1. Introduction
After Silberrad (Reference Silberrad1912) reported propeller erosion on the ships ‘Lusitania’ and ‘Mauretania’, Rayleigh (Reference Rayleigh1917) proposed cavitation as the likely cause for the damage. The model named after him describes a collapsing spherical, empty cavity. He found that such a collapse creates high pressures in the liquid that may be responsible for the damage observed on the steel propellers. Improved spherical bubble models for example by Hickling & Plesset (Reference Hickling and Plesset1963) and later on Keller & Miksis (Reference Keller and Miksis1980) include the pressure of non-condensable gas inside the bubble, liquid viscosity, surface tension and the energy loss from acoustic radiation. Boundaries that violate the spherical symmetry of the flow are not accounted for in these models. Early experimental investigations to understand the origin of erosion studied the effect of boundaries and geometries that break the spherical symmetry. For example, Shutler & Mesler (Reference Shutler and Mesler1965) studied the dynamics and collapse of a non-spherical spark-generated bubble near rigid plates, Benjamin & Ellis (Reference Benjamin and Ellis1966) used an evacuated free fall device, and Lauterborn & Bolle (Reference Lauterborn and Bolle1975) introduced laser-induced cavitation bubbles to increase control on location and timing of the bubble. They all observed directed liquid flows or jets that form during the shrinkage of the bubble. Blake & Gibson (Reference Blake and Gibson1987) developed a boundary element method to simulate the inviscid flow which helped to understand the mechanism of jet creation and the movement of the bubble towards the rigid boundary. Very good agreement between the experimental and the simulated bubble dynamics was found, see also Best & Kucera (Reference Best and Kucera1992).
As we know that an infinitely extended rigid boundary induces a jet towards the boundary, one may expect that a narrow diameter through-hole embedded in the boundary may not much alter the jetting dynamics. As a result, some of the liquid volume of the jet may be accelerated through the perforated boundary. This cavitation based liquid pump was first suggested by Khoo, Klaseboer & Hung (Reference Khoo, Klaseboer and Hung2005) in a numerical work using the boundary integral method to simulate a bubble near a plate with a small diameter through-hole. This was followed by experiments from Lew, Klaseboer & Khoo (Reference Lew, Klaseboer and Khoo2007) and later Karri et al. (Reference Karri, Ohl, Klaseboer, Ohl and Khoo2012b), who used a spark-generated bubble with a radius of several millimetres demonstrating the pumping mechanism. Dijkink & Ohl (Reference Dijkink and Ohl2008) down-scaled the technique for microfluidic application using considerably smaller, laser-generated bubbles.
Karri et al. (Reference Karri, Pillai, Klaseboer, Ohl and Khoo2011) revealed that in the presence of a gaseous domain on the opposite side of the perforated plate, the bubble dynamics results in high-speed liquid jetting into the gas. The jet fragments into a spray of many smaller droplets. Initially, they used a lithotripter to create a bubble with 
$R_{max}\approx 250$ 
$\mathrm {\mu }{\rm m}$, and later a spark discharge for a bubble with 
$R_{max}\approx 6\,{\rm mm}$ (see Karri et al. Reference Karri, Gonzalez-Avila, Loke, O'Shea, Klaseboer, Khoo and Ohl2012a). Further investigations of such jets and sprays were performed by Gonzalez-Avila, Song & Ohl (Reference Gonzalez-Avila, Song and Ohl2015) and Kannan, Balusamy & Karri (Reference Kannan, Balusamy and Karri2015) using a spark-generated bubble very close to a plate with a hole, with 
$R_{max}= 1$ and 8 mm, respectively.
Wang et al. (Reference Wang, Zhang, Liu and Wu2013) experimentally studied the dynamics of a spark-generated cavitation bubble with length scales of millimetres near a thin perforated plate. Numerical investigations were performed by Dawoodian, Dadvand & Nematollahi (Reference Dawoodian, Dadvand and Nematollahi2015) using the boundary element method to simulate a bubble in a tube near a thin plate with a hole. Continuous pumping of the liquid has been demonstrated by Cao, Liu & Qu (Reference Cao, Liu and Qu2017) with periodically created cavitation bubbles just above a channel of radius 
$r_{c}=2.5$ 
$\mathrm {\mu }{\rm m}$. The particular shape of the plate has an effect on the bubble and therefore on the pumping as demonstrated by Cui et al. (Reference Cui, Zhang, Wang and Wang2013), who conducted cavitation experiments with relatively large bubbles on top of a curved perforated plate. The experiments were later simulated by Moloudi et al. (Reference Moloudi, Dadvand, Dawoodian and Saleki-Haselghoubi2019) based on the boundary element method and, for certain parameters, revealed a pronounced jet in the opposite direction. However, the simulations stopped once the jet pierced the bubble and the resulting fluid transport past that instant was not obtained.
While there is strong experimental support for the pumping effect of bubbles near through-holes, there is a considerable lack in numerical simulations that include the jetting dynamics through the hole and the following bubble dynamics, and that study the influence of various geometrical parameters of the hole as well as viscosity. These aspects are addressed in the present work, which also accounts for surface tension and compressibility of the liquid and the gas.
We start with a description of the governing flow equations and continue with the implementation in a Volume-of-Fluid (VoF) solver in § 3. Selected simulation results are compared to experiments. Therefore, we describe a suitable experimental setup in § 4 that can reveal not only the bubble dynamics but also the liquid transport. The results section starts with a detailed description of the pumping mechanism (§ 5.1). Then, the influence of various parameters on the pumping effectiveness are revealed and discussed in § 5.2. We then take a closer look at two particular flow phenomena in § 6, namely the so-called needle jet and the reverse jet using and comparing simulations with experiments, before we discuss and summarise the results.
2. Governing equations
The simulation model describes two immiscible, compressible, viscous fluids, i.e. water and a non-condensable gas, and accounts for their surface tension. Mass and heat transfer across the fluid–fluid interface are neglected. The initial process of cavitation nucleation, e.g. from a laser or a spark discharge, is ignored. Instead we start the fluid mechanics model with a small gas bubble at a high pressure at time 
$t=0$.
The fluids satisfy the continuity equation (2.1) and the law of conservation of momentum (2.2) of Newtonian fluids,
\begin{gather} \frac{\partial\rho}{\partial t}+\boldsymbol{\nabla}\boldsymbol{\cdot}(\rho\boldsymbol u)=0, \end{gather}
\begin{gather} \rho\frac{{\rm D}\boldsymbol u}{{\rm D}t}=\rho\boldsymbol f-\boldsymbol{\nabla} p+\mu\left(\Delta\boldsymbol u+\frac{1}{3}\boldsymbol{\nabla}(\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol u)\right).\end{gather}
Here, the physical quantities are density 
$\rho$, time 
$t$, velocity 
$\boldsymbol u$, volume acceleration 
$\boldsymbol f$ (e.g. gravity) and the dynamic viscosity 
$\mu$. A liquid viscosity of 
$\mu =1$ mPa s and a gas viscosity of 
$\mu =0.0184$ mPa s are used, as well as a surface tension of 
$\sigma =70\,{\text {mN}}\,{\text {m}}^{-1}$. For both components, the Tait equation is used as an equation of state:
\begin{equation} p=(p_0+B)\left(\frac{\rho}{\rho_0}\right)^\gamma-B, \end{equation}
where the constants are 
$p_0=101\,325$ Pa, 
$B=303.6$ MPa, 
$\rho _0=998.21\,{\rm kg}\,{\rm m}^{-3}$, and 
${\gamma =7.15}$ for the liquid (water) and 
$p_0=10\,320$ Pa, 
$B=0$, 
$\rho _0=1\,{\rm kg}\,{\rm m}^{-3}$, and 
$\gamma =1.33$ for the gas. With 
$B=0$, the equation of state (2.3) describes the adiabatic change of state of an ideal gas.
