2.1 Liquid rheology

The liquids used in this study were (dimethylpolysiloxane) silicone oils (Clearco Products Co. Inc, USA) with low-shear kinematic viscosities of ${\it\nu}_{0}=100$ , 1000, 10 000, 1000 000 and 20 000 000 cSt (at $25\,^{\circ }\text{C}$ ). The shear-thinning nature of these liquids for ${\it\nu}_{0}\geqslant 1000$ cSt can be seen in figure 3 from the apparent viscosity ${\it\mu}_{app}$ versus shear rate $\dot{{\it\gamma}}$ data, obtained from tests conducted on a research rheometer (ARES-G2, TA Instruments). Fits to the rheological properties of these fluids have been formed using the Carreau model,

(2.1) $$\begin{eqnarray}\displaystyle \frac{({\it\mu}_{app}-{\it\mu}_{\infty })}{({\it\mu}_{0}-{\it\mu}_{\infty })}=[1+({\it\lambda}\dot{{\it\gamma}})^{2}]^{(n-1)/2}, & & \displaystyle\end{eqnarray}$$

where ${\it\mu}_{\infty }=0$ and values for the relaxation time ${\it\lambda}$ and power index $n$ are given in Table 1. Liquids with ${\it\nu}_{0}\geqslant 1000\,000$ cSt are found to be associated with relatively large relaxation times and $G^{\prime \prime }/G^{\prime }$ ratios approaching closer to unity, which indicates their high viscoelastic nature.

2.2 Parameter space

In accordance with previous studies (Ardekani et al. Reference Ardekani, Joseph, Dunn-Rankin and Rangel2009; Marston et al. Reference Marston, Vakarelski and Thoroddsen2011a ; Uddin et al. Reference Uddin, Marston and Thoroddsen2012), the main dimensionless numbers used in this work are the impact Stokes number, $St_{0}=2{\it\rho}_{s}R_{0}V_{0}/9{\it\mu}_{0}$ , Deborah number,  $De={\it\lambda}V_{0}/R_{0}$ and the capillary number,  $Ca={\it\mu}_{0}V_{0}/{\it\sigma}$ . Since a single-valued viscosity ${\it\mu}_{0}$ is assumed, the impact Stokes number and capillary number do not account for non-Newtonian effects. The film thickness was kept in the range of ${\it\delta}=4{-}6$  mm. Experiments for the relatively low viscosity liquids (100 and 1000 cSt silicone oils), however, were conducted for ${\it\delta}=37$  mm to result in fully immersed impacts. The release height was varied between $h_{r}=4.2{-}170$ cm, giving impact velocities of $0.31\leqslant V_{0}\leqslant 5.77~\text{m}~\text{s}^{-1}$ based on the film thicknesses used. These correspond to Stokes numbers ranging $8.18\times 10^{-3}\leqslant St_{0}\leqslant 305.3$ , Deborah numbers ranging $0\leqslant De\leqslant 401$ and capillary numbers of $1.44\leqslant Ca\leqslant 5.23\times 10^{6}$ depending on ${\it\mu}_{0}$ and ${\it\lambda}$ , respectively.

Figure 4. Series of images (top to bottom and left to right) showing cavitation during the impact from $h_{r}=67$  mm onto a glass plate covered with a 37 mm deep pool of ${\it\nu}_{0}=100$ cSt silicone oil. $R_{0}=10$  mm,  $V_{0}=0.77~\text{m}~\text{s}^{-1}$ ,  $St_{0}=264$ ,  $Ca=3.54$ . The insets present an enlarged image of the primary bubble entrapment (indicated by black arrows). The secondary bubbles are denoted with grey arrows in the first side-view image. A close-up of the bottom view at $t=125~{\rm\mu}\text{s}$ labelled with various cavitation structure features has also been provided. (See caption of figure 1 for description of $t_{0}$ ,  $h_{0}$ , *,  $\uparrow$ , $\downarrow$ and $h$ for this and all figures presented hereafter.)

3.1 Qualitative results

Figure 4 shows the impact of a sphere onto a glass wall covered with a 37 mm deep pool of ${\it\nu}_{0}=100$ cSt silicone oil (see caption for details). Since the sphere is released from above the surface of the liquid ( $h_{r}=67$  mm), we observe a main or ‘primary’ bubble entrapped on the sphere surface due to the lubrication pressure of the air (Marston et al. Reference Marston, Vakarelski and Thoroddsen2011a ). This is clearly seen at the bottom tip of the sphere in the first image (see black arrow). Other smaller (secondary) bubbles are also observed (see grey arrows), caused by the dynamic wetting of the sphere with the viscous liquid, as the main (outer) contact line moves up around the sphere.

As the sphere approaches the wall, these bubbles squeeze radially outwards and act as flow tracers. However, as soon as the sphere makes contact with the wall (taken as the reference point,  $t_{0}=0~{\rm\mu}\text{s}$ and $h_{0}=0~{\rm\mu}\text{m}$ ) and undergoes rapid deceleration, the liquid near the sphere surface depressurizes, causing the entrapped bubbles to expand and form more small bubbles around the region of contact. This region was shown by Marston et al. (Reference Marston, Yong, Ng, Tan and Thoroddsen2011b ) to be approximately equal to the Hertzian contact area, being the area on the tip of the sphere that flattens elastically during the collision period ( $t=0{-}31~{\rm\mu}\text{s}$ ). As the bubbles expand rapidly, the radial expanse of the cavitation increases reaching a maximum $r_{max}\approx 6.6$  mm at $t=63~{\rm\mu}\text{s}$ . By this time the sphere has reversed its direction of motion and the cavitation structure then starts to contract radially inwards. While bubbles on the outer edge collapse, those closer to the centre expand to form a distinct inner ring. Bubbles different from the rest of the structure having elongated interfaces are also observed immediately around the region of contact. At $t=219~{\rm\mu}\text{s}$ , bubbles collapsing at the outer extent reach the inner bubble ring, preventing it from further expansion. The inner ring then shrinks as the sphere moves further away and the entire structure disappears at $t=594~{\rm\mu}\text{s}$ . These observations are in agreement with those of Mansoor et al. (Reference Mansoor, Uddin, Marston, Vakarelski and Thoroddsen2014) and show cavitation to occur by depressurization in the liquid when the sphere makes contact and rebounds away from the wall, as predicted by the conventional theory. Though the primary bubble entrapped by the lubrication pressure of the air can be observed as a large expanded bubble during the collision, bubbles expanding from the air entrainment by the dynamic wetting of the sphere (secondary bubbles) are difficult to distinguish from cavitation bubbles surrounding the region of contact.

Figure 5. Bottom-view images of impacts onto a 37 mm deep pool of ${\it\nu}_{0}=1000$ cSt silicone oil from release heights of $h_{r}=47$ , 77 and 147 mm. $R_{0}=10$ mm,  $V_{0}=0.44~\text{m}~\text{s}^{-1}$ , $0.89~\text{m}~\text{s}^{-1}$ , $1.47~\text{m}~\text{s}^{-1}$ corresponding to $St_{0}=15.0$ , 30.4, 50.1,  $De=4.44\times 10^{-3}$ ,  $8.99\times 10^{-3}$ , $1.50\times 10^{-2}$ and $Ca=20.1$ , 40.7, 67.3, in (a), (b) and (c) respectively. Notice the decrease in size but increase in number of the secondary bubbles as the release height is increased (see insets for magnified views). The bubble entrapped by the lubrication pressure of air in each case has been marked by a white arrow.

