3.1. Typical features of a microsphere's oblique impact

Two different impact modes, namely the submergence and oscillation, were observed after impact, and the time series of the two typical impact processes representing these two impact modes is shown in figure 3. It is obvious that the sphere motion and liquid surface distortion after an oblique impact are significantly different from those after a vertical impact. The sphere experiences the sinking stage first, where the sphere moves downward and the TPCL slides upward along the sphere surface, and the contact angle $\theta$ is basically maintained at $\theta _a$ with a little variation. The sinking stage of a submerging sphere can be divided into three stages, including slamming, cavity developing and cavity collapse (Ji etal. Reference Ji, Song and Yao2017), and the cavity shape in our study is similar to the quasi-static cavity shape after the steel sphere's impact reported by Aristoff & Bush (Reference Aristoff and Bush2009) and Speirs et al. (Reference Speirs, Mansoor, Belden and Truscott2019). Because the time duration of the slamming stage is extremely short, the distortion of the liquid surface is small and cannot be observed clearly. During the cavity developing stage, the TPCL on the left-hand side of the sphere first moves below the undisturbed liquid surface compared with the right-hand side, which produces a half-cavity above the sphere's left-hand side (0.10–0.17 ms in figure 3$a$). Subsequently, the TPCL on the right-hand side moves below the undisturbed liquid surface and a non-axisymmetric full-cavity is formed (0.27–0.49 ms in figure 3$a$) with the meniscus curvature in the vertical plane on the right-hand side being larger than that on the left-hand side, while the distortion of the liquid surface after the vertical impact is axisymmetric without any half-cavities. The smaller the impact angle is, the longer the time the half-cavity persists. The cavity collapsing stage arises when the sphere reaches the position where the cavity becomes unstable (0.49 ms in figure 3$a$). Accordingly, the meniscus above the sphere pinches off quickly and the sphere submerges. A bubble is entrapped on the top of the sphere after submergence. It can be concluded from the motion of the bubble that the sphere rotates clockwise (backspin) during the oblique impact process (0.60–0.78 ms in figure 3$a$). An oscillating sphere only experiences the slamming and cavity developing stages during the sinking stage because its vertical velocity decreases to zero before it reaches the position where the cavity becomes unstable. Subsequently, the sphere begins to rise under the action of surface tension, and then enters the reverting stage while the direction of the sphere's horizontal velocity remains unchanged. Correspondingly, the TPCL is pinned on a certain position at the sphere surface with a gradually decreasing $\theta$, while the cavity retracts in the vertical direction (0.79–1.36 ms in figure 3$b$). The small $\theta _{r}$ (or large contact angle hysteresis) of the PMMA sphere results in its not being able to obtain enough kinetic energy at the reverting stage to bounce off the liquid surface (Jietal. Reference Ji, Song and Yao2017). When the sphere is near the horizontal of the undisturbed liquid surface, it moves slowly in the horizontal direction with its vertical motion stopped, and the sphere eventually floats at the liquid surface (1.36–1.82 ms in figure 3$b$). The non-axisymmetry of the cavity decreases gradually at the late stage of the impact. This occurs because of the larger Laplace pressure of the meniscus on the sphere's right-hand side attributed to the larger curvature compared with that on the left-hand side. This effect flattens the meniscus on the right-hand side at a faster rate compared with that on the left-hand side.

Figure 3. Time series of the oblique impact of PMMA microspheres on a water surface exhibiting different impact modes. ($a$) Submergence: $\alpha =43.8^{\circ }, d_p=306.0\ \mathrm {\mu }\textrm {m}, u_t=1.36\ \textrm {m s}^{-1}, u_0=2.84\ \textrm {m s}^{-1}, We=34$ and $Re=867$. ($b$) Oscillation: $\alpha =28.6^{\circ }, d_p=294.3\ \mathrm {\mu }\textrm {m}, u_t=1.30\ \textrm {m s}^{-1}, u_0=2.71\ \textrm {m s}^{-1}, We=30$ and $Re=796$.