The Rayleigh collapse time is the idealised time it takes for an empty cavity under the action of a constant external pressure to shrink to zero size. Lord Rayleigh solved the energy equation for an inviscid, incompressible fluid initially at rest once a portion of the liquid was instantaneously removed (see Rayleigh Reference Rayleigh1917). He obtained an equation for the bubble radius as a function of time that could be integrated to obtain the Rayleigh collapse time
\begin{equation} \tau\approx0.91468\,R_{max}\sqrt{\frac{\rho}{p_0}}. \end{equation}
This Rayleigh collapse time is approximately 
$\tau =46\,\mathrm {\mu }$s for typical bubbles of 
$R_{max}=0.5$ mm. The spherical bubble dynamics obtained from the VoF solver can be compared to the Keller–Miksis model (Keller & Miksis Reference Keller and Miksis1980):
\begin{equation} R\ddot R\left(1-\frac{\dot R}{c}\right)+\frac{3}{2}\dot R^2\left(1-\frac{\dot R}{3c}\right)+\frac{4\nu\dot R}{R}+\frac{2\sigma}{\rho R}+\frac{p_0-p_{g}}{\rho}\left(1+\frac{\dot R}{c}\right)-\frac{R\dot p_{g}}{\rho c}=0, \end{equation}which is a solution to the Navier–Stokes equation and accounts for compressibility in the limit of linear acoustic waves.
3. Numerical implementation
3.1. Numerical solver
For the simulations, a modified version of the VoF solver compressibleInterFoam from the open-source framework OpenFOAM 4.0 (2016) is used. It models two compressible, non-isothermal, immiscible fluids using a phase-fraction based interface capturing scheme. The solutions are obtained in 250 000 to 370 000 computational cells, each of which is assigned its respective flow equations. The solver solves for the five fields pressure 
$p$, non-gravitational pressure 
$p_{rgh}=p-\rho \,\boldsymbol g\boldsymbol {\cdot } \boldsymbol r$, velocity 
$\boldsymbol u$, temperature 
$T$ and phase fraction 
$\alpha$, which is a non-dimensional scalar field ranging between 0 and 1 that specifies the relative volume amount of a cell occupied by one of the fluids. In this case, 
$\alpha =1$ means that a cell is fully occupied by water and 
$\alpha =0$ means it is fully occupied by gas. Since gravitational effects are neglected, 
$\boldsymbol g$ is chosen to be 0 and thus 
$p_{rgh}=p$.
The solver is derived from CavBubbleFoam by Koch et al. (Reference Koch, Lechner, Reuter, Köhler, Mettin and Lauterborn2016) and Zeng et al. (Reference Zeng, Gonzalez-Avila, Dijkink, Koukouvinis, Gavaises and Ohl2018), which itself was a modification of the original compressibleInterFoam. CavBubbleFoam updates the compressibility field in every time step and neglects non-isothermal effects by removing the temperature equation. This simplification has been used successfully for cavitation bubble simulations by Zeng et al. (Reference Zeng, Gonzalez-Avila, Dijkink, Koukouvinis, Gavaises and Ohl2018).
In each time step, the equations for 
$\alpha$, 
$\rho$, 
$\boldsymbol u$ and 
$p$ are solved. In the following, the notation 
$\alpha _1:=\alpha,\alpha _2:=1-\alpha$ is used. Since phase transitions are ignored, each component satisfies the continuity equation separately (2.1), i.e.
\begin{equation} \frac{\partial(\alpha_j\rho_j)}{\partial t}+\boldsymbol{\nabla}\boldsymbol{\cdot}(\alpha_j\rho_j\boldsymbol\phi)=0, \end{equation}
where 
$\boldsymbol \phi$ denotes the interpolation of 
$\boldsymbol u$ at the cell faces since 
$\boldsymbol u$ is only defined at the centre of each cell and 
$j=1,2$ represents the two fluid components. By using the definition of the compressibility, 
${\rm d}\rho _j=\psi _j\,{\rm d}p$, (3.1) yields
\begin{equation} \frac{\partial\alpha_1}{\partial t}+\boldsymbol{\nabla}\boldsymbol{\cdot}(\alpha_1\boldsymbol\phi)+\frac{\alpha_1}{\rho_1}\psi_1\frac{Dp}{Dt}+\boldsymbol{\nabla}\boldsymbol{\cdot}(\alpha_1\alpha_2\boldsymbol\phi_{r})=0. \end{equation}
The last term on the left-hand side was added to counteract the smearing of the fluid–fluid interface due to numerical diffusion. The vector 
$\boldsymbol \phi _{r}$ is defined as
\begin{equation} \boldsymbol\phi_{r}=c_\alpha\sum_i\frac{\vert\phi_i\vert}{\vert\boldsymbol A_i\vert}\frac{\boldsymbol{\nabla}\alpha_{1,i}}{\vert\boldsymbol{\nabla}\alpha_{1,i}\vert}, \end{equation}
with the parameter 
$c_\alpha \leq 1$ and the cell face vector 
$\boldsymbol A_i$. Solving (3.2) yields an updated 
$\alpha _1$ for the new time 
$t$. The partial derivatives are discretised using low order discretisation schemes like the Euler scheme for temporal and the upwind scheme for spatial derivatives. Numerical errors of this solution may result in 
$\alpha _1$ being slightly below 1 within the bulk of the liquid. Once such a cell is exposed to a low pressure, we wrongly observe the creation of new gas regions. We overcome this problem by a correction of 
$\alpha _i$, i.e. cells with a value of 
$\alpha _i>0.9999$ are clamped to 
$\alpha _i=1$. This threshold value was chosen to remove the wrongly created gas regions without significantly affecting the main bubble dynamics.
Next the Navier–Stokes equation (2.2) is solved to update the velocity field 
$\boldsymbol u$. Since gravitational effects are neglected, the volume force term disappears (
$\boldsymbol f=0$). The pressure field is computed iteratively, using the pressure-implicit method with splitting of operators (PISO) scheme. For that, the Navier–Stokes equation (2.2) is written as
\begin{equation} M\boldsymbol u={-}\boldsymbol{\nabla} p, \end{equation}
where 
$M$ is a known tensor. Now the diagonal elements of 
$M$ are defined as 
$A$ and another quantity 
$\boldsymbol H=(M-A)\,\boldsymbol u$ is defined. Then, (3.4) can be written as
\begin{equation} \boldsymbol{\nabla}({-}A^{{-}1}\boldsymbol{\nabla} p)=A^{{-}1}\boldsymbol H-\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol u. \end{equation}
This is solved to update the pressure field 
$p$ and from there, the velocity and density fields are corrected. Discretisation errors cause the bubble mass, which is calculated from 
$m=\sum _i^{cells}\alpha _{2,i}\,\rho _{2,i}\,V_i$, to fluctuate. To counteract this, the density is corrected as 
$\rho _{2,i}'=({m_0}/{m})\,\rho _{2,i}$ to keep the bubble mass constant, where 
$m_0$ is the initial bubble mass calculated at 
$t=0$, as seen in Koch et al. (Reference Koch, Lechner, Reuter, Köhler, Mettin and Lauterborn2016).