This is not true for collisions occurring in more viscous liquids, whereby bubbles expanding from secondary bubbles and those opening from within the liquid to form cavitation bubbles can be observed to form in distinct regions at the sphere’s surface. Figure 5 shows the impact of a sphere onto a glass wall covered with 37 mm deep pool of ${\it\nu}_{0}=1000$ cSt silicone oil, for three different release heights $h_{r}=47$ , 77 and 147 mm. It can be observed that the air bubbles entrained by the dynamic wetting of the sphere become smaller in size and larger in number as the speed of impact onto the liquid surface increases (see insets). This phenomenon can be explained by the mechanism with which air entrainment occurs in liquid coating procedures (e.g. slot die coating), where a thin liquid film is applied onto a moving substrate. Air entrainment results from a series of processes which initiate when the dynamic contact line is unable to move quicker than a threshold speed perpendicular to itself (i.e. when the dynamic contact angle between the film and the substrate approaches $180^{\circ }$ ), producing sawtooth structures or air pockets (Blake & Ruschak Reference Blake and Ruschak1979). At this point, viscous forces pulling the liquid in the direction of substrate motion become dominant over surface tension forces acting in the opposite direction (Bhamidipati, Didari & Harris Reference Bhamidipati, Didari and Harris2012), which results in an interfacial instability, causing the film liquid to entrain the air pocket from all sides (Severtson & Aidun Reference Severtson and Aidun1996). When the substrate velocity (scaling with $V_{0}$ in our case) is increased, the viscous stresses also increase, enabling them to propagate upstream more quickly and causing the sawteeth to extend over a smaller area before pinch-off occurs (Bhamidipati et al. Reference Bhamidipati, Didari and Harris2012). This mechanism, which is dictated by the capillary number, results in the entrainment of smaller air bubbles as also noted in figure 5 (see figure caption).

For all three release heights, the primary (white arrows) and secondary bubbles squeeze radially outwards as the sphere approaches towards the glass wall to create an annular bubble structure, which has previously been misinterpreted as shear-induced cavitation by Seddon et al. (Reference Seddon, Kok, Linnartz and Lohse2012). While the deceleration of the sphere upon contact for $h_{r}=47$  mm is not enough for the liquid to open from these bubbles, an annular pattern of small bubbles connected in a foam-like structure is clearly seen for $h_{r}=147$  mm. In contrast, a distinct pattern of elongated cavity bubbles (Marston et al. Reference Marston, Yong, Ng, Tan and Thoroddsen2011b ) is observed immediately around the Hertzian contact area for all three release heights. While bubbles expanding from pressure reduction in the annular structure comprise of a mixture of air and vapour, those surrounding the contact region are mainly vapour filled. Also, since the liquid films are exposed to the atmosphere, the possibility of dissolved air cannot be neglected, which can result in optically similar cavity bubbles near the contact region and in the annular structure during the decompression stage.

Figure 6 shows the sphere impact from $h_{r}=25$ cm onto a glass plate covered with a 5 mm thick layer of ${\it\nu}_{0}=10\,000$ cSt silicone oil. The primary bubble, which initially spreads into an air sheet following the entry of the sphere into the liquid, is observed to deform into a ring of several smaller bubbles as the sphere approaches closer towards the wall ( $h=809~{\rm\mu}\text{m}$ ). These bubbles squeeze radially outwards in the final moments before contact, which is followed by a series of cavity formation events similar in nature to those observed from underneath the sphere in the 1000 cSt silicone oil film (figure 5). Since the secondary bubbles entrained are even smaller in size and larger in number, due to further increase in viscous forces experienced by the liquid, the annular cavitation structure formed is much more densely packed and noted to occur for a brief time period ( $t=0{-}30~{\rm\mu}\text{s}$ ). Meanwhile, the cavity pattern formed immediately around the contact area is composed of a distinct ring of fine bubbles (see inset for magnified view). The cavitation bubbles and the film liquid peel off from the sphere’s surface as it rebounds away from the wall to form a single larger cavity, which extends into a noticeably large hourglass shape ( $t=30~{\rm\mu}\text{s}$ –1.79 ms). Such an occurrence is not noted for impacts in lower viscosity liquids discussed in the preceding figures.

Figure 6. Images of an impact from $h_{r}=25$ cm onto a glass wall covered with a 5 mm thick layer of ${\it\nu}_{0}=10\,000$ cSt silicone oil. $R_{0}=10$  mm,  $V_{0}=2.19~\text{m}~\text{s}^{-1}$ , $St_{0}=7.43$ , $De=0.17$ ,  $Ca=993$ . The primary bubble entrapped on the sphere’s surface deforms by viscous shear flow as the sphere approaches the base wall (see insets for magnified views). The cavity is noted to peel off from the sphere’s surface while extending vertically to assume an hourglass shape.

Impacts from (a) $h_{r}=40$ cm and (b) $h_{r}=60$ cm onto a glass plate covered with a 5 mm thick layer ${\it\nu}_{0}=100\,000$ cSt silicone oil are shown in figure 7. In both cases the primary bubble entrapped on the sphere’s surface is squeezed radially outwards until it escapes into the atmosphere. The process however is so intense and instant that the bubble fragments at the sphere–bubble interface, leaving behind a cluster of much smaller bubbles in a circular patch at the point of close approach. We call these ‘remnant bubbles’ and indicate them by black arrows.

Figure 7. Images of impacts from (a) $h_{r}=40$ cm and (b) $h_{r}=60$ cm onto a 5 mm thick layer of ${\it\nu}_{0}=100\,000$ cSt silicone oil. $R_{0}=10$  mm,  $V_{0}=$ (a) $2.78~\text{m}~\text{s}^{-1}$ , (b) $3.42~\text{m}~\text{s}^{-1}$ corresponding to $St_{0}=0.94$ , 1.16,  $De=1.59$ , 1.95 and $Ca=12.6\times 10^{3}$ ,  $15.5\times 10^{3}$ , respectively. The black arrows indicate remnant bubbles composed in a circular patch at the point of close approach. The sphere rebounds without making contact for $h_{r}=40$ cm where the onset of cavitation is observed $181~{\rm\mu}\text{s}$ after time $t_{m}$ (reference point at which the sphere reaches its minimum gap distance $h_{m}$ , marked with the square symbol ▫ for this and all other figures presented hereafter). See also supplementary movie 1 available at http://dx.doi.org/10.1017/jfm.2016.229.

The inertia of the sphere is insufficient to fully penetrate the film and contact the base for $h_{r}=40$ cm (i.e. $St_{0}<St_{0}^{c}$ in § 3.2), as a significant portion of the kinetic energy is lost due to viscous dissipation, while the remaining is stored as elastic energy in the liquid film. The sphere is not expected to deform elastically in this manner since the deformation length scale, identified by Davis et al. (Reference Davis, Serayssol and Hinch1986) as $x_{1}=(4{\it\theta}{\it\mu}V_{0}R_{0}^{3/2})^{2/5}$ , is calculated as being small ( $x_{1}\approx 10~{\rm\mu}\text{m}$ ) relative to the minimum separation distance $h_{m}$ reached, even when the shear-thinning effects are neglected for $h\leqslant x_{0}$ . As the sphere reaches $h_{m}$ at time $t_{m}$ (taken as the reference point for non-contact cases), the remnant bubbles compress from the pressure in the squeeze flow, making them nearly invisible in the process. After this point the elastic energy stored in the film starts converting back to the kinetic energy of the sphere, enabling it to rebound. The liquid however does not cavitate immediately during the depressurization stage but at $t=121~{\rm\mu}\text{s}$ after reaching $h_{m}$ . To our knowledge, this is the first experimental evidence of cavitation by depressurization for a new class of non-contact cases during the impact of a sphere onto a wall covered with a film of viscoelastic liquid. Cavitation is first observed to originate from the same patch of remnant bubbles to form a dense foam-like circular structure, which then extends to a pattern composed of discrete bubbles. These bubbles generally increase in size as we move radially outwards from the point of closest approach. The circular foam-like structure is formed in the same region as the position of remnant bubbles observed when the sphere enters and descends ( $t=-909~{\rm\mu}\text{s}$ ) into the film, which confirms them to exist in a compressed form and not to squeeze radially outwards at $h_{m}$ . The cavity reaches its maximum radial extent at $t=1.67$ ms. As the sphere rebounds away from the wall, its motion is slowed down by gravity and viscous lubrication forces until it comes to a halt at $t=3.15$ ms. By this time the liquid has started to flow back into the cavitated region, which causes the cavity to shrink, and with further passage of time ( $t=10.75$ ms) the sphere gradually moves back towards the wall.