3.2. Motion behaviours of sphere and TPCL

As mentioned above, $\alpha$ and $We$ determine the impact processes in this study, and their effects on the motion behaviours of the sphere and TPCL are discussed in this section. Figure 4($a$) shows the trajectories of spheres after impacting on a water surface at different $\alpha$ values when $We=34$, while the sphere trajectories at different $We$ values when $\alpha =29.5^{\circ }$ are shown in figure 4($b$), whereby the sphere position and time are non-dimensionlized by the characteristic length $d_p$ and time $d_p/u_0$, respectively. The corresponding sphere velocity (normalized by $u_0)$ evolutions are shown in figure 4($c$). With the increase in $\alpha$ or $We$, the impact mode changes from oscillation to submergence. During the sinking stage, the sphere typically moves along its impact direction, with the trajectories deviating downward first and then upward after the early stage of the oblique impact process. The evolutions of sphere velocity at different $We$ or $\alpha$ values show a universal behaviour at the initial stage of impact (slamming stage), and then separate from each other at the late stage of impact, which means that the decay ratios of the sphere's kinetic energy (or momentum) have little difference at the slamming stage. When $\alpha$ decreases, the sphere velocity at the end of the sinking stage $u_p$ decreases, as shown in figure 4($c$). The decrease in $\alpha$ also makes the sphere's maximum trajectory deviation (${\rm \Delta} s_m)$ during the sinking stage increase, which means that the deviation of the sphere trajectory becomes more obvious, as shown in figure 4($d$). The decrease in $We$ decreases the sphere velocity at the end of the sinking stage and the maximum deviation of the sphere trajectory (as shown in figure 4c,d). The velocity of the oscillating sphere at the reverting stage is much smaller than its impact velocity. The vertical motion of the sphere stops, and the sphere slowly slides horizontally at the gas–liquid interface when it rises to the position of 0.8$d_p$ below the undisturbed liquid surface.

Figure 4. Sphere trajectories coloured based on the dimensionless time $T\ (=tu_0/d_p)$ during oblique impact on a water surface: ($a$) $We=34$ at different $\alpha$ values and ($b$) $\alpha =29.5^{\circ }$ at different $We$ values (see supplementary movies 1–8 available at https://doi.org/10.1017/jfm.2020.511). Panel ($c$) is the corresponding sphere velocity evolutions. The grey dashed lines represent the impact directions. The red line with arrows in panel ($a$) represents the sphere's maximum trajectory deviation (${\rm \Delta} s_m)$ during the sinking stage. Here, S and O represent the modes of submergence and oscillation, respectively, while the symbol $\times$ represents the end of the sinking stage. Panel ($d$) is the corresponding sphere velocities at the end of the sinking stage ($u_p)$ and ${\rm \Delta} s_m$ as a function of $\alpha$. The spheres shown in the figure have a $u_0/u_t$ of 1.89–3.70.

Figure 3 shows that the TPCL is basically in a plane (TPCL plane). Thus, the angle between the TPCL and the axis of the TPCL plane $\phi$ (as shown in figure 2$b$) and the inclination of the TPCL plane relative to the horizontal $\alpha _T$ are defined to characterize the wetting ratio of the sphere surface and the position of the TPCL on the sphere surface, and their evolutions with the normalized sphere displacement $s/s_p$ are given in figure 5($a$,$b$), respectively. For a submerging sphere, $\phi$ increases rapidly first (slamming stage), and the increase then becomes slower (cavity developing stage), and increases rapidly again to a final value near ${\rm \pi}$ (cavity collapsing stage). The evolutions of $\phi$ with $s/s_p$ have no obvious difference for submerging spheres at different $\alpha$ and $We$ values. By comparison, $\phi$ increases more slowly for an oscillating sphere, which is kept almost constant after increasing to a value near ${\rm \pi} -\theta /2$ at the late cavity developing stage. Accordingly, $\alpha _T$ increases first and then decreases during the sphere's oblique impact process, with a maximum value $<{\rm \pi} /2-\alpha _u$, which means that the TPCL plane is not perpendicular to the motion direction and the wetting of the sphere surface is non-axisymmetric along the motion direction; specifically, the wetted part of the sphere surface below the motion axis is larger than that above. When $\alpha$ decreases, the deflection angle of the TPCL plane relative to the equatorial plane (perpendicular to the impact direction) ${\rm \pi} /2-\alpha _u-\alpha _T$ increases, which means the non-axisymmetry of sphere wetting increases.