The updated 
$\boldsymbol u$ and 
$\rho$ are used to correct 
$p$, and are then corrected once again. Finally, the density field 
$\rho$ is updated using 
$\rho =\alpha _1\rho _1+\alpha _2\rho _2$, and the new compressibility fields 
$\psi _1$ and 
$\psi _2$ are obtained from
\begin{equation} \psi_j=\frac{1}{\gamma_j(p+B_j)} . \end{equation}3.2. Geometry
The geometry of interest is axisymmetric. It consists of two cylindrical reservoirs (each with a height of 5 mm and a radius of 5 mm) connected via a thin channel with a common axis of symmetry, as shown in figure 1. Both orifices of the channel are filleted (with a radius of curvature of 25 
$\mathrm {\mu }{\rm m}$) to avoid numerical instabilities from sharp edges. Because of the symmetry, only half of the geometry shown in figure 1 is calculated in the simulation. The flow equations are solved in the 
$x$–
$y$-plane, where the 
$y$-axis is the rotational axis of the system. This results in an effectively two-dimensional simulation, which greatly reduces the cost of computation.
Simulation geometry (not to scale) of a single bubble in the vicinity of a plate with thickness 
$l$ including bubble radius 
$R$ and stand-off distance 
$d$. The plate is perforated by a channel of radius 
$r_{c}$.
Within the OpenFOAM framework, the geometry is reduced to a thin wedge that is thinnest at the 
$y$-axis and is only divided into cells along the 
$x$- and 
$y$-directions.
The geometry is discretised into a square mesh, as seen in figure 2, which close to the location of the bubble is resolved with a cell width of 2.5 
$\mathrm {\mu }{\rm m}$. Outside of this region of interest, an unstructured mesh is used, where the cell size increases with distance from the bubble to approximately 100 
$\mathrm {\mu }{\rm m}$ width at the outer boundary of the geometry.
Computational mesh of the channel geometry with 
$r_{c}=50\,\mathrm {\mu }$m and 
$l=1$ mm. An unstructured mesh is used far from the bubble and at the filleted channel orifices. Close to the bubble and in the channel, a structured mesh is used.
3.3. Initial conditions
The simulations are started with a small spherical bubble seed on the axis of symmetry with 50 
$\mathrm {\mu }{\rm m}$ radius. The bubble interior is assigned 
$\alpha =0$ while the bubble exterior is assigned 
$\alpha =1$. The field 
$\alpha$ is then smoothed by solving the Poisson equation,
\begin{equation} \alpha'-4\times 10^{{-}11}\Delta\alpha'=\alpha , \end{equation}
where 
$\alpha '$ denotes the replacement of the initial field 
$\alpha$. This smoothing procedure reduces Rayleigh–Taylor instabilities during the early stage of rapid bubble expansion. The parameter is chosen to provide minimum smoothing but a numerically stable bubble surface during the early expansion phase.
The pressure field is set as 
$p=\alpha \,p_0+(1-\alpha )\,p_{g}$ with the atmospheric pressure 
$p_0$ and the initial gas pressure 
$p_{g}$. Here, 
$p_{g}$ is chosen such that the density of the gas is equal to the liquid density, 
$\rho _{g}=\rho _{l}$. Using this and the Tait equation (2.3) for the gas, a gas pressure of 
$p_{g}=16.88$ kBar is obtained. This is based on the assumption that for a laser generated bubble, the energy deposition by the laser pulse is much faster than the bubble expansion, so that the liquid seed vapourises almost instantaneously, before the bubble seed expands significantly. Initially, the velocity field is zero in the entire fluid domain.
The simulations were carried out as parallel processes using four CPU cores of an AMD Ryzen Threadripper 1900X at 3.8 GHz each with 64 GB of RAM. The time needed to run a simulation is roughly proportional to the square of the number of cells computed. For this work, the time taken ranged between 21 and 51 h to simulate 200 
$\mathrm {\mu }{\rm s}$ of the bubble dynamics after bubble initiation.
3.4. Boundary conditions
In the following, the directions 
$x$, 
$y$ and 
$z$ in the simulation results are considered synonymous with right, up and front, respectively. The leftmost boundary is imposed with the symmetryPlane boundary condition for all fields. Similarly, the boundary condition wedge is used for the front and back planes of the geometry. The fluid boundaries of the two large fluid regions located at the top, bottom and right in figure 2 represent an infinitely large fluid domain; therefore, their boundary conditions are 
$p=10\,1325$ Pa, 
$\boldsymbol {\nabla } u_i=0$ and 
$\boldsymbol {\nabla }\alpha =0$. The remaining boundary is the one separating the two fluid domains, which also bounds the channel that connects them. This is a rigid wall with a no-slip boundary condition, 
$\boldsymbol u=0$ and 
$\boldsymbol {\nabla } p=0$. There we also impose a boundary condition of 
$\alpha =1$. This can be justified to some extent by the experiments of Reuter & Kaiser (Reference Reuter and Kaiser2019), who found that a bubble collapsing very close to a wall in water does not touch the wall but is separated from it via a thin liquid film during its first collapse. Their experiments covered the stand-off distances 
$0.47<\gamma <1.07$, 
$\gamma ={d}/{R_{max}}$ here denoting the dimensionless stand-off parameter with the maximum bubble radius 
$R_{max}$ and the distance between the bubble centre and the wall 
$d$. In the present work, we assume this separation of the gas and the wall for the full range of 
$\gamma$ values studied. Whether this assumption holds true in an experimental setting may depend on the wettability and structure of the surface. If in such a setting the bubble came into contact with the wall, the contact angle would be expected to play a minor role on the bubble dynamics. Thus, we would still expect the formation of the main jet causing the pumping effect.
4. Experimental setup
The experiments use a laser-induced bubble in front of a perforated boundary. A schematic representation of the setup is shown in figure 3. It consists of a glass tank with dimensions 
$55\,{\rm mm} \times 55\,{\rm mm} \times 65\,{\rm mm}$ filled with deionised water. A microscope objective (Mitutoyo 
$\times\,50$ Plan Apochromat Objective, 436–656 nm, 0.55 NA, 13 mm WD) is sealed water-tight and incorporated into the bottom of the tank. This objective focuses a Nd:YAG laser pulse (Litron Nano SG-100-2; 
$\lambda = 532\,{\rm nm}$; 
${\rm FWHM} = 6\,{\rm ns}$, pulse energy 
$\approx$ 5 mJ) into the water to seed the bubble. A smaller L-shaped cuvette that accommodates a plate with a micro-perforation is submerged into the glass tank connected to a motorised three-axis stage. The plate is made out of a flexible polymeric material used for printed circuit boards (supplier: PCBWay). It has dimensions of 
$16\,{\rm mm} \times 12\,{\rm mm} \times 170$ 
$\mathrm {\mu }{\rm m}$. The perforation is a cylindrical drilled hole of 195 
$\mathrm {\mu }{\rm m}$ in diameter (measured with a light microscope). The bubble is produced axially in front of the perforation which is confirmed by a high-speed camera with a perpendicular view on the plate surface (Photron FASTCAM MINI AX200). To avoid the generation of spurious bubbles on the plate, which can happen as the laser energy is not entirely absorbed in the plasma, the laser is focused in parallel to the plate. This implies a partial blocking of the focusing cone that is dependent on the stand-off distance. To assure constant maximum bubble size over all stand-off distances, half of the laser beam is blocked at the input aperture side of the objective.