For $h_{r}=60$ cm, the sphere has enough kinetic energy to impact the glass wall after penetrating into the liquid film. The remnant bubbles in a circular patch (observed e.g. at $t=-394~{\rm\mu}\text{s}$ ) are again difficult to perceive in their compressed state ( $t=-91~{\rm\mu}\text{s}$ ), but then get squeezed out from the diminishing gap to create a dark circular area as physical contact is made ( $t_{0}=0~{\rm\mu}\text{s}$ , see panel 3 in the final column of figure 7). The dark spot, which corresponds to the Hertzian contact area, contrasts with the reflection of light from secondary bubbles in the surrounding region formed by air entrainment. These secondary bubbles are also present in the case of $h_{r}=40$ cm but are not as easily noticeable. Similar to contact cases in the less viscous liquid films shown in the preceding figures, cavitation for $h_{r}=60$ cm is observed as soon as the sphere impacts and starts to rebound away from the wall. In comparison to figure 6 however, the cavitation bubbles coalesce to form a single cavity structure, which extends into an hourglass shape ( $t=3.03$ ms). The bubbles expand radially in the process to form elongated interfaces (see bottom views), which are more prominently observed at the periphery of the structure. The annular cavitation structure, which is known to originate from the secondary bubbles, is not observed upon impact.

To understand the change in cavitation structures as the wetted collision changes from a non-contact to a contact case, we show a series of images in figure 8 taken from underneath the sphere for release heights ranging 36–70 cm ( $V_{0}=2.64{-}3.69~\text{m}~\text{s}^{-1}$ ). The images are taken when the cavitation structure reaches its maximum radial extent, from the moment the sphere either reaches its minimum gap distance ( $t_{m}$ ) or makes contact with the wall ( $t_{0}$ ). This time interval $t$ is generally noted to decrease as $h_{r}$ is increased (see figure for details). As seen from the images, the fine foam-like structure in a circular region, which originates from the remnant bubbles at the point of closest approach, is consistent among all three non-contact cases ( $h_{r}=36$ , 38 and 40 cm). The differences develop in the surrounding bubbles, which enlarge in size and become increasingly discrete entities ( $h_{r}=40$ cm) from being connected by small bubbles in a foam-like network ( $h_{r}=36$ cm). The discrete bubbles begin experiencing radial expansion and coalescence to form a single cavity bubble for contact cases starting with $h_{r}=50$ cm. More energetic impacts occurring for $h_{r}\geqslant 60$ cm intensify this phenomenon to result in the formation of elongated interfaces in the bottom views; these are generally prominent at the periphery of the structure, and are also found immediately around the Hertz contact area for $h_{r}=70$ cm. This area corresponds to the white region at the point of closest approach, which can be noticed to increase with the impact velocity. It is noteworthy to mention that while impacts here are categorized into non-contact and contact cases based on the visual appearance of a separation gap between the sphere and base wall before rebound, the impacting surfaces in the latter case can still be separated by an extremely thin liquid layer, as dictated by the elastohydrodynamic theory. A true contact in such cases may occur if the sphere has sufficient initial inertia to reduce the separation gap to a few nanometers (Serayssol & Davis Reference Serayssol and Davis1986), where additional effects such as surface roughness and interparticle surface forces become important.

Figure 8. Comparison of cavitation structures observed underneath impacts from various release heights onto a 5 mm thick layer of ${\it\nu}_{0}=100\,000$ cSt silicone oil. $R_{0}=10$  mm,  $V_{0}=2.64~\text{m}~\text{s}^{-1}$ ,  $2.71~\text{m}~\text{s}^{-1}$ , $2.78~\text{m}~\text{s}^{-1}$ ,  $3.12~\text{m}~\text{s}^{-1}$ , $3.42~\text{m}~\text{s}^{-1}$ ,  $3.69~\text{m}~\text{s}^{-1}$ corresponding to $St_{0}=0.89$ , 0.92, 0.94, 1.06, 1.16, 1.25 and $De=1.51$ , 1.55, 1.59, 1.78, 1.95, 2.11, respectively. The images are taken when the cavity reaches its maximum radial distance during the rebound stage of the sphere. The Hertzian contact area can be observed as a white circular region at the centre of the structure for $h_{r}=50$ , 60 and 70 cm.

Figure 9. Images of impacts onto a 5.5 mm thick layer of ${\it\nu}_{0}=1000\,000$ cSt silicone oil from (a) $h_{r}=90$ cm and (b) $h_{r}=120$ cm. $R_{0}=10$  mm,  $V_{0}=$ (a) $4.19~\text{m}~\text{s}^{-1}$ , (b) $4.84~\text{m}~\text{s}^{-1}$ giving $St_{0}=0.14$ , 0.16 and $De=32.9$ , 38.0, respectively. The remnant bubbles are observed to expand, coalesce and form a single large cavity bubble in the rebound stage of the non-contact case.

Investigating cavitation structures in the ultra-viscous liquid regime, figure 9 shows the impact of a sphere onto a 5.5 mm thick layer of ${\it\nu}_{0}=1000\,000$ cSt silicone oil for (a) $h_{r}=90$ cm (non-contact case) and (b) 120 cm (contact case). A circular patch of remnant bubbles at the point of closest approach is again observed in the bottom view ( $t=-545~{\rm\mu}\text{s}$ ) for the non-contact case (figure 9 a), which fades as the sphere reaches its minimum separation gap distance $h_{m}$ . Cavitation is noticed to initiate at $t=273~{\rm\mu}\text{s}$ in the form of small bubbles, which originate from the region of remnant bubbles observed earlier. The bubbles expand and coalesce as the sphere rebounds further away from the wall to form a single cavity bubble. Similar to the contact case shown in figure 7, a single cavity bubble having an hourglass shape is also formed on contact for $h_{r}=120$ cm in figure 9(b). The cavity surface, however, has a much smoother texture, which comprises of a few streak lines extending along the length of the cavity bubble, but many at the interface where the cavity attaches to the sphere (from above) and the wall (from below). The cavity reaches its maximum radial extent at $t=1.51$ ms and then starts shrinking as the sphere is brought to a halt by viscous forces and gravity, at a maximum rebound height of $h=3.45$  mm at $t=4.30$ ms. A portion of the rebound kinetic energy is again stored as elastic energy in the film, which pulls the sphere back towards the wall ( $t=8.96$ ms). The process repeats resulting in a series of sphere oscillations which come to a complete halt at $t=45.97$ ms. The cavity bubble shrinks to a structure composed of thin filaments, which break up one by one till the sphere is completely detached from the wall at $t=148.94$ ms.

Figure 10. Bottom-view images of impacts from various drop heights onto a 5.5 mm thick layer of ${\it\nu}_{0}=1000\,000$ cSt silicone oil. $R_{0}=10$  mm,  $V_{0}=4.07~\text{m}~\text{s}^{-1}$ , $4.30~\text{m}~\text{s}^{-1}$ , $4.53~\text{m}~\text{s}^{-1}$ , $4.63~\text{m}~\text{s}^{-1}$ , $4.84~\text{m}~\text{s}^{-1}$ , corresponding to $St_{0}=0.138$ , 0.145, 0.153, 0.157, 0.164 and $De=31.9$ , 33.7, 35.5, 36.3, 38.0, respectively. For each release height the pictures show the maximum radial distance reached by the cavity during sphere rebound. Notice the reduction in the maximum radial cavity distance in the contact case of $h_{r}=120$ cm.

