2 Experimental set-up

The experiments have been made with a high-frequency interferometric device, sketched in figure 1(a), originally designed by Lecoq et al. (Reference Lecoq, Feuillebois, Anthore, Anthore, Bostel and Petipas1993). A solid sphere of density $\unicode[STIX]{x1D70C}_{s}$ and radius $R$ settles under gravity in a silicon oil 47V1000 (from Bluestar silicons) of density $\unicode[STIX]{x1D70C}=978~\text{kg}~\text{m}^{-3}$ and kinematic viscosity $\unicode[STIX]{x1D708}=10^{-3}~\text{m}^{2}~\text{s}^{-1}$ , at $25\,^{\circ }\text{C}$ , towards the horizontal bottom wall of a cylindrical container with a 50 mm diameter and a 40 mm height. This bottom wall can be either a smooth glass wall or a textured surface made of a thiolen resin (Norland Optical Adhesive NOA 81 of elastic modulus $E_{w}\simeq 1.4~\text{GPa}$ ) obtained by photolithography and comprised of a forest of pillars with a height $e$ and a square base of side $2b$ (figure 1 b). These pillars are organised in a periodic square pattern of wavelength $L$ , thus with the surface fraction $\unicode[STIX]{x1D719}=(2b/L)^{2}$ . Different textures are prepared by varying the geometrical parameters within the ranges $20\leqslant e\leqslant 130~\unicode[STIX]{x03BC}\text{m}$ , $50\leqslant 2b\leqslant 100~\unicode[STIX]{x03BC}\text{m}$ and $140\leqslant L\leqslant 240~\unicode[STIX]{x03BC}\text{m}$ , leading to an aspect ratio and surface fraction of the pillars in the range $0.1<e/2b<2.5$ and $0.05\leqslant \unicode[STIX]{x1D719}\leqslant 0.3$ , respectively. The sphere radius has been varied in the range $4\leqslant R<8~\text{mm}$ so that the relative texture roughness lies in the range $10^{-3}\lesssim e/R\lesssim 10^{-2}$ . The sphere material is either steel ( $\unicode[STIX]{x1D70C}_{s}=7.8\times 10^{3}~\text{kg}~\text{m}^{-3}$ , $E_{s}=240~\text{GPa}$ ) or carbide tungsten ( $\unicode[STIX]{x1D70C}_{s}=15.6\times 10^{3}~\text{kg}~\text{m}^{-3}$ , $E_{s}=550~\text{GPa}$ ) of roughness $\unicode[STIX]{x1D716}\simeq 0.1~\unicode[STIX]{x03BC}\text{m}$ and $\unicode[STIX]{x1D716}\simeq 0.03~\unicode[STIX]{x03BC}\text{m}$ , respectively, which values are negligible when compared to the pillar height ( $\unicode[STIX]{x1D716}/e<10^{-3}$ ) and also smaller than the light wavelength, leading to a mirror polish surface for the sphere.

Figure 1. (a) Sketch of the experimental set-up, (b) geometrical parameters involved in the sphere/textured wall collision, (c) example of interferometric signal in a bouncing case.