Schematic experimental setup (not to scale): (a) optical path for bubble generation as well as Schlieren imaging through the glass tank; (b) cuvette with a perforated plate in side view submerged in the glass tank and position of bubble generation.
To visualise the pumping effect of the bubble, the L-shaped cuvette is filled with an aqueous 3 wt% NaCl solution which results in optical Schlieren at the mixing interface. Schlieren high-speed imaging is implemented to visualise the transport of two nearly identical liquids. For that, the water tank is illuminated by a collimated pulsed laser beam (Ekspla FemtoLux 3; 
$\lambda$ = 515 nm; 1 MHz) parallel to the plate. After passing the tank, the light gets focused by an objective where the undeflected part is blocked by a knife edge and the deflected light is directed into a second high-speed camera (Shimadzu Hyper Vision HPV-X2; camera 1 in figure 3, operated at 500 000 frames per second; shutter opening time 1 
$\mathrm {\mu }{\rm s}$; effective exposure time 0.3 ps; 
$7.8\,{\mathrm {\mu }{\rm m}}\,{\text {pixel}}^{-1}$). The imaging is also sensitive to density changes in the liquid from shock waves which are emitted during the creation and the collapse of the bubble and reflected at the boundaries.
5. Results
We start by demonstrating the effect of the presence of the channel in the rigid boundary on the bubble dynamics via simulations.
5.1. Jetting through the perforated boundary
Figure 4 compares the axisymmetric fluid dynamics of a bubble expanding and collapsing close to a rigid wall with an identical bubble close to a perforated wall. The bubble is seeded at a stand-off distance of 500 
$\mathrm {\mu }{\rm m}$ and reaches a maximum radius of approximately 475 
$\mathrm {\mu }{\rm m}$. This gives a dimensionless stand-off distance of 
$\gamma =1.05$. The cylindrical channel has a radius of 50 
$\mathrm {\mu }{\rm m}$ and a length of 1 mm, which is the same as the thickness of the wall separating the two fluid domains. In both cases, the bubble expands and then collapses, creating a jet towards the wall. As a result, the bubble re-expands as a torus close to the wall.
Simulations of a cavitation bubble close to a rigid boundary with stand-off parameter 
$\gamma =1.05$, comparison of the flow dynamics for a boundary without (a–d) and with a channel below the bubble (e–h). The left side of each frame shows the bubble gas in red and water in blue, the right side shows the velocity field in the liquid.
There are small but noticeable differences between the bubble dynamics in the two geometries. In the case with the hole, during the shrinking phase, a small indentation on the lower side of the bubble develops, which is not present in the continuous wall case. This feature is caused by the axial inflow from the opposing fluid domain through the channel and onto the lower bubble wall. The flow deforming the bubble is driven by the pressure difference during the shrinking phase of the bubble, i.e. the low pressure in the bubble and the atmospheric pressure far from the bubble below the boundary.
Overall, both bubbles form a jet starting from the wall-distant pole of the bubble. After this jet has pierced the lower bubble wall, in the continuous wall case, we find a stagnation point at the wall on the axis of symmetry. Further outside, a radially spreading flow along the continuous wall is formed. In contrast, in the perforated wall case, the stagnation point flow is lost and the liquid flows through the hole in the boundary. Viscous stresses oppose the flow through the boundary and some of the jet flow still feeds into the radially spreading flow along the wall. Overall, we find that for these particular chosen parameters, a rather small effect of the perforation on the familiar bubble shape or timing of the bubble dynamics during expansion and collapse is observed, albeit there is a fluid transport occurring through the plate.
To quantify this pumping effect of the bubble, the liquid flow velocity 
$u_y$ through the channel outlet, i.e. on the plate surface facing away from the bubble, is measured (just above the rounded edge, see figure 1) and averaged over the channel cross-section as 
$u_y=({1}/{{\rm \pi} r_{c}^2})\int _0^{r_{c}}\alpha _1\boldsymbol u_y(r)\boldsymbol {\cdot}\boldsymbol e_y\boldsymbol {\cdot }2{\rm \pi} r\,{\rm d}r$. The result is plotted in figure 5(a). Here, a positive sign of 
$u_y$ indicates a flow from the bottom water reservoir to the top and a negative sign, a flow in the downward direction. The red line in figure 5 depicts the volume equivalent bubble radius 
$R$ to help the reader to link the bubble dynamics to the channel flow. Figure 5(a) shows that during bubble expansion, the liquid is initially accelerated downwards to approximately 
$u_y= -10\,{\text {m}}\,{\text {s}}^{-1}$ before the velocity reduces to approximately 0 at the time of maximum bubble expansion (
$t=50$ 
$\mathrm {\mu }{\rm s}$). Then, rather symmetrically, the liquid is transported in the opposite direction and slowly accelerated during bubble shrinkage. Just prior to the collapse, a rapid acceleration sets in, reverting the flow direction and accelerating it to approximately 
$u_y= -18\,{\text {m}}\,{\text {s}}^{-1}$ at 
$t=105$ 
$\mathrm {\mu }{\rm s}$. At this time, the jet has developed and pushes the liquid through the channel. Examining the bubble dynamics between 
$t=100$ 
$\mathrm {\mu }{\rm s}$ and 
$t=200$ 
$\mathrm {\mu }{\rm s}$, we find a second oscillation cycle of the bubble with a reduced equivalent bubble radius. We want to remind the reader that the bubble is not spherical but has now transformed into a toroidal shape. While the fluid velocity 
$u_y$ is rather symmetrical for most of the expansion and shrinkage of the first bubble oscillation, i.e. downward during bubble expansion and upward during shrinkage, we see a different feature during the second oscillation cycle. Here the velocity 
$u_y$ remains negative for most of the second oscillation cycle and the flow reverses only shortly before the second collapse. As a consequence, we can expect that a net downward flow occurs during this part of the bubble oscillation. This expectation is confirmed in figure 5(b). It plots the liquid volume 
$V$ transported through the channel as a function of time, which is calculated as 
$V(t)={\rm \pi} r_{c}^2\int _0^tu_y(t')\,{\rm d}t'$. Again, a negative sign stands for a net liquid transport downwards through the channel. Figure 5(b) shows a roughly symmetrical flow during the first oscillation period as discussed above, pumping liquid downwards during bubble expansion and upwards during bubble shrinkage. The cumulative volume flow 
$V$ remains negative for this entire oscillation cycle. Upon bubble collapse, the developed jet creates a strong downward flow through the channel, which lasts until the maximum expansion of the second bubble oscillation cycle and reaches up to 7 nL at 170 
$\mathrm {\mu }{\rm s}$, before the flow is reversed and the net amount of pumped liquid is slightly reduced. It is instructive to compare the pumped liquid with the maximum bubble volume, which is approximately 500 nL, and with the volume of the channel, which is approximately 8 nL.