Figure 10 shows a series of cavities at their maximum radial extent as the release height onto the liquid film is systematically increased. As also noticed in figure 8, the time interval $t$ between reaching $t_{m}$ (for non-contact cases) or $t_{0}$ (for contact cases) and this state is found to decrease as $h_{r}$ is increased (see figure for details). Cavitation in the non-contact cases is observed to originate from remnant bubbles, which expand and coalesce during the rebound stage to form a single cavity bubble. Since the minimum gap distance decreases with increase in release height, the remnant bubbles at the point of close approach have an increasingly lesser room to expand longitudinally. This forms a region appearing grainy at the centre of the cavity structure, which increases in size till $h_{r}=105$ cm. For $h_{r}=110$ cm, the $h_{m}$ reached is small enough to squeeze the remnant bubbles radially outwards from the point of closest approach. Qualitatively, the maximum radial cavity extent is observed to decrease as the collision changes from a non-contact to a contact case at $h_{r}=120$ cm. This is simply because the cavity attaches to the glass wall in the latter, which resists its radial expansion as the sphere rebounds away. This trend is not observed for contact cases shown in figure 8 as the cavities formed therein are only attached around the Hertzian contact area (see figure 7 b).

Figure 11. Images of an impact from $h_{r}=150$ cm onto a 5 mm thick layer of ${\it\nu}_{0}=20\,000\,000$ cSt silicone oil. $R_{0}=10$  mm,  $V_{0}=5.42~\text{m}~\text{s}^{-1}$ ,  $St_{0}=9.16\times 10^{-3}$ ,  $De=376$ . The grey arrow indicates the escape direction of the primary bubble entrapment into the atmosphere for reference purposes. Notice that the deposition of remnant bubbles on the sphere’s surface is not uniform.

Figure 12. Side-view images of impacts onto a 5 mm thick layer of ${\it\nu}_{0}=20\,000\,000$ cSt silicone oil from $h_{r}=155$ , 160 and 170 cm. $R_{0}=10$  mm,  $V_{0}=5.51~\text{m}~\text{s}^{-1}$ , $5.59~\text{m}~\text{s}^{-1}$ , $5.77~\text{m}~\text{s}^{-1}$ corresponding to $St_{0}=9.31\times 10^{-3}$ ,  $9.45\times 10^{-3}$ ,  $9.75\times 10^{-3}$ and $De=383$ , 388, 401, respectively. The onset of shear-induced cavitation and cavitation by depressurization is marked by symbols Ⓢ and Ⓓ, respectively. See also supplementary movie 2.

Investigating cavitation structures in the most viscous liquid used in this study, figure 11 shows the impact onto a glass wall covered with a 5 mm thick layer of ${\it\nu}_{0}=20\,000\,000$ cSt silicone oil. The release height is $h_{r}=150$ cm for which the sphere rebounds without making contact with the base wall. Though the close approach region is again observed to contain remnant bubbles ( $t=-424~{\rm\mu}\text{s}$ ), their deposition is not uniform (as compared to those shown in figures 7 and 9) but rather concentrated in the direction of the primary bubble’s escape into the atmosphere (see grey arrow). A fact proved later in the rebound stage when the bubbles start to expand ( $t=242~{\rm\mu}\text{s}$ ) and coalesce in the process to form a single cavity bubble, which is noticed to grow faster ( $t=363~{\rm\mu}\text{s}$ ) in the region where the majority of the remnant bubbles are situated. With the passage of more time ( $t=666~{\rm\mu}\text{s}$ ) the cavity enlarges further to develop an inverted conical structure.

Cavitation structures formed for increased release heights of $h_{r}=155$ , 160 and 170 cm are shown in figure 12 (see caption for details). In contrast to all the preceding figures where cavitation is only observed upon contact or during the rebound stage for non-contact cases, here we observe the liquid to cavitate when the sphere has reached a sufficiently small separation distance while approaching towards the glass wall i.e. during pressurization (marked with symbol Ⓢ). This phenomenon known as shear-stress-induced cavitation was first predicted on theoretical grounds by Joseph (Reference Joseph1998). The model presented was considered by Seddon et al. (Reference Seddon, Kok, Linnartz and Lohse2012) in proposing a new formal requirement for the onset of cavitation (see § 4, (4.1)). This stated that a liquid squeezing radially outwards in the diminishing gap between the sphere and the wall could rip open in tension if the applied shear stress overcomes the pressure in the film, reducing it below the liquid vapour pressure.

We find shear-induced cavities formed in figure 12 not to be characterized by well-defined edges but rather a hazy interface, which makes observing them in the bottom views (not shown) difficult. The cavities grow while the sphere continues its approach towards the wall to either reach a minimum gap distance (for $h_{r}=155$ cm) or make wall contact (for $h_{r}=160$ and 170 cm). Before shear-induced cavitation initiates however, remnant bubbles deposited on the sphere’s surface begin squeezing radially outwards from the point of closest approach. These are subsequently caught up and get trapped at the expanding interface of the cavity bubble formed by depressurization (marked with symbol Ⓓ), when the sphere rebounds away from the wall.

Figure 13. Close-up views from the side and bottom of selected cavitation events shown in figure 12 for $h_{r}=155$ and 160 cm. The remnant bubbles that squeeze radially out during the sphere’s close approach towards the base wall and get caught up in the cavity walls can be seen as small pecks.

In contrast to perfect hourglass shaped cavity bubbles observed for impacts in less viscous liquids, figure 12 and close-ups in figure 13 collectively show strikingly different cavitation structures. The radial expansion and longitudinal extension of structures along their length occurs in a disproportionate manner. Considering the case of $h_{r}=155$ cm for example (figure 12), the cavity bubble elongates more extensively in the central region during its early stages of formation by depressurization ( $151{-}666~{\rm\mu}\text{s}$ ) to experience necking. As the cavity stretches further with time ( $666{-}909~{\rm\mu}\text{s}$ ), it starts to peel off from the sphere’s surface creating a grainy texture at the top of the cavity wall (see figure 13). The neck shrinks further meanwhile and the cavity region underneath also experiences a strong resistance to radial expansion due to the ongoing longitudinal extension. As soon as the depressurization in the cavity bubble overcomes the effect of radial shrinkage resulting from its longitudinal elongation, the structure starts expanding radially outwards, reaching its maximum radial extent at $t=1.97$ ms. Close-up images of this process are shown in figure 13. Remnant bubbles that are squeezed radially outwards from the point of closest approach but subsequently caught and carried up along the cavity wall, can also be observed as spots or shorts streaks due to elongation. Because the cavity–sphere interface peels off steadily, the longitudinal extension is noted to localize beneath the necking region. The concentration of remnant bubbles in this cavity wall region therefore reduces, which becomes apparent with the increasingly reflective area formed. Interestingly, even though the sphere makes no contact with the wall for $h_{r}=155$ cm, the cavity formed is still observed to be attached to it by means of small bubbles, which rip open from the wall under depressurization. These bubbles are also noticed to form around the base of the cavity bubble for contact cases as shown for $h_{r}=160$ cm.

After reaching its maximum radial extent ( $t=1.97$ ms), the cavity starts to shrink radially inwards again due to the ongoing elongation as the sphere rebounds further away. The rebound kinetic energy is enough to overcome viscous forces, allowing the sphere to escape the fluid film. When the peeling cavity–sphere interface rises to the height at which it is about to break open into the atmosphere ( $t=2.63$ ms) in the process, the cavity reaches its maximum elongation (marked by symbol $\unicode[STIX]{x2460}$ ). As soon as the seal breaks ( $t=2.82$ ms) allowing air to enter inside, the partially detached cavity shrinks further radially inwards and starts to contract back towards the wall (marked by symbol $\unicode[STIX]{x2461}$ ). After the sphere has detached completely and escaped the liquid film, the longitudinal contraction of the cavity becomes rapid ( $t=3.03$ ms). Since the process of peeling does not occur proportionately along the cavity–sphere interface, the grainy textured cavity wall formed is asymmetric, which detaches and retracts earlier from one side to give the structure a noticeable tilt (marked by symbol $\unicode[STIX]{x2462}$ ). The cavity contracts to a minimum length (marked by symbol $\unicode[STIX]{x2463}$ ) at $t=3.21$ ms where a portion of its kinetic energy is stored as elastic energy in the film. The release of this energy enables the cavity to elongate again ( $t=3.48$ ms, marked by symbol $\unicode[STIX]{x2464}$ ), hence completing a cycle of vertical oscillation, which repeats with decreasing amplitudes until all the energy is dissipated internally by viscous forces. The ultra-viscous liquid flows back slowly into the cavitated space amidst these events thus collapsing the cavitation bubble. This becomes evident with the passage of further time, for example, at $t=59.63$ ms (marked by symbol $\unicode[STIX]{x2465}$ ).