The cylindrical container is filled with oil (no free surface) and inserted in an interferometric device, where the immersed sphere is used as a reflector (figure 1 a). Before each experiment the sphere is held with a magnet at the centre of the top plane window (made of glass of optical quality) of the fixed container. Removing this magnet initiates the sphere settling and enables the pathway for the upper laser beam which reflects on the top of the settling sphere. This laser beam interferes with the other beam reflecting onto a plane mirror that is fixed under the bottom wall of the container. As the sphere moves, the interference fringes move accordingly. The light intensity of the fringes is then converted into an electric signal by a photodiode, and recorded with an oscilloscope at a sampling frequency of 25 MHz (figure 1 c). Detection of the successive maxima of this electric signal gives access to the sphere displacement $\text{d}h$ with a resolution of $\unicode[STIX]{x1D706}/2n\simeq 0.23~\unicode[STIX]{x03BC}\text{m}$ , where $\unicode[STIX]{x1D706}=0.633~\unicode[STIX]{x03BC}\text{m}$ is the wavelength of the He–Ne laser beam, and $n=1.4$ is the optical index of the silicon oil. In practice, the maximum signal frequency that can be detected by the set-up is 0.5 MHz, yielding a time resolution of 0.002 ms. The instantaneous sphere velocity $V(t)=-\text{d}h/\text{d}t$ can thus be measured in the range $10^{-4}\lesssim V\lesssim 10^{-1}~\text{m}~\text{s}^{-1}$ , with a relative error smaller than 1 % in the approach phase and that does not exceed 5 % in the collision and contact phase. The instantaneous position, $h(t)$ , is obtained by integration, with the reference $h=0$ determined a posteriori as the final sphere rest position on the wall when $V=0$ . It should be noted that when the sphere is resting on the surface, compressive stresses due to the apparent weight of the sphere deform the surface. For a glass wall, this deformation can be neglected. However, for a compliant material such as the NOA resin textures, we anticipate that the penetration of the sphere into the surface is non-negligible compared to the scale of observation. The static penetration depth of the sphere in the pillars is denoted $\unicode[STIX]{x1D6FF}_{s}$ (see figure 1 b), and its value will be discussed in § 4.1. As the sphere slows down on approach to the wall, the recorded light modulation slows down accordingly until its complete arrest in the case of a ‘sticking’ collision. In the bouncing case, the time modulation of the signal first displays a slowing down before an acceleration, which is the signature of the change of direction of the sphere (figure 1 c).

In the present experiments, we investigate the elastohydrodynamic regime close to the bouncing transition in the range $2.5\leqslant St\leqslant 5.8$ (see table 1), where the Stokes number is defined as $St=(2/9)\unicode[STIX]{x1D70C}_{s}RV_{T}/\unicode[STIX]{x1D702}$ , with $\unicode[STIX]{x1D702}=\unicode[STIX]{x1D70C}\unicode[STIX]{x1D708}$ the dynamic viscosity of the oil. Note that the present definition of $St$ is based on $V_{T}$ , the terminal velocity of the sphere in the corresponding unbounded fluid, in contrast to other studies that used the velocity of the sphere just prior to impact. Here, $V_{T}$ can be inferred from the viscous ‘Stokes’ drag force corrected from the small inertial effect known as Oseen’s correction (see Mongruel et al. Reference Mongruel, Lamriben, Yahiaoui and Feuillebois2010). The corresponding Reynolds number $Re=(2R)V_{T}/\unicode[STIX]{x1D708}=9(\unicode[STIX]{x1D70C}/\unicode[STIX]{x1D70C}_{s})St$ is indeed in the range $2.8\leqslant Re\leqslant 6.6$ for the two density ratios used here ( $\unicode[STIX]{x1D70C}_{s}/\unicode[STIX]{x1D70C}\simeq 8$ and 16).

Table 1. Parameter values for the different used spheres, of different materials and diameter $2R$ , settling in 47V1000 silicon oil. The Reynolds and Stokes numbers are based on the calculated terminal velocity $V_{T}$ of the sphere in the corresponding unbounded fluid.