(a) Velocity 
$u_y$ at channel exit (near the solid surface facing away from the bubble) and (b) liquid volume 
$V(t)$ transported through the channel, each overlaid with the bubble radius 
$R$ to show the correlation between the bubble dynamics and the flow through the channel.
5.2. Variation of parameters
The present fluid transport problem is rich in the choice of parameters, e.g. the geometry of the channel (length 
$l$, radius 
$r_{c}$, shape), the stand-off distance 
$d$ of the bubble and the properties of the liquid, to name a few. In this section, we vary some of these parameters individually and document their effect on the pumped liquid through the channel. The flow behaviour in the channel in tandem with the evolution of the bubble radius 
$R$ observed in figure 5 happens in a similar fashion in all the following cases studied, meaning that the first bubble collapse is accompanied by the formation of a microjet towards the perforated plate and a sudden acceleration to a high magnitude negative flow velocity 
$u_y$ in the channel.
5.2.1. Variation of the channel length
The three channel lengths, 
$l=200$ 
$\mathrm {\mu }{\rm m}$ (
$=0.42~R_{max}$), 
$l=400$ 
$\mathrm {\mu }{\rm m}$ and 
$l=1000$ 
$\mathrm {\mu }{\rm m}$ (
$=2.11R_{max}$), are studied. The main difference caused by this change is the viscous flow resistance of the channel flow. As a result, for shorter channels, we observe higher flow velocities 
$u_y$ (see figure 6(a) at 
$t=110$ 
$\mathrm {\mu }{\rm s}$), yet the duration of the flow driven by the liquid jet is also shorter. This is because after the first collapse, the inertia of the liquid jet transports gaseous fragments of the bubble along and through the channel. Once the bubble content reaches the lower end of the channel (where the liquid flow is measured), the liquid flow rate abruptly diminishes, which happens at approximately 
$t=110$ 
$\mathrm {\mu }{\rm s}$ for 
$l=200$ 
$\mathrm {\mu }{\rm m}$ and 
$t=120$ 
$\mathrm {\mu }{\rm s}$ for 
$l=400$ 
$\mathrm {\mu }{\rm m}$. Also, for shorter channels, a stronger backflow towards the collapsing bubble is created (approximately 
$t=150$ 
$\mathrm {\mu }{\rm s}$), which affects and reduces the net flow further. This is due to the second bubble collapse, when the channel begins to fill with liquid from the bottom again. During the third expansion (starting at approximately 
$t=180$ 
$\mathrm {\mu }{\rm s}$), the liquid is pushed downwards through the channel again. For 
$l=200$ 
$\mathrm {\mu }{\rm m}$, similar amounts of liquid are being moved up and down through the channel, creating a roughly periodic pattern in the volume transport curve 
$V(t)$ in figure 6(b). The bubble dynamics for short channels is discussed in greater detail during the comparison with the experiments, see § 6.2. Since the transported liquid volume 
$V$ increases with increasing channel length 
$l$, one may suspect that the transported volume scales with the channel volume (
$\sim l$). Therefore, we plotted the transported liquid volume normalised by the channel volume, 
$V_{n}(t)$, in figure 6(c). This reveals that 
$V(t)$ does not scale with the channel length 
$l$ and relatively smaller liquid volumes are pumped for longer channels.
Study of the liquid transport as a function of the length of the channel 
$l$ for 
$r_{c}=112.5$ 
$\mathrm {\mu }{\rm m}$, 
$d=500$ 
$\mathrm {\mu }{\rm m}$, 
$R_{max}=475$ 
$\mathrm {\mu }{\rm m}$: (a) velocity 
$u_y(t)$ at channel exit; (b) liquid volume 
$V(t)$ transported through the channel and (c) transported liquid volume normalised by the channel volume 
$V_{n}(t)$.
5.2.2. Variation of the channel radius
Next the radius of the channel 
$r_{c}$ is varied between 25 
$\mathrm {\mu }{\rm m}$ (or 
$0.05~R_{max}$) and 200 
$\mathrm {\mu }{\rm m}$ (or 
$0.42~R_{max}$). Figure 7(a) shows that for thinner channels, the liquid jet causes a higher velocity 
$u_y$, yet for a shorter duration compared to the wider channels (see 
$100\,\mathrm {\mu }{\rm s}\le t\le 150\,\mathrm {\mu }{\rm s}$). Similar to § 5.2.1, the faster decay in fluid velocity can be attributed to the stronger viscous drag in the thinner channels. For wider channels, an increasingly larger part of the bubble moves into the channel after collapse. In the case of 
$r_{c}=200$ 
$\mathrm {\mu }{\rm m}$, the bubble collapses downwards after the second expansion, entirely entering the channel. This stands in contrast to the previous cases, where the second bubble collapse is directed towards the axis of symmetry. The larger cross-sections allow a much larger liquid flux to pass through them compared to thinner channels (see figure 7b). If the flow velocity 
$u_y$ is almost unaffected by the channel width, which it is at least until the bubble collapse, we expect the transported volume 
$V$ to scale with the channel volume (
$\sim r_{c}^2$). This is indeed supported by figure 7(c), where the normalised transported liquid volume 
$V_{n}$ during the first bubble oscillation is approximately the same for each 
$r_{c}$. Yet, after jetting, 
$V_{n}$ is affected by the channel radius 
$r_{c}$ and reaches the largest magnitude value for 
$r_{c}=50$ 
$\mathrm {\mu }{\rm m}$.
Study of the liquid transport as a function of the radius of the channel 
$r_{c}$ for 
$l=1000$ 
$\mathrm {\mu }{\rm m}$, 
$d=500$ 
$\mathrm {\mu }{\rm m}$, 
$R_{max}=475$ 
$\mathrm {\mu }{\rm m}$: (a) velocity 
$u_y(t)$ at channel exit; (b) liquid volume 
$V(t)$ transported through the channel and (c) transported liquid volume normalised by the channel volume 
$V_{n}(t)$.
5.2.3. Variation of the channel shape
The channel shape is varied by fixing the radius at the upper end of the channel at 
$r_{c,upper}=50$ 
$\mathrm {\mu }{\rm m}$ and changing the radius at the lower end to 
$r_{c,lower}=25$ 
$\mathrm {\mu }{\rm m}$ to obtain a narrowing channel, and 
$r_{c,lower}=100$ 
$\mathrm {\mu }{\rm m}$ for a widening channel. The ends are connected via a straight line, creating channels of a truncated cone geometry. Figure 8(d) schematically depicts these three channel shapes. In the narrowing channel, a higher absolute pumping velocity is reached, as seen in figure 8(a). For the widening channel, a higher amount of fluid is transported through them (while the channel volume also increases), as seen in figure 8(b), but there is also a stronger backflow that acts against the net amount of fluid pumped downwards through the channel, see figure 8(b) at 
$150\,\mathrm {\mu }{\rm s}\le t\le 200\,\mathrm {\mu }{\rm s}$. Plotting the transported liquid volume divided by the respective channel volumes, 
$V_{n}$, reveals that the cylindrical channel transports the largest absolute volume over the first 200 
$\mathrm {\mu }{\rm s}$ (see figure 8c).