Figure 5. Evolutions of ($a$) the angular position of the TPCL $\phi$ and ($b$) the inclination of the TPCL plane $\alpha _T$ as a function of the normalized sphere displacement $s/s_p$ during oblique impact on a water surface, coloured based on the dimensionless time $T\ (=tu_0/d_p)$, where $s_p$ represents the total sphere displacement during the sinking stage.

This non-axisymmetric sphere wetting results in the non-axisymmetric pressure (positive at the upstream and negative at the downstream) and shear stress (mainly along the sides) distributions on the sphere surface, significantly influencing the forces acting on the sphere as well as the sphere motion. This non-axisymmetry first makes the hydrodynamic force $F_h$ which dominates the early sinking stage have a downward component $F_{hv}$ perpendicular to the motion direction. Meanwhile, it also generates a torque, which makes the sphere rotate clockwise (speed $w)$, producing a downward force $F_l$ perpendicular to the motion direction caused by the Magnus effect, as sketched in figure 6($a$). Therefore, the sphere trajectory is first deflected in the downward direction after oblique impact. Then, $F_h$ decreases quickly after the early sinking stage, while the force due to surface tension $F_s$ increases and becomes the dominant force at the late sinking stage. Owing to the upward deflection of the TPCL plane, $F_s$ has an upward component $F_{sv}$ perpendicular to the motion direction, which gradually exceeds the sum of $F_l$ and $F_{hv}$ at the late sinking stage, as sketched in figure 6($b$), thus causing the sphere trajectory to begin to deflect in the upward direction. At the reverting stage, $F_{sv}$ increases further and tends to be vertical because the TPCL plane tends to be horizontal, while $F_h$ becomes extremely small due to the small sphere velocity, as sketched in figure 6($c$). This causes the sphere's vertical velocity to decay rapidly and the horizontal velocity to decay slowly. Accordingly, the sphere moves slowly in the horizontal direction. Since the non-axisymmetry of sphere wetting increases with decreasing $\alpha$, the deviation of the sphere trajectory becomes larger when the impact angle decreases.

Figure 6. Schematics of the forces acting on the microsphere at different stages of the impact process: (a) early sinking stage where the hydrodynamic force $F_h$ dominates, (b) late sinking stage where the surface tension force $F_s$ dominates and (c) reverting stage where the surface tension force $F_s$ dominates. Here, $w$ is the rotation angular velocity of the sphere. The components perpendicular to the sphere motion direction of the hydrodynamic force and surface tension force are $F_{hv}$ and $F_{sv}$, respectively. The force due to the Magnus effect caused by the sphere rotation is $F_l$.

3.3. Penetration time and penetration depth

The impact processes at the impact angles of near $90^{\circ }, 75^{\circ }, 60^{\circ }, 45^{\circ }, 30^{\circ }$ and $15^{\circ }$, are chosen to give scaling arguments on the penetration time $t_p$ and the penetration depth $h_p$. The dependence of the dimensionless penetration time $T_p\ (=t_pu_0/d_p)$ of the microsphere on $We$ after impacting on the surfaces of water and propanol solution with different $\alpha$ values is shown in figure 7($a$,$b$). For oscillating spheres, $T_p$ is independent of $\alpha$ and increases as a function of $We$ with a slope of 1/2 in the log–log scale. Since the oscillating spheres have a small $We$, this relationship may be understood by considering the balance between momentum change of the sphere and the surface tension acting on it (Chen et al. Reference Chen, Heim, Golovko and Bonaccurso2012; Zhu et al. Reference Zhu, Liu, Wang, Niu, Lu and Chen2019), i.e. $m_p ({\textrm {d} u}/{\textrm {d} t})={\rm \pi} d_p\sigma \sin \phi \sin (\theta +\phi )$. We can scale the momentum term as $\rho _p d_p^3({d_p}/{t_p^2})$ and set the surface tension proportional to $\sigma d_p$, which makes the penetration time follow a characteristic time, i.e. $t_p\sim (\rho _pd_p^3/\sigma )^{1/2}$. Thus, we can find that $T_p\sim (DWe)^{1/2} (D =1.18$ in our experiment). The simulations conducted by Chen et al. (Reference Chen, Liu, Lu and Ding2018) on the impact of millimetre-sized spheres with $Bo=0.48$ elicited the characteristic penetration time of $t_p\sim (\sigma /(\rho _l g^3))^{1/4}\sim d_p/u_0We^{1/2}Bo^{-3/4}$, which corresponds to the time taken for a capillary–gravity wave to travel the capillary length $l_c\sim (\sigma /(\rho _l g))^{1/2}$. This expression was also suggested to characterize the pinch-off time of millimetre-sized spheres (Lee & Kim Reference Lee and Kim2008). Our study indicates that the effect of gravity on the evolution of the gas–liquid interface can be neglected during the impact of a microsphere with $Bo\ll 1$, which makes $t_p$ independent of $Bo$. For a submerging sphere, $T_p\approx ({s_p/d_p})/({u_a/u_0})$ where $u_a\approx (u_0+u_p)/2$ is the average velocity of the sphere during the sinking stage. Since the increase in $We$ causes a larger $u_p$ and thus results in a larger $u_a, T_p$ decreases with the increase in $We$. This decreasing trend of $T_p$ slows down with increasing $We$, and $T_p$ tends to be constant when $We$ is much larger than the critical Weber number for the sphere to submerge. This is because when $We$ is large enough, the hydrodynamic force is totally responsible for the energy loss of the sphere, and $u_a$ becomes constant since the decay ratios of the kinetic energies (velocity) of the spheres during the slamming stage have little difference at different $We$ values. The increase in $\alpha$ makes $T_p$ decrease due to a decreasing $s_p$.