3 Critical Stokes number for bouncing

Figure 2 shows the evolution of the sphere velocity, $V$ , as a function of the sphere position, $h$ , for a given sphere settling towards three different wall textures at the same Stokes number $St=4.7$ . As stated before, the Stokes number is defined using the terminal velocity of the sphere, and hence is fixed by the physical properties of the sphere ( $\unicode[STIX]{x1D70C}_{s},R$ ) and of the oil ( $\unicode[STIX]{x1D70C},\unicode[STIX]{x1D708}$ ). Thus, at a given Stokes number, the dynamics of the sphere far from the wall, i.e. at a distance from the wall larger than one sphere radius, is the same for the three experiments. By contrast, the near-wall dynamics of the sphere is dramatically influenced by the type of texture. In the case of a smooth wall, ( $e=0$ , $\unicode[STIX]{x1D719}=0$ ), the sphere velocity drops towards zero for vanishing distance, with a linear terminal regime that results from the lubrication flow in between the smaller and smaller gap, eventually dissipating all of the kinetic energy of the sphere (see Mongruel et al. Reference Mongruel, Lamriben, Yahiaoui and Feuillebois2010). This observation is consistent with the Stokes number here being subcritical ( $St\lesssim 10$ ). In the case of a textured wall ( $e\neq 0$ , $\unicode[STIX]{x1D719}\neq 0$ ), the velocity of the sphere is significantly larger, for a given distance $h$ , than the corresponding velocity for a smooth wall. This is a consequence of the fluid flow in between the pillars, which decreases the pressure in the gap and thus the resisting lubrication force on the sphere. The velocity in figure 2 appears to be shifted along the horizontal axis, with a shift length that is proportional to the pillar height, $e$ , but also depends on $\unicode[STIX]{x1D719}$ and $b/L$ (Chastel Reference Chastel2015). When the height of the textures is small enough (e.g. $e=21~\unicode[STIX]{x03BC}\text{m}$ for $\unicode[STIX]{x1D719}=0.05$ in figure 2), and despite this near-wall enhancement of velocity, the sphere velocity still drops towards zero at vanishing distance $h$ , but with a new ultimate regime which corresponds now to the lubrication flow in between the sphere and the top surface of the closest pillar (see Chastel & Mongruel Reference Chastel and Mongruel2016). Strikingly, when the height of the texture is large enough (e.g. $e=89~\unicode[STIX]{x03BC}\text{m}$ for $\unicode[STIX]{x1D719}=0.05$ ), the local velocity enhancement of the sphere is such that the contact with the top of the pillars arises with a non-zero impact velocity $V_{i}\simeq 5~\text{cm}~\text{s}^{-1}$ . Note that this impact velocity is significantly smaller than the terminal velocity $V_{T}\simeq 38~\text{cm}~\text{s}^{-1}$ due to the lubrication forces acting during the wall approach. In that case of non-zero impact velocity, the sphere rebounds, whereas bouncing was precluded in the two previous cases. Indeed, negative $h$ values observed in figure 2 correspond to the dynamical extra compression of the pillars during the collision.

Figure 2. Velocity $V$ as a function of position $h$ (with $h=0$ for the sphere at rest on the wall) for a steel sphere ( $2R=14~\text{mm}$ ) settling in 47V1000 silicon oil ( $St=4.7$ , $Re=5.3$ , $V_{T}=380~\text{mm}~\text{s}^{-1}$ ) towards an horizontal wall: smooth wall (▾ (magenta)), textured wall with surface fraction $\unicode[STIX]{x1D719}=0.05$ and height of pillars $e=21~\unicode[STIX]{x03BC}\text{m}$ (▪ (cyan)) or $e=89~\unicode[STIX]{x03BC}\text{m}$ (● (yellow)).

As the elastic modulus of the sphere is much larger than the elastic modulus of the textures, the elastic deformation of the sphere is negligible when compared to the elastic deformation of the textures. The non-zero kinetic energy of the sphere when touching the textured wall is stored by the elastic deformation of the textures. The dynamic deformation is maximum (here $h\simeq -0.01~\text{mm}$ ) when the sphere velocity vanishes. The stored elastic energy turns back into kinetic energy as the sphere velocity becomes increasingly negative and the sphere takes off the wall with a non-zero rebound velocity: $V_{r}\simeq -2~\text{cm}~\text{s}^{-1}$ at $h\simeq 0$ . The impact and rebound velocity measurements lead to the restitution coefficient $|V_{r}/V_{i}|\simeq 0.4$ , which is significantly smaller than one. The height of the micro-rebound of the sphere in the fluid is $h\simeq 8~\unicode[STIX]{x03BC}\text{m}$ . The sphere then returns to the wall with a vanishing velocity at contact so that no subsequent rebound is possible: the sphere comes to rest at the ‘attractor’ point $(h=0,V=0)$ . Note that the $V(h)$ curve in the bouncing case has the same spiral shape as predicted from the calculations of Lian et al. (Reference Lian, Adams and Thornton1996).