Study of the liquid transport as a function of the shape of the channel for 
$r_{c,upper}=50$ 
$\mathrm {\mu }{\rm m}$, 
$l=1000$ 
$\mathrm {\mu }{\rm m}$, 
$d=500$ 
$\mathrm {\mu }{\rm m}$, 
$R_{max}=475$ 
$\mathrm {\mu }{\rm m}$: (a) velocity 
$u_y(t)$ at channel exit; (b) liquid volume 
$V(t)$ transported through the channel; (c) transported liquid volume normalised by the channel volume 
$V_{n}(t)$ and (d) schematic of channel shapes used.
5.2.4. Variation of the liquid viscosity
Next we increase the viscosity of the liquid from that of water (
$\mu =1$ mPa s) by 1 and 2 orders of magnitude, respectively, while keeping the gas viscosity constant. Figure 9 summarises the results and reveals the expected drastic decrease of the absolute pumping velocity 
$u_y$ and pumped volume 
$V$, as well as a stronger damping of the flow in the channel. Interestingly, the backflow towards the bubble is strongly reduced for 
$\mu =100$ mPa s. This can be attributed to the strong viscous damping of the microjet. Unlike in the cases with lower viscosity, the jet drags no bubble gas into the channel that would create a backflow during the collapse later. The viscous damping also causes the bubble to have a 37 % larger collapse time during its first oscillation and reach a 42 % smaller rebound volume upon its second expansion than in the case of 
$\mu =1$ mPa s. In Appendix B, we estimate the total pumped volume as a function of the viscosity (B5) assuming a dissipating Hagen–Poiseuille flow. From that consideration, neglecting the effect of viscosity on the jetting, the pumped volume can be considered to be proportional to 
$r_{c}^4$ and 
$\mu ^{-1}$.
Study of the liquid transport as a function of the liquid viscosity 
$\mu$ for 
$r_{c}=50$ 
$\mathrm {\mu }{\rm m}$, 
$l=1000$ 
$\mathrm {\mu }{\rm m}$, 
$d=500$ 
$\mathrm {\mu }{\rm m}$, 
$R_{max}=475$ 
$\mathrm {\mu }{\rm m}$: (a) velocity 
$u_y(t)$ at channel exit and (b) liquid volume 
$V(t)$ transported through the channel.
5.2.5. Variation of the stand-off distance
Increasing the stand-off distance of the bubble seed from 
$d= -50$ 
$\mathrm {\mu }{\rm m}$ (
$\gamma = -0.11$) up to 
$d=750$ 
$\mathrm {\mu }{\rm m}$ (
$\gamma =1.58$) reveals a general trend that the velocity 
$u_y$ and the amount of liquid pumped 
$V$ decreases with the distance (see figure 10a). For the smallest distances 
$d$, considerably higher velocities of up to 
$u_y= -55\,{\text {m}}\,{\text {s}}^{-1}$ at 
$d= -50$ 
$\mathrm {\mu }{\rm m}$ are achieved, yet these distances also result in a stronger backflow during the second bubble oscillation period. This can be partly attributed to the bubble expanding into the channel after jet formation, pushing liquid down through it and dragging liquid up again during the subsequent collapse. Overall, the highest pumped volume through the channel 
$V$ is achieved for the smallest distances 
$d$ (see figure 10b). For large distances, i.e. 
$d=750$ 
$\mathrm {\mu }{\rm m}$, a net pumped volume of nearly zero is observed just prior to the first bubble collapse. This behaviour is expected for an inviscid, spherically oscillating bubble that does not translate, i.e. one that can only be fulfilled for large stand-off distances. Interestingly, the decrease in peak velocity and pumped volume up to the second collapse is not a monotonic function of 
$d$ but contains a local maximum around 
$d=500$ 
$\mathrm {\mu }{\rm m}$ (
$\gamma =1.05$, see figure 10a at 
$t=105$ 
$\mathrm {\mu }{\rm s}$ and 10b at 
$t=170$ 
$\mathrm {\mu }{\rm s}$). The bubble dynamics changes significantly as a function of the stand-off distance 
$d$. As the bubble is generated closer to the boundary, it becomes more deformed from a spherical shape until it obtains an almost hemispherical shape at maximum expansion with a small protrusion reaching into the perforation in the boundary. In the case of 
$d=100$ 
$\mathrm {\mu }{\rm m}$, no pronounced jetting is visible, instead an annular flow develops in the late stage of an almost hemispherical collapse. This flow interacting with the late microjet from above compete and mitigate jetting. For 
$d=0$, the bubble develops a considerably thinner jet that, due to its higher speed, pulls along gas from the bubble into and through the channel during bubble re-expansion. This change in jetting behaviour is discussed in greater detail in the following section. Figure 10(c) shows that for 
$d<0$, the velocity peak is reached at a later time, which can be linked to the bubble expanding further into the channel. There, the gas phase collapses only after a few microseconds after the collapse of the main bubble (see figure 11 at 103.0 
$\mathrm {\mu }{\rm s}$).
Study of the liquid transport as a function of the stand-off distance 
$d$ for 
$r_{c}=50$ 
$\mathrm {\mu }{\rm m}$, 
$l=1000$ 
$\mathrm {\mu }{\rm m}$, 
$R_{max}=475$ 
$\mathrm {\mu }{\rm m}$: (a) velocity 
$u_y(t)$ at channel exit; (b) liquid volume 
$V(t)$ transported through the channel and (c) velocity 
$u_y(t)$ at channel exit after jetting for 
$d= -50\ldots 100$ 
$\mathrm {\mu }{\rm m}$.
Simulation of cavitation bubble dynamics in the perforation of a rigid plate in the plane of the upper wall of the plate (
$\gamma =0$). The left side of each frame shows the bubble gas in red and water in blue, the right side shows the velocity field in the liquid.
6. Selected pumping regimes
After studying the pumping effect and the influence of parameters thereon in the preceding sections, we now examine the effect of the channel on the bubble dynamics for two distinct regimes in greater detail.