For the higher release height of $h_{r}=170$ cm, in which case the sphere impacts the wall, the radial extent of the cavity formed in the initial stages of depressurization ( $t=30{-}91~{\rm\mu}\text{s}$ ) is comparatively much larger (see figure 12). However, the effect of elongation during ( $t=91{-}515~{\rm\mu}\text{s}$ ) is so dominant that the cavity not only develops a smaller neck but also shrinks noticeably radially inwards. This causes the contact angle of the cavity–wall interface to increase significantly, which promotes the formation of an additional necking region near the wall as the cavity extends further ( $t=848~{\rm\mu}\text{s}$ ). The cavity then expands as the effect of depressurization becomes prominent, reaching its maximum radial extent at $t=1.70$ ms, which is succeeded by the sequence of development stages marked $\unicode[STIX]{x2460}$ – $\unicode[STIX]{x2465}$ explained in the preceding passage.

Figure 14. Close-up images from the side and bottom of shear-induced cavitation and cavitation by depressurization (onsets marked by Ⓢ and Ⓓ, respectively) for an impact onto a ${\it\nu}_{0}=20\,000\,000$ cSt silicone oil film from $h_{r}=165$ cm. $R_{0}=10$  mm,  ${\it\delta}=5$  mm,  $V_{0}=5.68~\text{m}~\text{s}^{-1}$ ,  $St_{0}=9.6\times 10^{-3}$ ,  $De=395$ . Areas indicated by white and grey dashed lines are further magnified. An additional bottom-view sketches the areas of shear-induced cavitation (solid white line) and cavitation by depressurization (dashed black line) and the site of impact (black region).

Figure 15. Summary of qualitative results in a three-dimensional ( $St_{0}$ ,  $Ca$ ,  $De$ ) plot, showing phase diagrams in the $St{-}Ca$ and $St{-}De$ planes. Phases in the horizontal $St{-}Ca$ plane are identified as: (i) annular cavitation structure with a distinct pattern of elongated bubbles around Hertzian contact area, (ii) dense annular cavitation structure with a ring of fine bubbles around contact region and (iii) formation of remnant bubbles from primary bubble fragmentation at film entry. The dashed line serves as a phase boundary between contact (blue) and non-contact cases (yellow) in the $St{-}De$ plane. The letters here correspond to cavity features: (a) dense foam-like circular structure surrounded by a pattern of discrete bubbles; (b) single large cavity bubble, which assumes an inverted conical structure in (c); (d) hourglass-shaped single large cavity bubble, characterized by a smooth texture in (e). The region in red indicates shear-induced cavitation. ${\it\delta}=5$  mm and 37 mm for ${\it\nu}_{0}\geqslant 10\,000$  cSt and ${\it\nu}_{0}\leqslant 1000$  cSt, respectively.

Investigating shear-induced cavitation structures in further detail, figure 14 shows close-up images taken from the side and bottom for the impact of a sphere from $h_{r}=165$ cm onto a 5 mm thick layer of ${\it\nu}_{0}=20\,000\,000$ cSt silicone oil film. From the images shown, shear-induced cavitation is observed to originate from the sphere’s bottom tip ( $t=-91~{\rm\mu}\text{s}$ ) upon reaching $h=102~{\rm\mu}\text{m}$ , while advancing towards the wall. As mentioned earlier, the cavity is observed to have a hazy interface from the side views, which makes it difficult to see in the images taken from underneath. Therefore, an additional close-up view taken from the bottom, outlining the region of impact, shear-induced cavitation and cavitation by depressurization (see caption for details), has also been provided. The bottom views reveal the cavity to have an irregular shape, which grows radially until the sphere makes contact with the wall ( $t_{0}=0~{\rm\mu}\text{s}$ ). Cavitation by depressurization initiates as soon as the sphere reverses its direction of motion ( $t=30~{\rm\mu}\text{s}$ ) to form a circular structure, which overlaps a large portion of the volume cavitated earlier by shear stress. The rebound also implies an absence of positive shear stress in the film, causing the shear-induced cavitation structure to shrink radially inwards. This resists the expansion of the cavity structure forming by depressurization to create an evident lag ( $t=30{-}61~{\rm\mu}\text{s}$ ) at the interface where the two cavities meet. No further resistance upon the complete collapse of the shear-induced cavity at $t=91~{\rm\mu}\text{s}$ allows the lag to be overcome and the structure to become perfectly circular. We also note the cavity walls in the corresponding side views to become relatively well defined at this point. As the sphere rebounds further away, the effect of elongation becomes prominent ( $t=151~{\rm\mu}\text{s}$ ), causing the cavity to shrink radially inwards and onset the formation of a necking region.

We summarize our qualitative results in figure 15 by producing a three-dimensional space ( $St_{0}$ ,  $Ca$ ,  $De$ ) plot and categorizing the principal observations into $St{-}Ca$ and $St{-}De$ phase diagrams on the horizontal and vertical planes, respectively. Phases in the $St{-}Ca$ plane are marked (i)–(iii) and distinct features in the $St{-}De$ plane are lettered a–e, (see caption for details).

Figure 16. (a–d) Cavity diameter $D_{cavity}$ versus time $t$ from impact onto the glass wall ( $t_{0}=0~{\rm\mu}\text{s}$ for contact cases) or from reaching the minimum separation distance ( $t_{m}=0~{\rm\mu}\text{s}$ for non-contact cases) in different silicone oil viscosities and increasing impact velocities $V_{0}$ (see legend). (e) Time to cavity initiation $t_{c}$ and (f) maximum cavity diameter $D_{max}$ versus impact velocity onto liquid films, corresponding to plots in (a–d). (a) $R_{0}=10$ mm,  ${\it\delta}=5$  mm,  ${\it\nu}_{0}=10\,000$ cSt ( $St_{0}=4.65{-}7.43$ ,  $De=0.11{-}0.17$ ), (b) ${\it\nu}_{0}=100\,000$ cSt ( $St_{0}=0.66{-}1.25$ ,  $De=1.12{-}2.11$ ), (c) ${\it\nu}_{0}=1000\,000$ cSt ( $St_{0}=0.10{-}0.16$ ,  $De=24.5{-}36.3$ ), (d) ${\it\nu}_{0}=20\,000\,000$ cSt ( $St_{0}=8.18\times 10^{-3}$ – $9.75\times 10^{-3}$ ,  $De=336{-}401$ ). The fits in (b) plot $V_{0}={\it\alpha}t_{c}+{\it\beta}$ (in SI units for $t_{c}\geqslant 0$ m), where ${\it\alpha}=-5397,-2450,-1559$ and ${\it\beta}=3.42$ , 4.54, 5.75 for ${\it\nu}_{0}=100\,000$ , 1 million and 20 million cSt, respectively.

3.2 Quantitative results

In figure 16(a–d) we plot data for the cavity diameter, $D_{cavity}$ as a function of time $t$ for impacts onto different viscosity silicone oil films with increasing impact velocities (see caption for details). Contrary to contact cases where the cavity is noted to form as soon as, or immediately after, the sphere touches the glass wall, a noticeable time period from when $h_{m}$ is reached is required in non-contact cases for cavity initiation during rebound. Plotting the time period taken for cavity initiation $t_{c}$ versus the impact speed $V_{0}$ for all trials in figure 16(a–d), we produce figure 16(e,f). Interestingly,  $t_{c}$ values are found to decrease linearly with increase in impact speed for all non-contact cases until reaching minimum values in the range of $0{-}30~{\rm\mu}\text{s}$ , for cases where the sphere impacts the glass wall. Reasonable linear empirical fits to the data have been found using ${\it\alpha}$ and ${\it\beta}$ as prefactors (see caption of figure 16 e,f).

