td>2.4×10<sup>-3</sup></td><td>16.5</td><td>9.3</td><td>17.5±2.4</td></tr><tr><td>Holenberg et al. (2012)</td><td>1.3×10<sup>-1</sup> <sup>[a]</sup></td><td>4.9×10<sup>1</sup></td><td>3.1×10<sup>-4</sup></td><td>16.3</td><td>11.1</td><td>13.1±1.8</td></tr><tr><td>Fig. 6</td><td>7.5×10<sup>-2</sup></td><td>1.13</td><td>1.5×10<sup>-1</sup></td><td>31.3</td><td>31.1</td><td>44.3±6.1</td></tr></tbody></table>

[a]: With $\Delta\rho$ calculated using eq. (4).

Similarly, fluid deformation above and below the sphere can also be followed using vertical spatio-temporal profiles. Fig. 9 shows three examples taken at distances of 33.2, 19.7 and 1.7 mm away from the central sphere axis. At the furthest distance from the sphere (Fig. 9a), bright bands move slightly upward by about 10 pixels (less than 1.5 mm) after the sphere has passed. Closer to the sphere’s path (Fig. 9b-c), the fluid structure is pushed downward before the sphere arrives and gently moves upward after it has passed.

Fig. 9 can also be used to qualitatively evaluate the elastic component of deformation. In fact, from the figure one can notice how some of the SAP grains are initially displaced from their original positions by the approaching sphere, only to return to their initial positions after the sphere passes. This reversible deformation is particularly evident in Fig. 9b-c, where the height of the black and white bands, after dropping briefly before the sphere arrives, returns to its un-deformed state when the sphere moves away. Indeed, immediately after the sphere passes, the light reflected by SAP grains located above the sphere marks a rapid upward movement that is originated from the elastic recoil of the fluid structure. This elastic recoil of the fluid structure is also apparent in Fig. 7a, where part of the fluid surrounding the sphere returns to its initial position immediately
<figure><img src="image_10.png" /><figcaption>Figure 9: Spatio-temporal vertical profiles of light intensity for the same experiment as Fig. 3. The position of the vertical lines along which the temporal evolution is shown are those indicated by red dashes lines in Fig. 3b. Their distance from the axis of fall of the sphere is: (a) 39.2 mm; (b) 19.7 mm; (c) 1.7 mm. Insets show zoomed areas where fluid deformation is more clear.</figcaption></figure>
after the sphere moves away. This recovery process can be quantified directly from Fig. 7a by analyzing the difference in light intensity along a horizontal pixel line at time $t$ compared to an initial time $t_0$ before the sphere’s arrival. This is shown in Fig. 10, where we plot the mean difference of a pixel line $\Delta_x$ with respect to its initial state as a function of time. Here, $t_0 = 4$ s, and $\Delta_x$ is normalized between its maximum value reached when the sphere appears completely in the spatio-temporal plot and its minimum value obtained from the average of $\Delta_x$ in the first 1.5 s after $t_0$. In Fig. 10, $\Delta_x$ remains relatively constant until around 6 s when the fluid begins to deform (red dashed line). The maximum difference corresponds to the appearance of the sphere in the plot (black dashed lines), after which it decreases until the deformation tail behind the sphere vanishes (red dashed line). Following the passage of the sphere, the difference between the final and initial states of the fluid structure is approximately 55 % (Fig. 10). For the case in Fig. 7a, this implies that approximately 45 % of the fluid structure along the $x$-direction returns its original undeformed state, highlighting the reversible nature of the deformation.

<figure><img src="image_11.png" /><figcaption>Figure 10: Temporal evolution of the normalized mean difference $\Delta_x$ between the light intensity of a pixel line at time $t$ in Fig. 7a and the same pixel line at time $t_0$. Black dashed lines delimit the presence of the sphere, while red lines the begging and the end of the deformation front.</figcaption></figure>

## 4. Results and discussion for the intermittent regime of motion ($d_s \ll d_g$)

In the intermittent regime of motion, the fall of the sphere is characterized by cycles of irregular motion at nearly constant velocity ("go events"), which are suddenly interrupted by periods where the sphere stops moving entirely ("stop events"). These latter intervals vary for each experiment: they occur at different heights in the fluid column and have different durations, up to involve almost the 60% of the total descending time (Sgreva et al., 2020).