6.1. Needle jet regime: small stand-off distance and long channel
Figure 4 reveals that for 
$\gamma \approx 1$, the shape of the bubble near a plane, unperforated boundary is rather similar to the shape observed near one which is perforated with a long and slender hole (
$l=1$ mm, 
$r_{c}=50$ 
$\mathrm {\mu }{\rm m}$). When the bubble is generated very close to an unperforated boundary, the jetting behaviour of the bubble transitions from the commonly known rather wide jets to a considerably thinner and faster jet, termed fast, thin jet (see Lechner et al. Reference Lechner, Lauterborn, Koch and Mettin2020) or needle jet (see Reuter & Ohl Reference Reuter and Ohl2021) that can reach a velocity with an order of magnitude of 
$1000\,{\text {m}}\,{\text {s}}^{-1}$, as revealed in simulations and experiments. We want to address the question whether this needle jet may also occur at a perforated boundary. With a hole in the boundary, the bubble can be initiated even closer and even within the channel, offering the possibility to probe negative stand-off distances. Figure 11 shows a bubble initiated on the boundary (
$\gamma =0$), where the bubble penetrates into the perforation during expansion. There, the upper part of the bubble obtains an approximately hemispherical shape and the lower takes on that of a cylinder filling the cross-section of the hole. As it starts to collapse, a circular kink forms at the upper part of the bubble (
$t=100.0$ 
$\mathrm {\mu }{\rm s}$). As the bubble shrinks, the kink remains roughly at the same distance from the wall, yet it converges in diameter (see 
$t=102.0$ 
$\mathrm {\mu }{\rm s}$ and 
$t=102.2$ 
$\mathrm {\mu }{\rm s}$ in figure 11). This has been connected with a wall-parallel radially convergent inflow focusing on the axis of symmetry (Lechner et al. Reference Lechner, Lauterborn, Koch and Mettin2020). Upon impact, this focused flow is responsible for the acceleration of liquid in the shape of a needle to high speeds along the axis of symmetry (see 
$t=102.3$ 
$\mathrm {\mu }{\rm s}$, 
$t=102.5$ 
$\mathrm {\mu }{\rm s}$ and 
$t=103.0$ 
$\mathrm {\mu }{\rm s}$ in figure 11). Please note that in addition to the prominent downward needle-like jet flow, in the perforation, a flow upwards is also created that competes against the spherically convergent main flow. The very thin downwards jet pierces the bubble with a velocity of 
$367\,{\text {m}}\,{\text {s}}^{-1}$, whereas the commonly known jet flow of a bubble at larger stand-off distances does not exceed approximately 
$100\,{\text {m}}\,{\text {s}}^{-1}$. It is important to note that the velocity of the fast jet is strongly dependent on the grid size, as reported by Lechner et al. (Reference Lechner, Lauterborn, Koch and Mettin2020), and thus may here be underestimated. The present observation shown in figure 11 resembles the bubble dynamics reported by Lechner et al. (Reference Lechner, Lauterborn, Koch and Mettin2020) and confirms that a needle jet can occur in the presence of a perforated boundary too.
The emergence of the needle jet phenomenon with decreasing 
$\gamma$ is easily revealed by plotting the maximum of the absolute velocity of the entire liquid domain, 
$u_{j}$, and the radius of the jet 
$r_{j}$ in figure 12(a,b). Following the curves from large 
$\gamma$-values to smaller 
$\gamma$ in figure 12(a), the velocity of the liquid jet 
$u_{j}$ decreases slowly at first from approximately 
$100\,{\text {m}}\,{\text {s}}^{-1}$ and then sharply rises to values of several 
$100\,{\text {m}}\,{\text {s}}^{-1}$, showing the lowest value at 
$\gamma =0.3$.
Study of the transition between jet and needle jet as a function of the normalised stand-off parameter 
$\gamma$: (a) jet velocity 
$u_{j}(\gamma )$; (b) jet radius 
$r_{j}(\gamma )$ and (c) maximum flow velocity at the channel exit 
$u_y(\gamma )$.
The radius of the jet 
$r_{j}$ is defined as the radius of the widest column along the axis of symmetry that is entirely filled with liquid after the jet has pierced the bubble (see figure 12b). This jet radius 
$r_{j}$ reveals a maximum at 
$\gamma =0.5$ but sharply drops to a very low value for 
$\gamma \lesssim 0.2$. Here, the jet becomes so thin that it is not fully resolved in the present simulations and splits into small droplets without fully penetrating the bubble. Then, a jet radius of 
$r_{j}=0$ is plotted in figure 12(b).
The velocity produced at the lower end of the channel 
$u_y$ (see figure 12c) shows a pronounced rise for 
$\gamma <0.5$; however, it is not as abrupt as during the formation of the needle jet. Judging from these results, the needle jet occurs at 
$\gamma \lesssim 0.2$, while the standard jet occurs at 
$\gamma \gtrsim 0.5$, with a transition region between 
$\gamma =0.2$ and 
$\gamma =0.5$, where no needle jet occurs, but the jetting behaviour changes.
6.2. Reverse jet regime: small stand-off distance and short channel
Next we investigate the bubble dynamics close to a short channel (
$l=170$ 
$\mathrm {\mu }{\rm m}$). We chose this regime because in the experiments, both sides of the plate can be captured by a single camera in the case of a thin plate. A second motivation was the report of a jet directed away from the wall by Cui et al. (Reference Cui, Zhang, Wang and Wang2013) that formed only near thin perforated plates which is also absent for unperforated walls. In this regime, we provide two comparisons between the simulated flow and the experiment in figure 13 with a stand-off distance of 
$d=108$ 
$\mathrm {\mu }{\rm m}$ and figure 14 with 
$d=202$ 
$\mathrm {\mu }{\rm m}$. In these figures, on the left and on the right, the simulated bubble and the experiment are depicted, respectively. We present not only the shape but also the transported liquid using a digital ink map in the simulations to visualise the net fluid transport from the unsteady flow field of cavitation bubbles as introduced by Reuter et al. (Reference Reuter, Gonzalez-Avila, Mettin and Ohl2017). This passive tracer field of different colours is advected with the simulated flow field. Initially, the liquid in the upper reservoir is coloured black and that in the lower with a bright grey. Additionally, the liquid in the perforation is coloured continuously from the top to the bottom from black to red to yellow to bright grey at the bottom. Adding the ink map, we can not only depict the bubble shape but also the origin of liquid at a later stage during the pumping. For example, the black blob in the lower reservoir in the simulation part of figure 13(a) is liquid that has been pumped from the upper into the lower reservoir.
Comparison between simulation (left) and experiment (right) of a cavitation bubble of maximum radius 
$R_{max}\approx 460$ 
$\mathrm {\mu }{\rm m}$ at distance 
$d=108$ 
$\mathrm {\mu }{\rm m}$ from a perforated wall, 
$l=170$ 
$\mathrm {\mu }{\rm m}$, 
$r_{c}=97.5$ 
$\mathrm {\mu }{\rm m}$. Bubble is shown in red, ink map of the liquid above the channel in black, liquid below the channel in white.
Comparison between simulation (left) and experiment (right) of a cavitation bubble of maximum radius 
$R_{max}\approx 460$ 
$\mathrm {\mu }{\rm m}$ at distance 
$d=202$ 
$\mathrm {\mu }{\rm m}$ from a perforated wall, 
$l=170$ 
$\mathrm {\mu }{\rm m}$, 
$r_{c}=97.5$ 
$\mathrm {\mu }{\rm m}$. Bubble is shown in red, ink map of the liquid above the channel in black, liquid below the channel in bright grey.
The experiment uses the higher index of refraction of the aqueous NaCl solution to visualise liquid transport via Schlieren imaging. This allows for contrasted high-speed recordings in regions of liquid entrainment. Note that due to the optical configuration, the liquid leaving the channel at the lower side can only be observed once it has propagated approximately 310 
$\mathrm {\mu }{\rm m}$ downwards into the bulk.