Figure 7. The dependences of the dimensionless penetration time $T_p\ (=t_pu_0/d_p)$ of the microsphere on $We$ after impacting on the surfaces of ($a$) water and ($b$) propanol solution with different impact angles, whereby the solid and hollow points represent the submergence and oscillation modes, respectively. The inset panel plots the data in the log–log scale with the same coordinate ranges. The maximum absolute and relative measurement errors of $T_p$ are $0.92$ and $19\,\%$, respectively.

The dependences of the dimensionless penetration depth $h_p/d_p$ of the microsphere with $We \boldsymbol{\cdot}\sin \alpha$ after impacting on the surfaces of water and propanol solution are given in figure 8($a$,$b$), respectively. It shows that the penetration depth of oscillating spheres keeps to the relationship of $h_p/d_p\propto We \boldsymbol{\cdot}\sin \alpha$. Given that the deviation of the sphere trajectory from its impact direction is typically small, we can find that $s_p\sim h_p/\!\sin \alpha$. Based on this, we can deduce that $s_p/d_p\propto We$. The same relationship was also found in the numerical studies on the vertical impact of millimetre-sized spheres by Kintea et al. (Reference Kintea, Breitenbach, Gurumurthy, Roisman and Tropea2016) and Chen et al. (Reference Chen, Liu, Lu and Ding2018). Since the decay ratios of the kinetic energies of the spheres during the slamming stage have little difference regarding the impacts at different $\alpha$ and $We$ values, the remaining kinetic energy after the slamming stage is consumed mostly by the work expended by the surface tension given that $u_p$ of an oscillating sphere is relatively very small. Accordingly, $\sigma d_ps_p\propto \rho _sd_p^3u_0^2$, from which the relationship between $s_p/d_p$ and We listed above can be deduced. For submerging spheres, $h_p/d_p$ is maintained almost constant and does not change as a function of $We$ at the same $\alpha$. The penetration depth of submerging spheres is determined by the evolution of the cavity (Aristoff & Bush Reference Aristoff and Bush2009). Given the similarity of the TPCL motions of submerging spheres in figure 5($a$), it can be inferred that the evolutions of the liquid surface after the impact of microspheres with the same impact angle at a certain range of $We$ are self-similar. It seems that the cavity cannot be maintained stable owing to the large curvature near the TPCL when the sphere reaches a certain depth, resulting in the pinch-off of the cavity and the submergence of microsphere (Aristoff & Bush Reference Aristoff and Bush2009). Thus, a microsphere submerges at a certain penetration depth (Vella & Li Reference Vella and Li2010), while a millimetre-sized sphere submerges at a certain time (Lee & Kim Reference Lee and Kim2008). Furthermore, the change of $h_p/d_p$ of the submerging sphere with $\alpha$ is small when $\alpha$ is large, while it decreases when $\alpha \le 45^{\circ }$. The average $h_p/d_p$ of an oblique submergence mode with $\alpha =17 \pm 3^{\circ }$ is approximately $75\,\%$ of that of a vertical submergence mode in figure 8($a$). For the submerging microspheres after impacting on the surfaces of water and propanol solution, $h_p/d_p$ is approximately $1.84$ and $1.65$, respectively, which is larger than that of a millimetre-sized sphere reported by Chen et al. (Reference Chen, Liu, Lu and Ding2018).