In Fig. 11, we show the velocity field during both a "go" and a "stop" event for a sphere-to-grain size ratio of $d_s/d_g=2.1$. Although the sphere's velocity is more disturbed by fluid grains than in the previous motion regime, the fluid velocity field during the
“go” phase remains similar to that observed in the continuous fall case (Fig. 3a). As before, regions of fluid that move and those that remain stationary can be identified. Similar observations can be made by examining the evolution over time of a vertical cross-section illuminated by the laser sheet (Fig. 12a). During a “go” event, the fluid is initially pushed downward by the approaching sphere, then flows upstream after the sphere passes. This is highlighted by the rise of bright and dark bands right after the falling sphere’s track (e.g., on the right side of the falling sphere track in Fig. 12a).

As in the continuous fall regime, the lateral deformation can be detected from the spatio-temporal evolution of a horizontal pixel line at 125 mm from the top. During the “go” event (Fig. 13a), bright bands on the side of the sphere are distorted as the sphere approaches the height of the pixel line. When the sphere moves further downward, the bright bands return to their initial position. In the case of Fig. 13a, the width of the fluidized region is estimated from the figure to be $l_d=14.3$ mm. Given the sphere diameter $d_s=12$ mm, the width of the yielded region from the center of the sphere is $L_f=20.3$ mm. For the same fluid sample (i.e. same rheological properties), eq. (9) gives $L_f=19.0\pm2.6$ mm, which agrees with the value measured directly from the spatio-temporal profile. On the other hand, Fig. 13b shows that once the sphere stops, the portion of fluid at its side remains motionless. Bright and dark lines show no evident temporal evolution during the period in which the sphere remains at rest. These aspects suggest that the intermittent motion consists of a sequence of steady falls and stops.

We can explain observations in the intermittent regime by considering the characteristics of soft glassy materials. In these materials, flow occurs through a sequence of elastic (reversible) deformations and local plastic (irreversible) rearrangement events, during which stresses are relaxed (Mansard and Colin, 2012). The higher the local shear stresses (e.g., in the case of a larger sphere), the more common these rearrangement events become. Additionally, since these events are elastically coupled, i.e. the stress released after a plastic event is transmitted through the material by elastic interactions, a single event can trigger entire avalanches (Balmforth et al., 2014). In our case, Fig. 12b shows a very slow motion in front of the sphere during the stop phase. Initially, this motion can be attributed to the elastic compression of one grain under the weight of the sphere. The resulting deformation during the stop phase can be described by $\epsilon = \Delta l/d_g$, where $\Delta l$ indicates the total compression of the grain, as shown in the figure. The maximal percentage deformation measured is $\epsilon \approx 45\%$. This value is too large to be attributed solely to the elastic compression of a single grain. Instead, the elastic deformation likely affects several grains simultaneously and may also occur alongside a few plastic rearrangements. We cannot determine the exact number of plastic events during the stop phase because these events occur in all directions, while our data is limited to the illuminated 2D vertical plane. However, a change in dynamics becomes evident once a threshold of maximal compression is reached or a sufficient number of plastic rearrangement events are triggered in the compressed region. When this happens, an avalanche-type event occurs, during which stresses are relaxed, allowing the sphere to move downward again.

Moreover, another difficulty in experiments in the intermittent regime, as opposed to the continuous fall regime, arises from the sphere’s irregular motion, which occurs not only in the vertical direction but also perpendicularly to the illuminated plane (i.e., out-of-plane motion). This can cause the sphere to fall completely out of the illuminated 2D plane, especially if its falling velocity is relatively low, as is typical in the regime of irreg-
<figure><img src="image_12.png" /><figcaption>Figure 11: Sphere settling through fluid sample n.3 in the intermittent motion regime ($d_s=12$ mm, $d_g=5.7$ mm). (a) A “go” event where both the sphere and the surrounding fluid are in motion. $Y=1.2 \times 10^{-1}$, $Bi=6.15$, $Re=3.3 \times 10^{-4}$. Snapshot taken 39.4 s after the sphere was released. Vectors length is kept constant, and velocity magnitude is indicated by the colormap. (b) Motionless fluid during a “stop” event. Snapshot at time $t=72$ s. Here, vector length and color indicate the magnitude of the velocity field (vector length is ten times the pixel displacement measured).</figcaption></figure>

ular motion. Consequently, the number of experiments suitable for analysis was greatly reduced, as it was challenging to obtain experiments where the sphere remained within the illuminated plane long enough to collect more deformation data. To address this, the experimental setup should be modified to better accommodate this specific motion regime (e.g., by employing a thicker laser sheet), although this might be less effective for