Following cavity initiation,  $D_{cavity}$ values in figure 16(a–d) are generally noted to increase and reach a maximum value $D_{max}$ , followed by a gradual reduction with time due to the incoming fluid. An exception to this trend occurs for $V_{0}=5.77~\text{m}~\text{s}^{-1}$ ( $h_{r}=170$ cm) in the 20 million cSt silicone oil film, where the cavity formed upon impact shrinks instantly and expands again (as explained in the preceding section). Figure 16(f) plots $D_{max}$ versus $V_{0}$ values for instances in figure 16(a–d) where the cavity diameter reaches a maximum value in the time frame of the videos recorded. Since $D_{max}$ values also depend noticeably on the degree of cavity attachment to the glass wall (e.g. as shown in figure 10), we classify each type for both sphere contact and non-contact cases into distinct categories (see legend). For contact cases the cavity either attaches only around the Hertzian contact area (e.g. figure 7 b) marked as partial cavity attachment or entirely at its base (e.g. figure 9 b) referring to a complete cavity attachment. The cavity does not attach to the wall in non-contact sphere cases with the exception of the impact at $V_{0}=5.51~\text{m}~\text{s}^{-1}$ in the 20 million cSt film (as shown in figure 13,  $h_{r}=155$ cm).

The increasing viscosity of films up to 20 million cSt adds resistance to cavity expansion under depressurization in the form of fluid friction, requiring higher impact velocities to obtain a given $D_{max}$ value. While $D_{max}$ increases with $V_{0}$ for ${\it\nu}_{0}=10\,000$  cSt, the values become steady as soon as contact cases start occurring for ${\it\nu}_{0}=100\,000$ cSt. When ${\it\nu}_{0}=1000\,000$ cSt, a decline in $D_{max}$ is noted for the contact case due to the added resistance to radial expansion offered by a complete cavity–wall attachment.

Figure 17. (a–d) Separation height $h$ plotted as a function of time $t$ for realizations in figure 16(a–d). $t=0~{\rm\mu}\text{s}$ is taken as a reference point corresponding to the moment of impact with the glass wall (for contact cases) or reaching the minimum separation distance (for non-contact cases). The inset provides a magnified view in one of the plots. (e) Minimum separation height $h_{m}$ plotted against the impact velocity $V_{0}$ from results in (a–d). Notice the linear decay in $h_{m}$ as $V_{0}$ increases for all non-contact cases. The fits plot $V_{0}={\it\alpha}h_{m}+{\it\beta}$ (in SI units for $h_{m}\geqslant 0$ m) where ${\it\alpha}=-1027,-1203,-1767$ and ${\it\beta}=2.83$ , 4.49, 5.66 for ${\it\nu}_{0}=100\,000$ , 1 million and 20 million cSt, respectively.

Figure 17(a–d) shows plots of the separation height $h$ between the sphere’s bottom tip and the glass wall versus time $t$ (see caption for details) for realizations in figure 16(a–d). The sphere is noted to reach a minimum position (set at $t=0~{\rm\mu}\text{s}$ ) in non-contact cases before rebounding away from the wall. This refers to the minimum separation height $h_{m}$ , which has been plotted for increasing impact velocities and film viscosities in figure 17(e). The data shows $h_{m}$ values to decrease linearly with increase in impact velocity until becoming zero for contact cases. Larger film viscosities require higher impact velocities to achieve a given $h_{m}$ value due to the increase in energy dissipation by internal fluid friction.

Figure 18. The (a) normalized minimum separation distance $h_{m}/{\it\delta}$ and (b) normalized cavity initiation time $t_{c}V_{0}/{\it\delta}$ plotted against the impact Stokes number $St_{0}=2{\it\rho}_{s}R_{0}V_{0}/9{\it\mu}_{0}$ corresponding to results in figures 16(e,f) and 17(e), respectively. The fits plot (a) $h_{m}/{\it\delta}={\it\alpha}ln(St_{0})+{\it\beta}$ where ${\it\alpha}=-0.46,-0.62,-0.58$ and ${\it\beta}=-1.60\times 10^{-2},-1.16,-2.70$ and (b) $t_{c}V_{0}/{\it\delta}={\it\eta}ln(St_{0})+{\it\zeta}$ where ${\it\eta}=-0.18,-0.91,-3.44$ and ${\it\zeta}=0.04,-1.65,-15.96$ for ${\it\nu}_{0}=100\,000$ , 1 million and 20 million cSt, respectively.

Non-dimensionalizing the results in figures 16(e) and 17(e), figure 18 plots the (a) normalized minimum seperation distance $h_{m}/{\it\delta}$ and (b) normalized cavity initiation time $t_{c}V_{0}/{\it\delta}$ against the impact Stokes number $St_{0}=2{\it\rho}_{s}R_{0}V_{0}/9{\it\mu}_{0}$ , respectively. The results show $h_{m}/{\it\delta}$ and $t_{c}V_{0}/{\it\delta}$ values to decrease as the impact Stokes number is increased for a given film viscosity. Also,  $St_{0}$ values at which the first contact case occurs decline as the viscosities are increased for a steady film thickness of ${\it\delta}=5$  mm. From the empirical fits to the data plotted (see caption for details), the critical impact Stokes number $St_{0}^{c}$ for contact cases to begin occurring is calculated to be 0.97, 0.15 and $9.5\times 10^{-3}$ for ${\it\nu}_{0}=100\,000$ , 1 million and 20 million cSt, respectively.

Figure 19. Shear rate ${\it\gamma}_{rz}$ versus separation distance $h$ at the line of closest approach (a,c,e,g) and rebound (b,d,f,h) corresponding to plots in figure 17(a–d).

Performing a regression analysis of the data shown in figure 17(a–d) and differentiating the $h$ profiles with respect to $t$ , we calculate the sphere tip velocity $V_{z}$ as a function of time. Using the lubrication length scale in accordance with Ardekani et al. (Reference Ardekani, Joseph, Dunn-Rankin and Rangel2009), the average radial velocity from the continuity equation can be expressed as $V_{r}\approx V_{z}\sqrt{R_{0}h}/h$ . We then obtain estimates of the apparent horizontal shear rate $\dot{{\it\gamma}}={\it\gamma}_{rz}=V_{r}/h\approx V_{z}\sqrt{R_{0}h}/h^{2}$ as the sphere approaches towards and rebounds away from the wall with time. Plots of ${\it\gamma}_{rz}$ versus $h$ in this manner are presented in figure 19 (see caption for details) for all realizations in figure 17(a–d).

Shear rates during the approach stage (figure 19 a,c,e,g) for contact cases are noted to increase with a decrease in $h$ , reaching maximum values in the range of $6.5\times 10^{5}~\text{s}^{-1}\lesssim {\it\gamma}_{rz}\lesssim 2.7\times 10^{7}~\text{s}^{-1}$ for $h\leqslant 35~{\rm\mu}\text{m}$ . In the non-contact cases, ${\it\gamma}_{rz}$ increases until suddenly coming to a decline as the sphere comes to a halt upon reaching $h_{m}$ . Maximum values here were found to reach as high as $10^{5}~\text{s}^{-1}$ at close approach in some instances. In the rebound stage (figure 19 b,d,f,h), the direction of fluid flow reverses as indicated by the negative ${\it\gamma}_{rz}$ values. Shear rates in contact cases decline as the separation distance increases while in non-contact cases, an immediate increase to reach a maximum is noticed before the values start to decline as the sphere rebounds further away.

3.3 Velocity field measured by particle image velocimetry analysis

Results for the apparent horizontal shear rate ${\it\gamma}_{rz}$ in the preceding section are based on average values of the radial velocity $V_{r}$ derived from $V_{z}$ . In order to obtain more reliable results directly from the velocity fields, we conducted further experiments employing particle image velocimetry (PIV) techniques.