Figure 2 shows that bouncing is observed when the impact velocity is non-zero. Hence, the bouncing of a given sphere in a given fluid (i.e. at a given Stokes number) can be triggered by the wall texture as a result of the local velocity enhancement, e.g. here, beyond a critical value of the pillar height $e$ for a given pillar fraction $\unicode[STIX]{x1D719}$ . Moreover, for a given $e$ , bouncing occurs at Stokes number values decreasing with decreasing $\unicode[STIX]{x1D719}$ , as a result of the local velocity enhancement being more pronounced for smaller $\unicode[STIX]{x1D719}$ values. The different cases are summarized in figure 3 where the bouncing and non-bouncing (‘sticking’) domains are reported in the ( $e/2b,St$ ) plane for three $\unicode[STIX]{x1D719}$ values ranging from 0.3 down to 0.05. The critical Stokes number for bouncing, $St_{c}$ , is thus situated at the boundary between those two domains and appears as a dashed line in figure 3. The results show that $St_{c}$ depends both on $e/2b$ and $\unicode[STIX]{x1D719}$ with measured values ranging typically from 3 to 5 for the texture geometries investigated here. These values are significantly smaller than the value $St_{c}\simeq 10$ reported previously in the case of smooth surfaces (Joseph et al. Reference Joseph, Zenit, Hunt and Rosenwinkel2001; Gondret et al. Reference Gondret, Lance and Petit2002). This shows quantitatively that the wall texture influences the rebound of particles on a wall in a fluid. This influence is a direct consequence of the modification brought by the wall texture to the dynamics of approach of the sphere just prior to touching the wall.

Figure 3. Bouncing (▴ (blue), ▵ (blue)) or sticking (● (red), ○ (red)) collision in the ( $e/2b,St$ ) plane for the settling of a steel (filled symbols) or tungsten carbide (open symbols) sphere of different diameters and thus different Stokes numbers $St$ in 47V1000 silicon oil towards a wall with micro-pillars of different aspect ratio $e/2b$ and different surface fraction (a) $\unicode[STIX]{x1D719}=0.30$ , (b) $\unicode[STIX]{x1D719}=0.15$ , (c) $\unicode[STIX]{x1D719}=0.05$ . The dashed lines are only a guide for the eye.

Let us discuss two phenomena that may arise in the present sphere/wall collisions: non-Newtonian behaviour of the oil and cavitation. Indeed, the high viscosity oil used here is known to be shear thinning for high shear rates $\dot{\unicode[STIX]{x1D6FE}}$ (Marston, Yong & Thoroddsen Reference Marston, Yong and Thoroddsen2010). It has a Newtonian behaviour with a constant viscosity for low enough shear rates ( $\dot{\unicode[STIX]{x1D6FE}}\lesssim 2500~\text{s}^{-1}$ ) but exhibits a viscosity decrease beyond and, e.g. for $\dot{\unicode[STIX]{x1D6FE}}\sim 10^{4}~\text{s}^{-1}$ , the viscosity value is approximately 85 % of the low shear viscosity. The shear rate $\dot{\unicode[STIX]{x1D6FE}}$ in the lubricated squeezed flow of thickness $h$ between the sphere and the wall can be estimated to be maximum at a distance $r_{0}=(2Rh)^{1/2}$ from the axis of the sphere, where the tangential flow velocity is equal, from mass conservation, to $Vr_{0}/2h$ . The highest value of the shear rate, $\dot{\unicode[STIX]{x1D6FE}}\sim Vr_{0}/2h^{2}$ , can be estimated for the minimum gap thickness $h=e$ in the case of a textured wall, and $h=\unicode[STIX]{x1D716}$ in the case of a ‘smooth’ wall. For typical values of sphere impact (e.g. $2R=14~\text{mm}$ ), $\dot{\unicode[STIX]{x1D6FE}}\sim 2400~\text{s}^{-1}$ for a typical textured wall ( $e=100~\unicode[STIX]{x03BC}\text{m}$ , $V=40~\text{mm}~\text{s}^{-1}$ ) and $\dot{\unicode[STIX]{x1D6FE}}\sim 7000~\text{s}^{-1}$ for a typical ‘smooth’ plane ( $\unicode[STIX]{x1D716}=1~\unicode[STIX]{x03BC}\text{m}$ , $V=0.2~\text{mm}~\text{s}^{-1}$ ). We thus conclude that the variation of the oil viscosity during impact is negligible in our case. Concerning cavitation, this phenomenon was observed during the rebound of a sphere from a wetted surface by Marston et al. (Reference Marston, Yong, Ng, Tan and Thoroddsen2011). According to the criterion of Joseph (Reference Joseph1998), cavitation is only expected when the tensile stress that can be estimated to be $2\unicode[STIX]{x1D702}\dot{\unicode[STIX]{x1D6FE}}$ exceeds an atmospheric pressure of approximately $10^{5}~\text{Pa}$ , which is not the case in our impact experiments, given the above values of $\dot{\unicode[STIX]{x1D6FE}}$ . Thus, the present experiments are not affected by non-Newtonian behaviour or cavitation.

