## 1. Introduction

The impact of a solid body into water comprises a complex series of events that occur both above and below the water surface and depend on the configuration of the body. The phenomena of interest are the associated splash, the cavities and the jets (Prosperetti & Oguz Reference Prosperetti and Oguz1993; Truscott, Epps & Belden Reference Truscott, Epps and Belden2014), which have wide relevance in fields ranging from water-skipping animals to air-to-sea projectiles and even to planetary crater formation (Lohse *et al.* Reference Lohse, Bergmann, Mikkelsen, Zeilstra, van der Meer, Versluis, van der Weele, van der Hoef and Kuipers2004; Hu & Bush Reference Hu and Bush2010; Van Der Meer Reference Van Der Meer2017). The nature of the splash and its closure (initial-stage events) often have long-lasting implications for the underwater events that follow (Thoroddsen *et al.* Reference Thoroddsen, Etoh, Takehara and Takano2004; Aristoff *et al.* Reference Aristoff, Truscott, Techet and Bush2010; Mansoor *et al.* Reference Mansoor, Marston, Vakarelski and Thoroddsen2014). An important factor that determines the characteristics of splash is the surface wettability (contact angle) of the impactor. For instance, increased wettability can induce an increase in the splash formation threshold (Duez *et al.* Reference Duez, Ybert, Clanet and Bocquet2007), whereas a superhydrophobic surface coating can lead to the formation of a drag-reducing, underwater cavity during water entry (Vakarelski *et al.* Reference Vakarelski, Klaseboer, Jetly, Mansoor, Aguirre-Pablo, Chan and Thoroddsen2017). The airflow rushing into this cavity also plays a significant role in the dynamics of the splash curtain that forms above the free surface (Thoroddsen *et al.* Reference Thoroddsen, Thoraval, Takehara and Etoh2011; Vincent *et al.* Reference Vincent, Xiao, Yohann, Jung and Kanso2018; Eshraghi, Jung & Vlachos Reference Eshraghi, Jung and Vlachos2020).

The dimensional parameters considered in prior studies are usually the density (or pressure) of the air above the water surface, the impact velocity, the projectile's shape and temperature, and the liquid properties (Enriquez *et al.* Reference Enriquez, Peters, Gekle, Schmidt, Lohse and van der Meer2012; Truscott *et al.* Reference Truscott, Epps and Belden2014; Mathai, Govardhan & Arakeri Reference Mathai, Govardhan and Arakeri2015; Peters *et al.* Reference Peters, Madonia, Lohse and van der Meer2016; Mansoor *et al.* Reference Mansoor, Vakarelski, Marston, Truscott and Thoroddsen2017; Aly & Asai Reference Aly and Asai2018; Zhang *et al.* Reference Zhang, Liu, Qu, Wang and Agarwal2018). Within this parameter space, a variety of splash and cavity types are possible, *viz*. the quasi-static seal, shallow seal, deep seal and surface seal. In the inertial regime (moderate to high-speed impact), the crucial parameter is the Froude number $Fr$, which determines the specific type of water entry (Aristoff & Bush Reference Aristoff and Bush2009). Within the air-entraining regime of water entry (Truscott *et al.* Reference Truscott, Epps and Belden2014; Hao, Zhang & Huang Reference Hao, Zhang and Huang2018), the collapse of the subsurface cavity displays a non-universal, Froude-number-dependent power-law exponent (Bergmann *et al.* Reference Bergmann, van der Meer, Stijnman, Sandtke, Prosperetti and Lohse2006) that approaches a 1/2 scaling in the limit of large $Fr$ (Lohse *et al.* Reference Lohse, Bergmann, Mikkelsen, Zeilstra, van der Meer, Versluis, van der Weele, van der Hoef and Kuipers2004; Duclaux *et al.* Reference Duclaux, Caillé, Duez, Ybert, Bocquet and Clanet2007; Bergmann *et al.* Reference Bergmann, Van Der Meer, Gekle, Van Der Bos and Lohse2009*b*). Similarly, the pinch-off depth of the cavity displays two distinct scaling regimes with Froude number, separated by discrete jumps (Gekle *et al.* Reference Gekle, van der Bos, Bergmann, van der Meer and Lohse2008).