Figure 8. Variations of the dimensionless penetration depth $h_p/d_p$ of the microsphere with $We \boldsymbol{\cdot}\sin \alpha$ after impacting on the surfaces of ($a$) water and ($b$) propanol solution with different impact angles, whereby the solid and hollow points represent the submergence and oscillation modes, respectively. The maximum absolute and relative measurement errors of $h_p/d_p$ are $0.22$ and $14\,\%$, respectively.

3.4. Criterion for different impact modes

According to the theoretical prediction by Johnson & Reid (Reference Johnson and Reid1975), a large PMMA sphere will bounce off the liquid surface when the impact angle is smaller than $18^{\circ }/D^{1/2}=16.6^{\circ }$. However, the PMMA microspheres did not rebound in our experiment, due to the difference of dominant impact forces between the microspheres (surface tension and hydrodynamic force) and large spheres (gravity and fluid force). Figures 9($a$) and 9($b$) show the phase diagrams with respect to $We$ and $\alpha$ for the oblique impact of microspheres on the surfaces of water and propanol solution, respectively. There is a clear boundary between the observed two impact modes of submergence and oscillation, which represents the variation of the critical Weber number $We_{cr}$. Correspondingly, the submerging process near this boundary is referred to as the critical sinking process. The phase diagram shows that $We_{cr}$ decreases as $\alpha$ increases. Herein, we establish a scaling model based on the energy balance during the critical sinking process to derive a criterion which can distinguish different impact modes.

Figure 9. Phase diagrams of oblique impact of PMMA microspheres on the surfaces of ($a$) water and ($b$) propanol solution with respect to $We$ and $\alpha$ coloured based on $Re$, where the solid squares and hollow circles represent the submergence and oscillation modes, respectively. The maximum absolute and relative measurement errors of $We$ are $7.8$ and $23\,\%$, respectively, and the maximum absolute and relative measurement errors of $\alpha$ are $4.3^{\circ }$ and $8.3\,\%$, respectively. The black dashed lines in ($a$) and ($b$) correspond to the predictions of (3.5) when $We_{cr\_n}$ equals to 14.92 and 13.78, respectively.

As mentioned above, the gravity and buoyancy can be neglected compared to the hydrodynamic force and surface tension during impact because $Bo\ll 1$ for a microsphere. For the critical sinking process, the kinetic energy of the sphere can be considered to be zero when it submerges. Thus, the initial kinetic energy of the sphere is consumed by the work expended by the hydrodynamic force $F_h$ and the force due to surface tension $F_s$ during the sinking stage.

Here, $F_h$ mainly works at the slamming stage and can be expressed as $F_h=\frac {1}{2}C_h({{\rm \pi} }/{4}) d_p^2\rho _l u^2$. Shiffman & Spencerd (Reference Shiffman and Spencerd1945a,Reference Shiffman and Spencerdb) theoretically analysed the impact forces on a sphere at the initial stage of the impact, and provided the expression of the resistance coefficient $C_h$. Subsequent experimental and theoretical research studies indicated that the fluid flow around the sphere with $Re>10^2$ at the slamming stage could be assumed as a potential flow, and $C_h$ can be considered as a single variable function of $h_p$ (as well as $\phi )$ (Miloh Reference Miloh1991; Moghisi & Squire Reference Moghisi and Squire1981). Additionally, it was considered that the momentum loss ratio of the sphere at the slamming stage was unrelated to $We$ (Shepard & Abraham Reference Shepard and Abraham2014). In consideration of the submergence processes evaluated by this study, the evolutions of $\phi$ with $s/s_p$ were almost the same. Thus, it can be assumed that the momentum (kinetic energy) losses of the sphere at the slamming stage have little difference regarding the submergence processes at different $\alpha$ and $We$ values, as shown in figure 4($c$). Therefore, the work expended by $F_h$ during the sinking process was evaluated according to $W_h\sim a({{\rm \pi} }/{8})\rho _l d_p^3u_0^2$, where $a$ is a constant.