Figure 4. (a) Time evolution of the velocity $V(t)$ of a steel sphere ( $2R=14~\text{mm}$ ) settling in 47V1000 silicon oil ( $St=4.7$ , $Re=5.3$ , $V_{T}=380~\text{mm}~\text{s}^{-1}$ ) and colliding with a textured wall ( $\unicode[STIX]{x1D719}=0.05$ , $e=89~\unicode[STIX]{x03BC}\text{m}$ ). (b) Corresponding time evolution of the sphere position $h(t)$ with $h=0$ for the sphere at final rest and top of the textures at $h=\unicode[STIX]{x1D6FF}_{s}\simeq 2.5~\unicode[STIX]{x03BC}\text{m}$ . $V_{i}\simeq 48~\text{mm}~\text{s}^{-1}$ , $V_{r}\simeq -13~\text{mm}~\text{s}^{-1}$ , $\unicode[STIX]{x1D6FF}_{max}\simeq 10~\unicode[STIX]{x03BC}\text{m}$ , $\unicode[STIX]{x1D70F}_{c}\simeq 1~\text{ms}$ .

In the next section, we focus on the influence of the texture on the collision dynamics.

4 Collision dynamics

In the bouncing case of figure 2, the time evolution of the sphere velocity, $V(t)$ , and of the sphere position, $h(t)$ , are displayed in figure 4. As already mentioned in § 2, the sphere position is defined with a reference origin $h=0$ taken from the final position of the sphere at rest. The top position of the textures thus corresponds to a small positive value $h=\unicode[STIX]{x1D6FF}_{s}$ calculated from the model presented below. The maximal deformation of the texture $\unicode[STIX]{x1D6FF}_{max}$ during the sphere/wall collision is measured in figure 4(b) from the minimum of the sphere position $h=\unicode[STIX]{x1D6FF}_{s}-\unicode[STIX]{x1D6FF}_{max}$ when the sphere velocity vanishes (figure 4 a). The corresponding time appears as the vertical middle dashed line in figure 4. The time $t_{i}$ (respectively $t_{r}$ ) at which the sphere touches (respectively takes off) the top of the texture at $h=\unicode[STIX]{x1D6FF}_{s}$ can be determined from figure 4(b) as the intersection of the $h(t)$ experimental curve with the estimated $h=\unicode[STIX]{x1D6FF}_{s}$ line. The collision duration $\unicode[STIX]{x1D70F}_{c}$ is measured as $\unicode[STIX]{x1D70F}_{c}=t_{r}-t_{i}$ and the impact and rebound velocity are measured from figure 4(a) as $V_{i}=V(t_{i})$ and $V_{r}=V(t_{r})$ , respectively. In the typical bouncing of figure 4, the maximal sphere penetration is $\unicode[STIX]{x1D6FF}_{max}\simeq 10~\unicode[STIX]{x03BC}\text{m}$ , the collision duration is $\unicode[STIX]{x1D70F}_{c}\simeq 1~\text{ms}$ , the impact and rebound velocities are $V_{i}\simeq 48~\text{mm}~\text{s}^{-1}$ and $V_{r}\simeq -13~\text{mm}~\text{s}^{-1}$ , leading to a restitution coefficient value of approximately 0.27.

Figure 5. Sketch of the sphere penetrating into the pillars and definition of the geometrical parameters used in the modelling.