A spinning projectile during water entry can produce several interesting features. For example, helical striations have been seen on the cavity surface during the water entry of a rotating projectile (Shi, Itoh & Takami Reference Shi, Itoh and Takami2000). Also, imparting a transverse spin to a water-entering projectile can cause the development of non-axisymmetric cavities. These show interesting similarities to the cavities formed by spheres with a half-hydrophilic and half-hydrophobic surface (Truscott & Techet Reference Truscott and Techet2009*b*,Reference Truscott and Techet*a*). A modification to the problem presented above is achieved when a background rotational motion is provided to the pool prior to water entry. Although rotation in the carrier liquid has been found to have a profound influence on a variety of fluid dynamical phenomena (Alvarez-Lacalle, Casademunt & Eggers Reference Alvarez-Lacalle, Casademunt and Eggers2009; Bergmann *et al.* Reference Bergmann, Andersen, Van der Meer and Bohr2009*a*; Jiang *et al.* Reference Jiang, Zhu, Wang, Huisman and Sun2020; Mathai, Lohse & Sun Reference Mathai, Lohse and Sun2020), its effect on the water entry of spheres has not been studied.

In the present work, we explore the familiar water entry phenomenon in the presence of background pool rotation, using experiments, numerical simulations and theoretical analysis. The liquid flow condition here essentially mimics the vortex core of a whirlpool (Stepanyants & Yeoh Reference Stepanyants and Yeoh2008), i.e. an azimuthal flow resembling a rigidly rotating liquid. We begin with a description of the experimental set-up and the numerical method. We reveal how the background liquid rotation alters both the splash and the cavity dynamics, triggering an earlier transition from deep seal to surface seal. Lastly, we provide scaling arguments to explain the observed cavity evolution and pinch-off dynamics.

## 2. Experimental set-up and procedure

The experimental set-up consisted of a cylindrical rotating water tank and an electromagnetic dropper (see schematic in figure 1*a*). The spheres were allowed to fall vertically into the water tank from a predetermined height to achieve the desired impact velocity. The sphere release was conducted using an electromagnet dropper. The contact point aligned with the vertical line passing through the sphere's centre of mass, which minimized rotation of the sphere during the release. The cylindrical container made of Plexiglas (diameter $D = 150\ \textrm {mm}$, wall thickness $h = 5\ \textrm {mm}$) was driven by a motor with a constant angular velocity, $0\ \textrm {rad}\,\textrm {s}^{-1} \leqslant \omega \leqslant 8{\rm \pi} \ \textrm {rad}\,\textrm {s}^{-1}$, about the central, vertical axis. The water impact projectile was a stainless-steel sphere with radius $R_0$ that varied from 5 mm to 15 mm. The surface of the sphere was coated with a hydrophobic coating, which provided a static contact angle of $\theta = 145 \pm 5^{\circ }$. The steel sphere was released from a height $H_{0}$ using the electromagnetic dropper, and impacted the lowest point of the parabolic free surface at an impact velocity $U_0 \approx \sqrt {2gH_0}$, where $g$ is the gravitational acceleration. The true impact velocity is calculated through an analysis using high-speed imaging, and ranges from $0.9\ \textrm {m}\,\textrm {s}^{-1}$ to $3.2\ \textrm {m}\,\textrm {s}^{-1}$. The corresponding Reynolds number is $Re\equiv U_0 R_0/\nu \sim {O}(10^4)$, where $\nu$ is the kinematic viscosity of water. The air pressure above the pool was atmospheric. Images were recorded using a high-speed camera (Photron Mini AX100) at speeds up to $10^4$ frames per second. For the optical configuration used here, image distortion in the vertical direction can be neglected. A ray-tracing model, in conjunction with a grid-based calibration method, was employed to correct for the optical distortion in the horizontal direction.

When the liquid viscosity is low, two independent parameters may be conveniently chosen to fully define the sphere impact problem (Oguz & Prosperetti Reference Oguz and Prosperetti1990). Here, we use the Froude and Bond numbers, defined as $Fr\equiv U_{0}^{2}/(gR_{0})$ and $Bo \equiv \rho g R_0^2/\sigma$, respectively, where $\rho$ is the density of water and $\sigma$ the surface tension of the air–water interface. In addition, we introduce a dimensionless rotational parameter $\mathcal {S} \equiv \omega R_{0}/U_{0}$, which is equivalent to the inverse Rossby number (Warn *et al.* Reference Warn, Bokhove, Shepherd and Vallis1995). The Weber number $We \equiv \rho U_{0}^{2}R_{0}/\sigma$ will also be listed; however, since $We$ can be expressed as the product of $Fr$ and $Bo$, $We = Fr\,Bo$, it does not serve as an additional control parameter in the present work. A representative snapshot of the underwater cavity that forms after the sphere impacts the rotating pool is shown in figure 1(*b*). A splash crown is visible above the free surface. Below the surface, the sphere has entrained an attached air cavity that pinches off at a later instant, due to the competing effects of inertial, hydrostatic and centripetal forces.