For this purpose the ultra-viscous liquids were seeded with $10~{\rm\mu}\text{m}$ diameter hollow glass beads (Potters Industries Inc.) and mixed thoroughly to ensure a uniform composition was achieved before being left for a week to set into evenly levelled films. The use of high magnification ( $4\times$ ) ensured a narrow depth-of-field, which coupled with the high capture rate (33 018 f.p.s.) rendered a pseudo-high-speed micro-PIV, where the plane of interest was directly in alignment with the focus of the bottom tip of the sphere. Consecutive images from the recorded videos were then analysed using using a sequential frame cross-correlation in MATLAB and DaVis 7.2 software (LaVision GmbH).

Figure 20. Particle image velocimetry measurements for a non-contact impact from $h_{r}=65$ cm onto a ${\it\delta}=4$  mm thick layer of ${\it\nu}_{0}=1000\,000$ cSt silicone oil. $R_{0}=10$  mm,  $V_{0}=3.56~\text{m}~\text{s}^{-1}$ ,  $St_{0}=0.12$ ,  $De=27.9$ . The left half of the sphere displays the velocity vectors (see scale bar) while the right half maps the radial component of velocity $V_{r}$ (see colour bar). Plots of $V_{r}$ versus height $Z$ (mm) from the base wall at a radial distance of $r=3$  mm (vertical dashed line) have also been shown. The horizontal dashed line corresponds to $H(r,t)=h(t)+R_{0}-\sqrt{R_{0}^{2}-r^{2}}$ at $r=3$  mm. See also supplementary movie 3.

Figure 20 shows results for the PIV analysis of a tungsten carbide sphere ( $R_{0}=20$  mm) impacting onto a 4 mm thick film of ${\it\nu}_{0}=1000\,000$ cSt silicone oil when released from a height $h_{r}=65$ cm. The presence of PIV particles in the film here provides additional points for the liquid to cavitate from during depressurization in comparison to films without beads used in figure 9. These appear as a few number of small bubbles that originate at the surface of the sphere and the base wall, in the vicinity of the significantly larger single cavity bubble (see also supplementary movie 3), and are noted to have a negligible effect on the surrounding flow field. The velocity fields where the vectors represent the absolute velocity,  $|v|$ are superimposed on the left half of the raw images in order that the sphere, cavity and wall edges can be observed. Colour maps of the radial velocity component,  $V_{r}$ are superimposed on the right half where positive values correspond to flow away from the sphere’s centre and negative values towards it. Plots of $V_{r}$ versus height $Z$ (mm) above the wall at $r=3$  mm (see caption) have also been included to portray the changes incurred by the radial velocity component at a given a radial distance.

The velocity fields obtained in the squeeze flow regime during the approach stage are noted to be in agreement with Engmann, Servais & Burbidge (Reference Engmann, Servais and Burbidge2005) and Uddin et al. (Reference Uddin, Marston and Thoroddsen2012). At $t=-787~{\rm\mu}\text{s}$ for example, the radial velocity component is observed to be prominent in a central band, the vertical position of which increases with radial distance. An approximately parabolic profile for $V_{r}$ at $r=3$  mm further implies $V_{r,max}$ to occur at $z\sim h(r,t)/2$ , as also predicted by Uddin et al. (Reference Uddin, Marston and Thoroddsen2012). The band is noted to draw towards the axis of symmetry as the sphere approaches closer to the wall ( $t=-212~{\rm\mu}\text{s}$ ). Moments before the sphere comes to a halt ( $t=-60~{\rm\mu}\text{s}$ ) and reaches its minimum gap distance,  $V_{r}$ diminishes rapidly as noticed by the almost vertical line, indicating a nearly zero radial flow $V_{r}\approx 0$ across the gap. This flow direction reverses ( $t=242~{\rm\mu}\text{s}$ ) once the sphere starts to rebound, forming a completely inverted parabolic $V_{r}$ profile in the negative half. The expanding cavity bubble ( $t=454~{\rm\mu}\text{s}$ ), observed briefly after the onset of cavitation, opposes the radial influx of film liquid into the extending sphere–wall gap, as indicated by the velocity vectors near the cavity surface. The opposition weakens (see colour maps) as the rate of radial cavity expansion decreases with time ( $t=454~{\rm\mu}\text{s}$ –1.64 ms) until eventually becoming zero when the cavity starts to shrink radially inwards ( $t=2.30$ ms). Further away from the cavity interface at $r=3$  mm, the fluid inflow is noted to decrease incessantly ( $t=454~{\rm\mu}\text{s}$ –2.30 ms) as the sphere moves away from the wall.

Figure 21. (a,b) Shear rate ${\it\gamma}_{rz}$ versus separation distance $h$ , (c,d) apparent viscosity ${\it\mu}_{app}$ versus $h$ and (e,f) apparent Stokes number $St_{app}=2{\it\rho}_{s}R_{0}V_{z}/9{\it\mu}_{app}$ versus normalized gap height $h/{\it\delta}$ profiles obtained from direct measurements of the velocity fields at $r=2$ and 3 mm (see legend), by the PIV analysis of the realization in figure 20 ( $V_{0}=3.56~\text{m}~\text{s}^{-1}$ ,  ${\it\mu}_{0}=978$ Pa s,  $St_{0}=0.12$ ,  $De=27.9$ ). Estimated shear rate values derived from $V_{z}$ using ${\it\gamma}_{rz}\approx V_{z}\sqrt{R_{0}h}/h^{2}$ (Ardekani et al. Reference Ardekani, Joseph, Dunn-Rankin and Rangel2009) and the empirical scaling ${\it\gamma}_{rz}\approx V_{z}\sqrt{R_{0}h}/3h^{2}$ , have also been inscribed in (a) for comparison purposes. (a,c,e) and (b,d,f) correspond to the approach and rebound stages, respectively.

Given the experimental configuration in this study, the horizontal shear is ${\it\gamma}_{rz}\equiv \partial V_{r}/\partial z$ . In accordance with Ardekani et al. (Reference Ardekani, Joseph, Dunn-Rankin and Rangel2009) and Uddin et al. (Reference Uddin, Marston and Thoroddsen2012) this shear rate at a given radial distance can be calculated as ${\it\gamma}_{rz}\approx V_{r,max}/H(r,t)$ where:

(3.1) $$\begin{eqnarray}\displaystyle H(r,t)=h(t)+R_{0}-\sqrt{R_{0}^{2}-r^{2}} & & \displaystyle\end{eqnarray}$$

is the separation gap distance across the curvature of the sphere in time.

From the PIV analysis of the realization in figure 20, figure 21(a,b) shows plots of ${\it\gamma}_{rz}$ at $r=2$ and 3 mm versus the separation distance $h$ at sphere tip. Results derived from $V_{z}$ values where ${\it\gamma}_{rz}\approx V_{z}\sqrt{R_{0}h}/h^{2}$ (Ardekani et al. Reference Ardekani, Joseph, Dunn-Rankin and Rangel2009) are also added for comparison purposes. While these derived shear rate values are found to be higher by approximately 0.5–1 orders of magnitude than those obtained directly from the velocity fields during both the approach and rebound stages, they capture the trend noted in the PIV results with reasonable accuracy. The shear profiles produced in a similar manner in figure 19, based on the scaling from Ardekani et al. (Reference Ardekani, Joseph, Dunn-Rankin and Rangel2009), can hence be deemed in accordance with actual values only at a qualitative level. Performing fits to the PIV results to form an empirical scaling, an acceptable agreement can be obtained for ${\it\gamma}_{rz}\approx V_{z}\sqrt{R_{0}h}/3h^{2}$ . From the PIV measurements of the non-contact case in figure 20, we note the shear rates in the approach stage to rise from ${\it\gamma}_{rz}\sim 800~\text{s}^{-1}$ at $h\approx 2.6$  mm to ${\it\gamma}_{rz}\sim 3000~\text{s}^{-1}$ at $h\sim 0.60$  mm, before declining steeply as the sphere slows down to reach its minimum gap distance. For the reversed flow direction in the rebound stage, shear rate values increase rapidly to reach an average maximum of ${\it\gamma}_{rz}\sim -2500~\text{s}^{-1}$ at $h\sim 0.40$  mm, followed by a continuous decline to ${\it\gamma}_{rz}\sim -80$ and $-125~\text{s}^{-1}$ at $r=2$ and 3 mm respectively, when the sphere reaches $h\sim 2.1$  mm.