These plots enable the quantitative measurements of both the maximal penetration depth, $\unicode[STIX]{x1D6FF}_{max}$ , and the collision duration, $\unicode[STIX]{x1D70F}_{c}$ . The collision duration of a sphere with a wall has been already measured with a force transducer by Falcon et al. (Reference Falcon, Laroche, Fauve and Coste1998) in the dry case and by Legendre et al. (Reference Legendre, Zenit, Daniel and Guiraud2006) in the fully immersed case, but this technique did not give access to the penetration depth. Falcon et al. (Reference Falcon, Laroche, Fauve and Coste1998) also discussed in detail the influence of gravity during the collision and presented a gravity-modified Hertz interaction law. Before presenting our measured values of $\unicode[STIX]{x1D6FF}_{max}$ and $\unicode[STIX]{x1D70F}_{c}$ with varying $\unicode[STIX]{x1D70C}_{s}$ , $R$ , $V$ , $e$ and $\unicode[STIX]{x1D719}$ , we first describe our modelling of the static and dynamic penetration of a sphere into a textured wall, taking into account the gravity effect.

4.1 Modelling the collision of the sphere with a textured wall

Given the materials used here for the sphere and the textured wall ( $E_{w}\ll E_{s}$ ), we will consider in the modelling that only the wall is deformable whereas the sphere can be considered as perfectly rigid. The model developed in the following is very close to the original one developed by Hertz for the force contact law and collision dynamics of a sphere of radius $R$ and mass $m=(4/3)\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}_{s}R^{3}$ with a smooth wall (Johnson Reference Johnson1985). Let $\unicode[STIX]{x1D6FF}_{0}(t)$ be the penetration of the bottom apex of the sphere in the pillars (figure 5). For a given penetration $\unicode[STIX]{x1D6FF}_{0}$ , the sphere/wall contact exists within a disk area of radius $a\simeq (2R\unicode[STIX]{x1D6FF}_{0})^{1/2}$ and the local sphere penetration is $\unicode[STIX]{x1D6FF}(r)=\unicode[STIX]{x1D6FF}_{0}[1-(r/a)^{2}]$ at the radial distance $r$ from the bottom apex of the sphere (figure 5). By assuming that the deformation of the pillars remains in the elastic regime, the pillar deformation $\unicode[STIX]{x1D6FF}(r)/e$ generates the normal stress $\unicode[STIX]{x1D70E}(r)$ such that $\unicode[STIX]{x1D70E}(r)=E_{w}\unicode[STIX]{x1D6FF}(r)/e$ . By taking into account the surface fraction of pillars, $\unicode[STIX]{x1D719}$ , the total force exerted by the pillars onto the sphere is thus given by $F=\int _{0}^{a}\unicode[STIX]{x1D719}\unicode[STIX]{x1D70E}(r)2\unicode[STIX]{x03C0}r\,\text{d}r$ , which leads to the normal force

(4.1) $$\begin{eqnarray}F=\unicode[STIX]{x03C0}\unicode[STIX]{x1D719}E_{w}\frac{R}{e}\unicode[STIX]{x1D6FF}_{0}^{2}.\end{eqnarray}$$

Note that this force contact law is nonlinear with the sphere penetration $\unicode[STIX]{x1D6FF}_{0}$ due to the increasing contact area, but it differs from the nonlinear Hertz scaling $F\sim \unicode[STIX]{x1D6FF}_{0}^{3/2}$ . This force contact law can be first used in the case of a static contact to calculate the penetration $\unicode[STIX]{x1D6FF}_{0}=\unicode[STIX]{x1D6FF}_{s}$ for a sphere at rest. In that case, the normal force $F$ exerted by the compressed pillars on the sphere balances the apparent gravity forces of the sphere $(4/3)\unicode[STIX]{x03C0}R^{3}(\unicode[STIX]{x1D70C}_{s}-\unicode[STIX]{x1D70C})g$ , which leads to

(4.2) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{s}=R\left[\frac{4e(\unicode[STIX]{x1D70C}_{s}-\unicode[STIX]{x1D70C})g}{3\unicode[STIX]{x1D719}E_{w}}\right]^{1/2}.\end{eqnarray}$$

Depending on the different spheres and the textures used in the present experiments, the static sphere penetration varies in the range $0.8\leqslant \unicode[STIX]{x1D6FF}_{s}\leqslant 3~\unicode[STIX]{x03BC}\text{m}$ .