## 3. Numerical method

In addition to the experiments, we performed boundary integral (BI) simulations based on potential flow theory (Oguz & Prosperetti Reference Oguz and Prosperetti1993; Peters *et al.* Reference Peters, Madonia, Lohse and van der Meer2016; Li *et al.* Reference Li, van der Meer, Zhang, Prosperetti and Lohse2020) to better and quantitatively understand the experimental observations. Considering the rotating flow background, we defined a cylindrical coordinate system $Or\theta z$, which was fixed to the rotating cylindrical tank. The origin $O$ was set at the centre of the free surface, and the $z$ axis direction pointed opposite to that of $g$. The BI can be reduced to two-dimensional simulations here, under the assumption of axisymmetry of the developing cavity. Since a non-inertial coordinate system was adopted, the centrifugal force must be taken into consideration.

The Coriolis force can be neglected in the simulations, as it acts normal to the $rz$ plane and is also of low magnitude when compared to the inertial forces. Additionally, the BI formulation allowed for the inclusion of air as a second ideal fluid phase. The simulations were first validated against quiescent liquid cases: then rotating liquid simulations were performed. Although the BI simulations enable us to accurately model the mechanisms governing the subsurface cavity evolution, they can only serve as a qualitative model for the splash closure.

## 4. Results and discussion

### 4.1. Regime transition of cavity type

To investigate the effect of the rotating liquid background on the transient cavity dynamics, we compare the experimental results in the quiescent liquid condition to those of the rotating background, at different values of $\mathcal {S}$ while maintaining $Fr$ and $Bo$ constant (see figure 1*c*,*d*). A typical water entry sequence in quiescent liquid is shown in figure 1(*c*), where $Fr = 39$ and $Bo = 30$, and consequently $We = Fr \, Bo = 1170$. Note that the effect of rotation becomes more dramatic with increasing sphere radius $R_0$, which will be discussed later in § 4.2. The initial impact creates a splash and cavity. A crown-like splash curtain forms above the water surface, which remains open during the entire sequence. The underwater cavity that is created is first pushed out by the descending sphere. Later, it contracts due to the hydrostatic pressure, leading to a pinch-off at $t = 77.0\ \textrm {ms}$. This type of pinch-off was referred to as a ‘deep seal’ in prior work, as it occurs at a significant depth below the water surface when compared to other cavity-sealing phenomena (Lohse *et al.* Reference Lohse, Bergmann, Mikkelsen, Zeilstra, van der Meer, Versluis, van der Weele, van der Hoef and Kuipers2004; Aristoff & Bush Reference Aristoff and Bush2009; Tan & Thomas Reference Tan and Thomas2018). The corresponding cavity is referred to as a deep seal cavity. Note that the pinch-off depth is about half the height of the whole cavity, which is in agreement with prior studies (Oguz & Prosperetti Reference Oguz and Prosperetti1990; Duclaux *et al.* Reference Duclaux, Caillé, Duez, Ybert, Bocquet and Clanet2007; Aristoff & Bush Reference Aristoff and Bush2009; Bergmann *et al.* Reference Bergmann, Van Der Meer, Gekle, Van Der Bos and Lohse2009*b*).

In comparison, for the rotating liquid case, a splash curtain is faintly observable above the free surface ($t = 12.8\ \textrm {ms}$ in figure 1*d*). However, the diameter of this splash is significantly lower than that in the quiescent liquid case. By $t = 25.6\ \textrm {ms}$, we find that the splash curtain has already closed. This process is commonly referred to as ‘surface seal’, since the splash crown is pulled radially inwards before finally closing above the free surface (Aristoff & Bush Reference Aristoff and Bush2009; Truscott *et al.* Reference Truscott, Epps and Belden2014). We note that, for the quiescent liquid case, a surface seal type of cavity cannot be expected until a high $Fr$, the threshold for which was estimated as $Fr_c$ = $(1/6400)(\rho /\rho _a)^2\approx 100$ (Birkhoff & Isaacs Reference Birkhoff and Isaacs1951), where $\rho _a$ is the density of air. Thus, the presence of a rotating background flow triggers an early transition from the deep seal to the surface seal type of cavity. Once the surface seal has been triggered, the events that succeed are markedly different (Marston *et al.* Reference Marston, Truscott, Speirs, Mansoor and Thoroddsen2016). The enclosed cavity in the rotating liquid case first undergoes a reduction in pressure due to its expanding volume (from $t = 25.6\ \textrm {ms}$ to $t = 38.4\ \textrm {ms}$). This pressure reduction causes the cavity to be pulled below the free surface, which is often followed by the formation of a Rayleigh–Taylor fingering instability at the apex of the enclosed cavity (see also Aristoff & Bush Reference Aristoff and Bush2009).