439

440

441

442

443
<figure><img src="image_13.png" /><figcaption>Figure 12: Spatio-temporal vertical profile of light intensity for the intermittent regime. This figure shows the time evolution of a vertical profile centered on the sphere for the experiment in Fig. 11. The inset displays the motion of the sphere between two “stop” events. Zoomed-in views of two stop events are shown in A and B. $\Delta l$ indicates the maximal compression of swollen SAP grains in front of the sphere before it starts to fall again.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Figure 13: Spatio-temporal horizontal profile of light intensity for the intermittent regime. This figure shows the time evolution of a horizontal profile at a distance of (a) 125 mm from the top during a “go” event, and (b) 150 mm from the top during a “stop” event. The experiment is the same of Fig. 11.</figcaption></figure>

the continuous fall regime. This modification would allow for further experiments and analyses of the fluid deformation in the irregular motion regime, which is a direction for future investigations.

## 5. Conclusion

We conducted an experimental study on the deformation of a macroscopic yield-stress mixture of water and hydrogels during the settling of a sphere. Using PIV measurements and spatiotemporal images of laser-illuminated fluid cross-sections, we determined the extent of the yielded region around the sphere and analyzed local grain motion. The
experiments were carried out in two different motion regimes: the continuous fall regime, characterized by a rapid, steady fall of the sphere, and the intermittent regime, characterized by successive periods of no motion and irregular fall.

For the continuous fall regime, we provide a comparison between the yield surface obtained from PIV and the maximum lateral deformation (perpendicular to the direction of fall) obtained through spatio-temporal images. The 2D shape of the yield surface from PIV is similar to that reported for microgels and can be approximated with an ovoid spheroidal shape. However, defining the boundaries of this surface strongly depends on the sensitivity of the experimental technique used and the minimum value of $\dot{\gamma}_2$ assumed as the threshold between solid-like and liquid-like regions. On the other hand, given the macroscopic structure of the fluid, analyzing vertical and horizontal spatiotemporal images proved effective in quantitatively estimating the lateral extent of deformation. This approach allowed us to define a scaling law for the fluidized lateral section $L_f$ that depends only on Y and on the diameter of the falling sphere. This relation also provides accurate estimates of the maximum deformation width for similar experiments in the literature carried out with Carbopol (Putz et al., 2008; Holenberg et al., 2012).

Furthermore, fluid elasticity was found to play an important role in the flow within the un-yielded region around the sphere. For all fluid samples used (with different grain size distributions and rheological properties) and in both motion regimes, a strong fore-aft asymmetry was always observed, characterized by the presence of a negative wake behind the sphere.

Finally, in the intermittent motion regime the motion is characterized by a sequence of steady falls and stops. The transition from a stationary to a moving sphere is associated with achieving maximum compression or triggering sufficient irreversible plastic events within the compressed fluid region. When either condition occurs during the “stop” phase, it initiates a sudden avalanche-like event where stresses are relaxed, allowing the sphere to resume its downward motion.

These results highlight the role of elasticity and fluid structure on the motion of a yield-stress fluid, with relevance to various applications. An example is in geophysical contexts where many geological processes occur in heterogeneous systems with complex rheology. Specifically, our results can provide insights into magmatic systems, helping to identify conditions under which gas bubbles may either become trapped and accumulate in magma bodies or escape and move efficiently, potentially fluidizing large mushy regions of these systems. This has a direct impact on the dynamics of explosive eruptions (Cashman et al., 2017), making the proper description of the system's dynamics highly relevant for natural hazard assessment.

Future work will focus on determining the elastic properties of individual grains of the SAP mixture to establish the shear modulus associated with the rheological transition, and thus quantify the role that grain elasticity plays in bulk deformation. Moreover, improving visualization techniques will be crucial to extend the analysis to 3D deformation and accurately quantify the number of plastic events necessary to initiate sphere motion in the intermittent motion regime.

## 6. Supplementary Material

See supplementary material for a detailed description on the preparation, water content and solid volume fraction of the fluid samples used in the experiments; and for the
497 terminal velocity measurements.