Next, we consider a settling sphere impacting a horizontal textured wall at a velocity $V_{i}$ , as measured in the experiments (figure 4). Before contacting the wall, the deceleration of the sphere is solely due to its hydrodynamic interaction with the wall, i.e. by the resistance of the lubricated interstitial fluid to be squeezed out of the gap between the sphere and the wall. During the contact, a fluid flow between the pillars still occurs, but for simplicity we assume here that the main contribution to the further deceleration of the sphere comes from the resistance of the pillars to elastic compression. We also neglect any other source of dissipation during the collision, such as vibrations. Under these assumptions, the conservation equation for the sphere energy can be written as

(4.3) $$\begin{eqnarray}\frac{1}{2}mV(t)^{2}+\frac{\unicode[STIX]{x03C0}}{3}\unicode[STIX]{x1D719}E_{w}\frac{R}{e}\unicode[STIX]{x1D6FF}_{0}(t)^{3}-\frac{4}{3}\unicode[STIX]{x03C0}R^{3}(\unicode[STIX]{x1D70C}_{s}-\unicode[STIX]{x1D70C})g\unicode[STIX]{x1D6FF}_{0}(t)=\frac{1}{2}mV_{i}^{2}.\end{eqnarray}$$

In (4.3), the first term is the instantaneous kinetic energy of the sphere with velocity $V(t)=\text{d}\unicode[STIX]{x1D6FF}/\text{d}t$ , the second term is the instantaneous elastic energy corresponding to $\int _{0}^{\unicode[STIX]{x1D6FF}(t)}F\,\text{d}\unicode[STIX]{x1D6FF}$ and the third term is the instantaneous reduced gravitational energy. If we neglect any source of dissipation during collision, the sum of these three terms must be equal to the initial kinetic energy $(1/2)mV_{i}^{2}$ of the sphere when just touching the textures. The maximal penetration depth of the sphere is reached at the time at which the sphere velocity $V(t)$ is zero. At that time, the elastic energy is maximal and corresponds to the initial kinetic energy plus the additional gravitational energy.

Let us first consider the case where the gravitational energy is zero or negligible compared to the initial kinetic energy. Hence, the corresponding value of the maximal penetration, $\unicode[STIX]{x1D6FF}_{m0}$ , reads as

(4.4) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{m_{0}}=\left(\frac{2\unicode[STIX]{x1D70C}_{s}eR^{2}V_{i}^{2}}{\unicode[STIX]{x1D719}E_{w}}\right)^{1/3}.\end{eqnarray}$$

The corresponding time of collision, $\unicode[STIX]{x1D70F}_{c_{0}}$ , can be calculated from (4.3) without the gravity term, as $\unicode[STIX]{x1D70F}_{c_{0}}=2\unicode[STIX]{x1D70F}\int _{0}^{1}(1-\tilde{t}^{3})^{-1/2}\,\text{d}\tilde{t}$ where $\tilde{t}=t/\unicode[STIX]{x1D70F}$ and $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D6FF}_{m_{0}}/V_{i}$ is the leading-order time scale. The integral term, denoted $I_{0}$ , can be expressed with $\unicode[STIX]{x1D6E4}$ functions as $I_{0}=\unicode[STIX]{x03C0}^{1/2}\unicode[STIX]{x1D6E4}(4/3)/\unicode[STIX]{x1D6E4}(5/6)\simeq 1.4$ so that the time of collision can be written as

(4.5) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{c_{0}}\simeq 2I_{0}\left(\frac{2\unicode[STIX]{x1D70C}_{s}eR^{2}}{\unicode[STIX]{x1D719}E_{w}V_{i}}\right)^{1/3}.\end{eqnarray}$$

When the static penetration $\unicode[STIX]{x1D6FF}_{s}$ is of the same order of magnitude as $\unicode[STIX]{x1D6FF}_{m_{0}}$ , the apparent weight of the sphere cannot be neglected during the collision. In that case, the maximal penetration depth of the sphere $\unicode[STIX]{x1D6FF}_{max}$ is given from the complete equation (4.3) by

(4.6) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{m_{0}}=\unicode[STIX]{x1D6FF}_{max}\left[1-3\left(\frac{\unicode[STIX]{x1D6FF}_{s}}{\unicode[STIX]{x1D6FF}_{max}}\right)^{2}\right]^{1/3}.\end{eqnarray}$$

For $\unicode[STIX]{x1D6FF}_{s}=0$ (no gravity), one recovers $\unicode[STIX]{x1D6FF}_{max}=\unicode[STIX]{x1D6FF}_{m_{0}}$ , whereas $\unicode[STIX]{x1D6FF}_{max}>\unicode[STIX]{x1D6FF}_{m_{0}}$ for $\unicode[STIX]{x1D6FF}_{s}\neq 0$ .