We vary $Fr$ and $\mathcal {S}$ systematically over a wide range ($100 \leq Fr \leq 205$ and $0 \leq \mathcal {S} \leq 0.047$) and characterize the splash and transient cavity dynamics at a fixed $Bo = 3.4$. A phase diagram indicating the dependence of the observed cavity type on $Fr$ and $\mathcal {S}$ is presented in figure 2(*a*). At a relatively low $Fr$ and low $\mathcal {S}$, we observe the deep seal. With increasing $\mathcal {S}$, the cavity closure undergoes a transition from deep seal to surface seal. The transitional $Fr$ decreases ever more steeply with increasing background rotation, until, for $\mathcal {S} \geq 0.045$, we always observe the surface seal cavity type. It is verified that the transitional $Fr$ in the quiescent liquid condition ($\mathcal {S} = 0$) is comparable to the threshold proposed by Birkhoff & Isaacs (Reference Birkhoff and Isaacs1951). Note that the data in figure 2(*a*) are obtained only for $Bo = 3.4$; a change in $Bo$ alters the transitional boundary of the $Fr$–$\mathcal {S}$ phase space presented here. Mapping out the full non-dimensional $Fr$–$\mathcal {S}$–$Bo$ parameter space would require even more extensive sets of experiments, which are beyond the scope of the present work. Next, we resort to the BI simulations to obtain the cavity shapes for different values of $\mathcal {S}$. Figure 2(*b*) shows the cavity profiles, in this case without the airflow modelled. With increasing $\mathcal {S}$ the cavity neck becomes narrower, thereby aiding in the transition to the surface seal regime. Yet, remarkably, the effects of rotation seem localized to near the free surface, and the cavity profiles nicely overlap for larger depths inside the pool.

### 4.2. Rayleigh–Plesset approach

To better understand the experimental results, we adapt the Rayleigh–Plesset equation (Plesset & Prosperetti Reference Plesset and Prosperetti1977) for an axisymmetrically evolving cavity in cylindrical coordinates ($r$, $\theta$, $z$) (Oguz & Prosperetti Reference Oguz and Prosperetti1990; Lohse *et al.* Reference Lohse, Bergmann, Mikkelsen, Zeilstra, van der Meer, Versluis, van der Weele, van der Hoef and Kuipers2004; Bergmann *et al.* Reference Bergmann, Andersen, Van der Meer and Bohr2009*a*; Lohse Reference Lohse2018). Based on the BI simulation results, the axial velocity $U_{z}$ can be neglected in comparison to the radial $U_{r}$ and azimuthal $U_{\theta }$ components. Applying the continuity equation, we then obtain $rU_{r} = R\dot {R}$, where $R$ denotes the radius of the cavity wall. Integrating the Rayleigh–Plesset equation radially with respect to $r$ from $R$ to $R_{\infty }$, we obtain

where $P_{\infty }$ is the pressure (in water) at a distance $R_{\infty }$, at which the flow may be regarded as quiescent, and $P$ the air pressure inside the cavity. In the high-Reynolds-number and high-Weber-number limit of our experiments, the first and second terms on the right-hand side of (4.1) can be safely ignored. Finally, we assume that the azimuthal velocity in the pool remains unchanged beyond the vicinity of the developing cavity, i.e. $U_{\theta } \approx \omega R$. These approximations lead to

where the characteristic length and time scales used in this non-dimensional representation are $R_0$ and $R_0/U_0$, respectively. On the right-hand side, the dimensionless rotational number $\mathcal {S}$ appears as an additional pressure term, which speeds up the closure of the cavity.