Given the shear-thinning nature of the fluid used, these shear rates can correspond to apparent viscosities that are significantly different from the zero shear viscosity value,  ${\it\mu}_{0}$ . This effect is depicted in figure 21(c,d) where the Carreau model and parameters from Table 1 are used to plot the apparent viscosity,  ${\it\mu}_{app}$ (from shear values at a given radial distance), at $r=1$ and 2 mm as a function of gap height $h$ . We note the local viscosity in the approach stage to decline at a similar rate at both radial positions to reach a minimum of ${\it\mu}_{app}\sim 28$ Pa s ( ${\it\gamma}_{rz}\sim 3000~\text{s}^{-1}$ at $h\sim 0.60$  mm,  ${\it\mu}_{app}/{\it\mu}_{0}\sim 0.03$ ), followed by a steep increase as the shear rates decrease. A minimum of ${\it\mu}_{app}\sim 32$ Pa s is noted, corresponding to the maximum shear rate in the rebound stage at $h\sim 0.40$  mm. With further increase in gap height, the local viscosity increases at a distinctly higher rate at $r=2$  mm than 3 mm, reaching ${\it\mu}_{app}\sim 300$ and 220 Pa s respectively, when $h\sim 2.1$  mm.

Figure 22. PIV measurements for the contact case of a sphere dropped onto a 4 mm thick layer of ${\it\nu}_{0}=20\,000\,000$ cSt silicone oil from $h_{r}=140$ cm. $R_{0}=10$  mm,  $V_{0}=5.23~\text{m}~\text{s}^{-1}$ ,  $St_{0}=8.84\times 10^{-3}$ , $De=363$ . Similar to figure 19, the left and right half of the sphere display the velocity vectors (see scale bar) and colour maps of the radial velocity component $V_{r}$ , respectively. Plots on the right side show the variation in $V_{r}$ versus distance $Z$ (mm) from the base wall at $r=2$  mm (see vertical dashed line). The horizontal dashed line marks the distance to the sphere edge from the base wall at $r=2$  mm, using (3.1). See also supplementary movie 4.

Figure 23. (a,b) Shear rate ${\it\gamma}_{rz}$ versus the separation distance $h$ , (c,d) ${\it\mu}_{app}$ versus $h$ and (e,f) $St_{app}$ versus normalized gap height $h/{\it\delta}$ obtained from the PIV measurements at $r=1$ and 2 mm of the realization in figure 22 ( $V_{0}=5.23~\text{m}~\text{s}^{-1}$ ,  ${\it\mu}_{0}=19\,580$ Pa s,  $St_{0}=8.84\times 10^{-3}$ ,  $De=363$ ). Shear rate results derived from $V_{z}$ using the empirical scaling ${\it\gamma}_{rz}\approx V_{z}\sqrt{R_{0}h}/3h^{2}$ and that provided by Ardekani et al. (Reference Ardekani, Joseph, Dunn-Rankin and Rangel2009) have also been included in (a). The approach and rebound stages have been assigned the (a,c,e) and (b,d,f), respectively.

Since the apparent viscosity varies substantially, a modified Stokes number,  $St_{app}=2{\it\rho}_{s}R_{0}V_{z}/9{\it\mu}_{app}$ , based on ${\it\mu}_{app}$ is more pertinent to the flow configuration, as also suggested by Uddin et al. (Reference Uddin, Marston and Thoroddsen2012). From the plots of modified Stokes number versus normalized gap height produced accordingly in figure 21(e,f), we note $St_{app}$ (i.e. the instantaneous ratio of sphere inertia to viscous forces) to increase during approach, despite the decline in sphere velocity and reach a maximum of $St_{app}\sim 2.4$ . The effect of decreasing $V_{z}$ , however, becomes prominent for $h/{\it\delta}\lesssim 0.23$ and 0.38 at $r=2$ and 3 mm, leading to the onset of $St_{app}$ decline. A maximum of only $St_{app}\sim 0.7$ is noted in the rebound stage, which reduces to become almost negligible as the sphere slows down, while ${\it\mu}_{app}$ rises with further increase in gap distance.

In addition to the PIV analysis of the non-contact case in figure 20, figure 22 shows velocity vectors,  $V_{r}$ colour maps and plots against length $Z$ (mm) at $r=2$  mm for the contact case of a sphere impacting a ${\it\delta}=4$  mm thick layer of ${\it\nu}_{0}=20\,000\,000$ cSt silicone oil from $h_{r}=140$ cm. The glass beads here do not have any discernible effect on the inception or structure of the cavity in comparison to observations made in non-seeded films. As expected, the radial velocity profiles are noted to be approximately parabolic, which invert due to flow reversal in the rebound stages (see arrows). The cavity bubble formed by depressurization following the onset of cavitation shrinks radially inwards and experiences necking ( $t=333$ , 515 and $666~{\rm\mu}\text{s}$ , see explanation in § 3.1), resulting in the influx of nearby fluid. Radial expansion begins to occur when the effect of depressurization becomes dominant, which is observed from the velocity vectors and the corresponding $V_{r}$ colour map ( $t=969~{\rm\mu}\text{s}$ ) to initiate in the cavity section beneath the necking region.

Similar in format to figure 21, figure 23 plots (a,b) ${\it\gamma}_{rz}$ versus $h$ , (c,d) ${\it\mu}_{app}$ versus $h$ and (e,f) $St_{app}$ versus $h/{\it\delta}$ at $r=1$ and 2 mm, from the PIV analysis of the realization in figure 21. Shear values derived from $V_{z}$ in (a), based on the approximation from Ardekani et al. (Reference Ardekani, Joseph, Dunn-Rankin and Rangel2009), are again only in qualitative agreement with those obtained by particle tracking measurements while being approximately $O(0.5{-}1)$ higher in magnitude. Results from the empirical scaling (obtained in figure 21 a,b) are however in accordance with the PIV measurements, indicating $V_{r}\approx V_{z}\sqrt{R_{0}h}/3h$ to be the pertinent scaling for the average radial velocity in this study. The PIV measurements show shear values to increase incessantly in the approach stage to reach maximums of ${\it\gamma}_{rz}\sim 65\,000~\text{s}^{-1}$ and $30\,000~\text{s}^{-1}$ at $h\sim 0.05$  mm for $r=1$ and 2 mm, corresponding to apparent viscosities of ${\it\mu}_{app}\sim 8$ Pa s and 15 Pa s, respectively. Declining shear rates are noted throughout the rebound stage which translate to apparent viscosities increasing at a higher rate for the smaller radial distance of $r=1$  mm, as also noticed in figure 21(c,d). Maximum values here are found to be ${\it\mu}_{app}\sim 550$  Pa s and 250 Pa s at $h=2.16$  mm, for $r=1$ and 2 mm.

In contrast to figure 21(e,f), the modified Stokes number for the contact case in figure 23(e,f) increases uninterruptedly during sphere approach at $r=1$  mm, to reach $St_{app}\sim 5.8$ at $h/{\it\delta}\sim 0.012$ . Values at $r=2$  mm meanwhile plateau at $St_{app}\sim 3.3$ , signifying a higher dominance of viscous effects further away for $h/{\it\delta}\lesssim 0.1$ . A similar occurrence is observed in the rebound stage for $h/{\it\delta}\lesssim 0.05$ , where maxima of $St_{app}\sim 2.5$ and 1.7 at $r=1$ and 2 mm decline with increase in normalized gap height. Lower values of $St_{app}$ for the rebound compared to approach stages in this and figure 21(e,f) are expected due to energy losses by viscous dissipation in the fluid film.