498 Acknowledgments

499 We would like to thank Prof. Kei Kurita for providing Fig. S1. NRS was supported by the Initial Training Network No. 642029-ITN CREEP, an Horizon 2020 - Marie Sklodowska-Curie Action. This research was also supported by Labex PALM (ANR-10-LABX-0039- PALM).

503 Data Availability

504 The data that support the findings of this study are openly available at
505 https://gitlab.univ-lorraine.fr/sgreva1/sgreva_data.git

506 Appendix A. Rheological Analysis

507 The detailed rheological analysis of all the fluid samples used in our experiments can be found in Sgreva et al. (2020) and Sgreva (2020b). For clarity, we include here the relevant flow curves and rheological properties of the four fluid samples described in section 2.1.

511 The effective rheology of the the fluids employed here is reported in Fig. A.14. The data points refer to falling sphere experiments presented in terms of the effective stress induced by the net weight of the falling sphere in the fluid, i.e. $\sigma_{\text{eff}} = (gd_s\Delta\rho)/3$, and the effective shear rate $\dot{\gamma}_{\text{eff}} = (v_s/d_s)^n$, where $n = 0.5$ is the shear-thinning exponent, and $v_y$ is the sphere's falling velocity. Since for experiments involving relatively large spheres wall effects can arise (Sgreva et al., 2020), in Fig. A.14 we only considered cases where $v_s/v^* > 0.8$, with $v^*$ being the sphere's terminal velocity in an unbounded medium. Each fluid sample's data can be linearly fitted (solid lines in Fig. A.14), showing that the effective rheology of our SAP mixtures follow the Herschel-Bulkley model with a power index of 0.5, as described by eq. (4). Furthermore, these linear fits enable the estimation of the yield stress ($\sigma_y$) and consistency ($K_v$) for each fluid. This can be done by rewrite eq. (4) as

$$ \frac{1}{3}(gd_s\Delta\rho) = 7\sigma_y + 8.52 K_v \left(\frac{v_s}{d_s}\right)^n. \qquad (A.1) $$

523 The yield stress is then obtained from Fig. A.14 from the intercept of the linear fits of with the y-axis, that is $\sigma_y = \text{intercept}/7$. Similarly, the consistency is estimated from the slope of the fit, that is $K_v = \text{slope}/8.52$. Yield stress and consistency values for each fluid sample are reported in Fig. A.15 and in Table 1.

527 Appendix B. Shear rate-defined and stress-defined yield surfaces

528 A yielded surface based on the second invariant of the shear rate should be interpreted only as a first approximation of the real surface subjected to solid-liquid transition. This is because a shear-rate-based yielded surface is calculated directly from the velocity gradients of the flow field, ignoring the stress field experienced by the fluid during the
<figure><img src="image_15.png" /><figcaption>Figure A.14: Stress induced by the net weight of the sphere in the fluid ($\sigma_{eff}$) as a function of effective shear rate, $\dot{\gamma}_{eff}$, for the four fluid samples used in the experiments.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Figure A.15: Yield stress and consistency for the four fluid samples used in the experiments.</figcaption></figure>

sphere's passage. The flow (i.e., the unjamming) of a yield-stress fluid occurs when a specific yield criterion is met. Ovarlez et al. (2010) demonstrated that this yield criterion is well described by the Von Mises criterion. The criterion postulates that yielding of a ductile material begins when the second invariant of the deviatoric stress, $J_2$, reaches a critical value. According, the fluidization of a region in our fluid occurs when the fluid yield stress is overtaken by $J_2$, that is

$$ \sqrt{\frac{1}{2} J_2} \geq Y. \qquad (B.1) $$

However, in experiments, it is significantly easier to measure the velocity fields and work with the rate of the deformation tensor instead of measuring the stress field. A
yield surface defined on $\dot{\gamma}_2$ (shear rate defined) and one based on $J_2$ (stress defined) are equivalent only for inelastic and unidirectional flows (Cheddadi et al., 2012; Fraggedakis et al., 2016). In our case, the flow around the sphere is definitely more complex and hides a stress tensor where several components might be not null (i.e., elastic stresses are present throughout the material). When fluid elasticity cannot be neglected, the elastic contribution to the deviatoric stress tensor, which defines the fluidization according to the Von Mises criterion, becomes significant.

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