In light of (4.2), one can rationalize the cavity behaviours that were observed experimentally. Firstly, we note that both the hydrostatic term $-\tilde {z}/{Fr}$ and the rotational term $(\mathcal {S}^2\tilde {R}^2)/2$ contribute to speeding up the cavity collapse. The latter is unchanged with depth, and (assuming $R \sim R_0$) is of relevance only up to a shallow depth estimated as ${z} \leq (\omega ^2 R_0^2/g)/2$. For the most extreme rotation rate in experiments, i.e. $\mathcal {S} = 0.04$ at $Bo = 3.4$, this yields a region of influence $z \sim R_0$. Beyond this depth below the free surface, the dynamics is dominated by the hydrostatic term. Thus, the rotating liquid seems to influence the cavity profiles only up to a shallow depth, a result that is also corroborated by our BI simulations (figure 2*b*).

Further to the increased pressure term $(\mathcal {S}^2\tilde {R}^2)/2$ in (4.2), the airflow through the narrowing splash curtain also contributes to the early surface seal in the rotating liquid case. We turn our focus to the splash radius $R_{sp}(t)$ near the free surface. Here the hydrostatic term can be safely neglected, but, instead, the Bernoulli pressure reduction due to air entering the cavity becomes important. The volume expansion rate of the cavity can be expressed as $\dot {\mathcal {V}}=\textrm {d} (\int {\rm \pi}R^2\,\textrm {d}z)/\textrm {d}t$. Since it is known that the role of rotation is localized to near the free surface (see figure 2*b*), we can assume that $\dot {\mathcal {V}}$ is unchanged with $\mathcal {S}$. Therefore, the continuity constraint necessitates that the mean airflow velocity near the free surface $U_{za} \propto 1/R_{sp}^2$. The corresponding under-pressure is $\Delta P \propto (\rho _a R_{sp}^{-4})/2$, which indicates that even a slight reduction in the splash radius can induce a cascading effect due to the inherent aerodynamic coupling, leading to the early surface seal. The surface seal time (defined as the time interval between the impacting moment and the surface seal moment) decreases with increasing $\mathcal {S}$. However, the precise moment of surface seal is difficult to estimate from side-view images in the rotating liquid cases. It would require additional recordings from above the free surface, which will be part of a future investigation. The role of the incoming air is further exemplified in the BI simulations of the pressure field in the air at an instant prior to the surface closure (see figure 2*c*). The pressure in the narrow neck region is significantly lower than that in the surrounding regions. In contrast, a simulation that ignores the airflow effect gives a noticeably wider opening near the free surface (magenta curve).

### 4.3. Underwater pinch-off of cavity

While the background liquid rotation triggers an earlier transition from a deep seal to a surface seal, there exists a range of Froude numbers for which the transition is not triggered (see deep seal regime in figure 2*a*). However, even within the deep seal regime, the rotation induces changes to the underwater cavity dynamics. Since the effect of background liquid rotation on cavity dynamics is more pronounced for the larger $R_0$ cases, as discussed in § 4.2, we used larger spheres ($Bo = 13.4$ and 30) to study the underwater cavity dynamics in the deep seal regime. These $Bo$ values, although larger than the $Bo = 3.4$ in figure 2(*a*), help demonstrate the dramatic effect of background liquid rotation on the subsurface cavity dynamics. We define the pinch-off time $t_{p}$ as the time interval between the moment the sphere touches the initial air–water interface and the moment of the cavity collapse. In figure 3(*a*), we plot the non-dimensional pinch-off time $t_{p}^{*}=t_{p}U_0/R_0$ as a function of $Fr^{1/2}$, with $\omega$ varied from $0\ \textrm {rad}\,\textrm {s}^{-1}$ to $8{\rm \pi} \ \textrm {rad}\,\textrm {s}^{-1}$. Prior studies (Glasheen & McMahon Reference Glasheen and McMahon1996; Duclaux *et al.* Reference Duclaux, Caillé, Duez, Ybert, Bocquet and Clanet2007; Truscott & Techet Reference Truscott and Techet2009*b*) have shown that the non-dimensional pinch-off time follows a square-root relation $t_{p}^{*}=k_t{Fr}^{1/2}$, where $k_t$ is a constant. As evident from figure 3(*a*), the scaling $t_{p}^{*} = k_t{Fr}^{1/2}$ is robust for the rotating flow cases as well. The prefactor $k_t$ ranges from 1.63 to 2.09, which is comparable to the value reported in prior work ($\approx$1.726; Truscott & Techet Reference Truscott and Techet2009*b*). However, the prefactor $k_t$ decreases noticeably with increase in $\omega$ (see inset to figure 3*a*). Beyond the $Fr$ range of 5.6–35 presented here, since the cavity undergoes surface seal, the deep seal time definition is somewhat ambiguous and hence will not be reported.