The force due to surface tension can be calculated with $F_s={\rm \pi} d_p\sigma \sin \phi \sin (\theta +\phi )$, and the work expended by $F_s$ can be expressed as

(3.1)\begin{equation} W_s=\int_0^{s_p}{\rm \pi} d_p \sigma\sin\phi\sin(\theta+\phi) \cos\left(\frac{{\rm \pi}}{2}-\alpha_u-\alpha_T\right)\, \textrm{d} s, \end{equation}

where ${\rm \pi} /2-\alpha _u-\alpha _T$ is the angle between the opposite direction of $F_s$ and the direction of $u$. Here, $F_s$ mainly works at the late stage of the critical sinking, and it can be concluded from the experimental data that ${\rm \pi} /2-\alpha _u-\alpha _T$ is basically less than $35^{\circ }$ at the late stage of the sinking process, which means that $\cos ({\rm \pi} /2-\alpha _u-\alpha _T)$ is basically larger than 0.82 at different impact angles. Here we assume $\cos ({\rm \pi} /2-\alpha _u-\alpha _T)\approx 1$ for simplification and (3.1) can be reduced as

(3.2)\begin{equation} W_s\approx{\rm \pi} d_p\sigma s_p\int_0^1\sin\phi\sin(\theta+\phi)\, \textrm{d} \left(\frac{s}{s_p}\right). \end{equation}

If we let $b=\int _0^1\sin \phi \sin (\theta +\phi )\, \textrm {d} (s/s_p)$, then $W_s\sim b{\rm \pi} d_p\sigma s_p$. As figure 5($a$) shows, the evolution of $\phi$ as a function of $s/s_p$ is almost the same for all of the submergence processes, which indicates that $b$ can be considered as a constant in this study. Research on the vertical impact of microspheres (Ji et al. Reference Ji, Song, Wang and Yao2019) show that $b$ is only related to $\theta$. Thus, the energy balance of the critical sinking process yields

(3.3)\begin{equation} \frac{{\rm \pi}}{12}\rho_sd_p^3u_0^2\sim a\frac{{\rm \pi}}{8}\rho_l d_p^3u_0^2+b{\rm \pi} d_p\sigma s_p. \end{equation}

Since $s_p\sim h_p/\!\sin \alpha$, we obtain

(3.4)\begin{equation} We_{cr}\sim \frac{24bh_p/d_p}{(2D-3a)\sin\alpha}. \end{equation}

For submerging spheres, we can know from figure 8 that the difference between the $h_p/d_p$ of the impact processes at different $\alpha$ values is small. Thus, (3.4) reduces to the critical sinking criterion for the vertical impact of microspheres when $\alpha =90^{\circ }$, and it can be expressed as

(3.5)\begin{equation} We_{cr}\sim \frac{We_{cr\_n}}{\sin\alpha}, \end{equation}

where $We_{cr\_n}$ is the critical Weber number for the vertical impact of microspheres, which can be determined by experiment or the criterion proposed by our previous study (Ji et al. Reference Ji, Song, Wang and Yao2019). The formula shows that $We_{cr}$ for the oblique impact of the microsphere is approximately $1/\!\sin \alpha$ times that for the vertical impact, which is mainly owing to the increasing work expended by the surface tension force (as well as the surface energy changes of the system) during the sphere's oblique impact process. By calculating the average of the maximum $We$ of oscillating spheres and the minimum $We$ of submerging spheres with $\alpha =88 \pm 2^{\circ }$, it can be found that $We_{cr\_n}$ equals $14.92\pm 0.62$ and $13.78\pm 0.33$ for the impacts on the surfaces of water and propanol solution, respectively. Correspondingly, $We_{cr}=14.92/\!\sin \alpha$ and $We_{cr}=13.78/\!\sin \alpha$ for the impacts on these two liquids surface, respectively, as shown by the dashed lines in figure 9($a$,$b$). Although $Re$ varies from 115 to 1408 in our experiment, (3.5) can accurately distinguish the two observed impact modes, thus confirming that $Re$ has a small effect on the impact process when $Re$ is large. When $\alpha$ is small (e.g. ${\sim }15^{\circ }$ in figure 9), the obvious decrease in the penetration depth and increase in the deviation of TPCL plane from the velocity direction will increase the prediction error of (3.5). The detailed sphere dynamics after impacting on the liquid surface with a very small impact angle still remains to be revealed.