In the gravity case, the time of collision, $\unicode[STIX]{x1D70F}_{c}$ , can be calculated from the complete equation (4.3) as $\unicode[STIX]{x1D70F}_{c}=2\unicode[STIX]{x1D70F}\int _{0}^{1}[1-\tilde{t}^{3}+3(\unicode[STIX]{x1D6FF}_{s}/\unicode[STIX]{x1D6FF}_{m_{0}})^{2}\tilde{t}]^{-1/2}\text{d}\tilde{t}$ . The integral term, denoted $I_{g}$ , can be calculated numerically for each value of the ratio $\unicode[STIX]{x1D6FF}_{s}/\unicode[STIX]{x1D6FF}_{m_{0}}$ . Finally, the time of collision reads as

(4.7) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{c}\simeq \unicode[STIX]{x1D70F}_{c_{0}}\frac{I_{g}}{I_{0}}.\end{eqnarray}$$

Note that the above analysis is restricted to low enough impact velocities, i.e. $V_{i}\ll c^{\ast }$ where $c^{\ast }=(E_{w}/\unicode[STIX]{x1D70C}_{s})^{1/2}$ is a mixed characteristic velocity which does not correspond to the sound speed either in the sphere or in the pillars. As $(E_{w}/\unicode[STIX]{x1D70C}_{s})^{1/2}>400~\text{m}~\text{s}^{-1}$ for the present experiments, this condition is largely fulfilled.

Figure 6. Sphere velocity $V$ as a function of the sphere position $h$ (with $h=0$ for the sphere at final rest) for settling in 47V1000 silicon oil towards textured walls with different surface fractions $\unicode[STIX]{x1D719}$ and heights $e$ of pillars. (a) Steel sphere ( $2R=14~\text{mm}$ and $St=4.7$ ) with $\unicode[STIX]{x1D719}=0.05$ and $e=57~\unicode[STIX]{x03BC}\text{m}$ (♦ (magenta)) or $e=89~\unicode[STIX]{x03BC}\text{m}$ (● (yellow)), (b) tungsten carbide sphere ( $2R=8~\text{mm}$ and $St=4.9$ ), with $\unicode[STIX]{x1D719}=0.15$ , and $e=60~\unicode[STIX]{x03BC}\text{m}$ (▴ (red)) or $e=117~\unicode[STIX]{x03BC}\text{m}$ (▸ (blue)).

Note also that the predicted scalings given for the maximal penetration depth and time of collision without gravity effect by (4.4) and (4.5) are significantly different from the classical Hertz scalings $\unicode[STIX]{x1D6FF}_{m_{0}}\sim (\unicode[STIX]{x1D70C}_{s}/E_{w})^{5/2}RV^{4/5}$ and $\unicode[STIX]{x1D70F}_{c_{0}}\sim (\unicode[STIX]{x1D70C}_{s}/E_{w})^{5/2}RV^{-1/5}$ . The present model predicts in particular the scaling with the texture characteristics, i.e. the texture height $e$ and the surface density $\unicode[STIX]{x1D719}$ , which are not embedded in the theory of Hertz. In addition, the influence of gravity on this scaling is described by (4.6) and (4.7).

Figure 7. (a) Dimensionless maximal penetration depth $\unicode[STIX]{x1D6FF}_{m_{0}}(\unicode[STIX]{x1D719}/eR^{2})^{1/3}$ as a function of the dimensionless impact sphere velocity $V_{i}/c^{\ast }$ . Experimental values from $\unicode[STIX]{x1D6FF}_{max}$ measurements and (4.5) for steel and tungsten carbide spheres onto different textures (data symbols) together with model (4.3) (——). (b) Corresponding dimensionless collision duration $\unicode[STIX]{x1D70F}_{c_{0}}c^{\ast }(\unicode[STIX]{x1D719}/eR^{2})^{1/3}$ . Experimental values from $\unicode[STIX]{x1D70F}_{c}$ measurements and (4.6) (same data symbols) together with model equation (4.4) (——).