Lastly, we reveal the dynamics of the cavity wall at the pinch-off depth as it accelerates towards the singularity of the pinch-off. Close enough to the pinch-off point, the cavity radius $R$ is small, while the reference radius $R_{\infty }$ is very large. Therefore, the logarithmic part of the inertial term in (4.2), i.e. $\ln (\tilde {R}/\tilde {R}_{\infty})$, diverges. This necessitates the condition that ${\textrm {d}(\tilde {R}\dot {\tilde {R}})}/{\textrm {d}\tilde {t}} = 0$. Integrating this, we obtain $R = k_R \sqrt {R_0 U_0} \, \tau ^{1/2}$, where $\tau =(t_p-t)$ denotes the time to pinch-off. In figure 3(*b*), we plot $R/R_0$ as a function of $(\tau U_0/R_0)^{1/2}$ for various values of $\mathcal {S}$. The data collapse nicely with a good agreement to the $1/2$ power-law prediction. The prefactor of the fit remains nearly constant ($k_R = 0.28\pm 0.01$) across the cases. When we focus on the final stage of the collapse ($(\tau U_0/R_0)^{1/2}<2$), the direct fitting between $R/R_0$ and $\tau U_0/R_0$ gives a power-law exponent of approximately 0.54 for all $\mathcal {S}$ cases, which is close to the value recently found in experiments (0.55; Yang, Tian & Thoroddsen Reference Yang, Tian and Thoroddsen2020). This is also consistent with previous experiments (Bergmann *et al.* Reference Bergmann, van der Meer, Stijnman, Sandtke, Prosperetti and Lohse2006) and the corresponding slow asymptotic theory (Eggers *et al.* Reference Eggers, Fontelos, Leppinen and Snoeijer2007). Therefore, the effect of the background liquid rotation is insignificant during the final stages of the cavity evolution. The inset to figure 3(*b*) shows that the normalized pinch-off depth $H_p/R_0$ monotonically decreases with increasing $\mathcal {S}$. This trend can again be traced back to the increased pressure in the liquid and the reduced cavity pressure due to the inrushing airflow (see § 4.2).

## 5. Conclusions

In summary, we have presented a combined experimental, numerical and theoretical investigation of the transient cavity dynamics following the impact of a heavy sphere into a rotating pool of water. Background liquid rotation triggers an early transition from a deep seal to a surface seal regime. We characterized this regime transition in terms of the Froude number $Fr$ and a dimensionless rotational number $\mathcal {S} \equiv \omega R_0/U_0$. With increasing $\mathcal {S}$, the transitional $Fr$ marking the change of the cavity type decreases. The reasons for this can be traced back to the additional pressure term arising due to the background rotation, in combination with the Bernoulli pressure reduction due to the airflow through the splash neck above the free surface. We used BI simulations to demonstrate the crucial role of air on the splash and cavity dynamics. By comparing two simulations modelled with and without air, we show that incoming airflow can dramatically affect the splash closure time. It is revealed that the pressure buildup in the rotating liquid accelerates the closure phenomenon, thereby inducing the early transition from deep seal to surface seal. Below the transitional $Fr$, we investigated the effect of background rotation on the dynamics of the deep seal cavity. Remarkably, the non-dimensional pinch-off time retains a $1/2$ power-law dependence on $Fr$ despite the centripetal effects, but with a noticeably reduced prefactor with increasing $\omega$.

We used the axisymmetric Rayleigh–Plesset equation to predict the radius evolution during the final moments before the pinch-off, which yields a $\tau ^{1/2}$ dependence, where $\tau$ is the time to pinch-off. The predictions are found to be in good agreement with the experimental measurements.

The current work has revealed that the effects of a rotating liquid background are mainly confined to the free surface and to shallow depths. Yet, these initial-phase modifications have noticeable effects on the later dynamics of water entry.

## Acknowledgements

We thank S. Maheshwari, Q. Zhou and L. Jiang for useful discussions.

## Funding

We acknowledge financial support by the Natural Science Foundation of China under grant nos. 11988102, 11861131005, 91852202 and 11672156.

## Declaration of interests

The authors report no conflict of interest.