We start the discussion with the simulated bubble dynamics for a rather small stand-off distance of 
$d=108$ 
$\mathrm {\mu }{\rm m}$ (
$\gamma =0.23$), as shown in figure 13. In this regime, in the case of an unperforated boundary, a needle jet is observed (see § 6.1) as well as when the boundary is perforated with a long channel. However, in the presence of a short channel, different dynamics is observed. During early bubble expansion (
$t=3$ 
$\mathrm {\mu }{\rm s}$), some of the liquid from the upper reservoir is pushed through the hole into the lower reservoir. This liquid is visible as a small black and coloured blob below the boundary. During the later expansion of the bubble (
$t=21$ 
$\mathrm {\mu }{\rm s}$), a small part of the bubble invades through the hole into the lower reservoir. The two parts of the bubble are now exposed to different flow fields. While the upper mostly hemispherical part of the bubble continuously expands to a maximum size at 
$t=43$ 
$\mathrm {\mu }{\rm s}$, the lower part pinches off from the main bubble near the exit of the hole. The stagnation pressure from this pinch-off drives a liquid jet back through the hole into the upper part of the bubble (see 
$t=61$ 
$\mathrm {\mu }{\rm s}$ in figure 13). This reverse jet then impacts at the apex of the upper bubble and eventually injects liquid from the lower reservoir into the upper one at 
$t=79$ 
$\mathrm {\mu }{\rm s}$. A careful inspection reveals that the upper hemispherically shaped bubble develops a kink approximately 100 
$\mathrm {\mu }{\rm m}$ from the surface, starting from 
$t=73$ 
$\mathrm {\mu }{\rm s}$. The location of the kink converges towards the axis of symmetry at 
$t=79$ 
$\mathrm {\mu }{\rm s}$. Yet, the reverse jet prevents a further convergence of this annular flow towards the axis of symmetry. The main bubble collapses at 
$t=94$ 
$\mathrm {\mu }{\rm s}$, while the pinched-off bubble in the lower reservoir already collapses at approximately 
$t=73$ 
$\mathrm {\mu }{\rm s}$. The pinch-off (
$t=43$ 
$\mathrm {\mu }{\rm s}$) of the bubble protruding through the plate creates a complex mixing flow below with vortices and bubble fragments transported deeper into the lower reservoir. Overall, we find a good agreement between experiments and simulations. It is important to note that the bubble in the simulations collapses earlier than in the experiments. While we have no definite proof, a likely explanation is the elastic deformation of the boundary seen in the experiments and not accounted for in the simulation.
A comparison for a larger stand-off distance of 
$d=202$ 
$\mathrm {\mu }{\rm m}$ is shown in figure 14 (
$\gamma =0.44$). This results in a decreased invasion of the bubble into the lower reservoir during the bubble expansion as compared to that in figure 13. Consequently, a smaller part of the protuberant bubble is pinched off below the wall in the range of 
$30\,\mathrm {\mu }{\rm s}< t<40\,\mathrm {\mu }{\rm s}$. As a consequence, the reverse jet forms a thinner tip that partly breaks up into small droplets at 
$t=50$ 
$\mathrm {\mu }{\rm s}$. The vortex flow created upon pinch-off below the wall is more pronounced, which is likely due to a weaker disturbance of the flow by oscillating gas fragments as compared to figure 13.
Further simulations using the same geometry as in figures 13 and 14 reveal the dependence of the pumping behaviour on the stand-off distance 
$d$ in figure 15. The backflow is clearly visible in the channel velocity 
$u_y$ as a positive spike occurring prior to the bubble collapse at 
$t\approx 100$ 
$\mathrm {\mu }{\rm s}$ in figure 15(a). This stronger backflow is mostly due to the rather small channel length (see also figure 6). The highest amount of pumped liquid can be achieved at 
$d=300$ 
$\mathrm {\mu }{\rm m}$ or 
$\gamma =0.63$ (see figure 15b), which is in contrast to the 
$\gamma$-dependence of the amount of pumped liquid for a longer channel reported in figure 10.
Study of the liquid transport as a function of the stand-off distance 
$d$ for 
$r_{c}=97.5$ 
$\mathrm {\mu }{\rm m}$, 
$l=170$ 
$\mathrm {\mu }{\rm m}$, 
$R_{max}=475$ 
$\mathrm {\mu }{\rm m}$: (a) velocity 
$u_y(t)$ at channel exit and (b) liquid volume 
$V(t)$ transported through the channel.
7. Discussion and conclusion
The simulations and experiments in this work present the flow through a perforation in a thin rigid plate produced by a cavitation bubble collapsing in front of the perforated plate. The effect of several geometric and liquid parameters are studied and overall reveal clear trends on the transport of liquid from the bubble side to the opposite plate side, i.e. the pumping of liquid. A longer channel increases the flow resistance and as a result reduces the liquid exchange between the two fluid domains, i.e. not only the pumping of fluid but also the backflow. For increasingly longer channels, the flow increasingly resembles that of a bubble close to an unperforated wall. Mixing of fluids rather than pumping can be achieved with shorter channels. If a part of the bubble penetrates through the perforation during bubble expansion, that part collapses earlier than the main body of the bubble, resulting in an early jetting flow from the lower fluid domain (see figure 6). We have termed the flow as reverse jet and confirmed it in experiments (see figures 13 and 14). The experiments use a Schlieren technique to reveal the mixing of rather similar liquids, i.e. water and an aqueous NaCl solution. While for the mixing of two liquids it may find applications, for the unidirectional pumping it should be reduced. This can be done by increasing the channel length such that the bubble does not expand beyond the lower end of the channel.
A straight cylindrical channel results in larger amounts of liquid being pumped as compared to simple sloped channels. Yet we expect that others geometries such that of a Tesla valve (see Forster et al. Reference Forster, Bardell, Afromowitz, Sharma and Blanchard1995) may reduce the backflow and increase the pumped volume. An optimisation of the geometry is open for future studies. Even without conducting a study, an optimal value of the channel width 
$r_{c}$ for the highest amount of pumped liquid is to be expected, since for the extreme cases of 
$r_{c}\rightarrow 0$ and 
$r_{c}\rightarrow \infty$, no liquid pumping will occur, i.e. non-perforated wall and spherical bubble collapse with an absence of jetting, respectively. This has been confirmed and narrowed down in the present study. For the regular jet, the jet radius 
$r_{j}$ is in the range of 
$0.1\,\ldots \,0.5R_{max}$ (see figure 12). Figure 7 shows that for 
$r_{c}=50$ 
$\mathrm {\mu }{\rm m}$ (
$\approx 0.1R_{max}$), overall, the highest flow velocities through the channel are reached and the amount of liquid pumped relative to the channel volume reaches a maximum. Thus an optimum in pumping is achieved when the channel radius is similar to the jet radius. Varying the stand-off distance 
$d$ of the bubble near a long channel (
$l=1$ mm) reveals two distinct regimes of strong pumping behaviour: one being around 
$\gamma =1$, featuring the typical jet, the other at very small stand-off distances, where the needle jet is observed. This is very similar to the continuous rigid boundary for sufficiently small stand-off distances.
Since the occurrence of the three regimes reverse jet, needle jet and standard jet (jetting without the former two particular jets) depends foremost on the stand-off distance 
$d$ and the channel length 
$l$, we summarise this in the regime map in figure 16. It shows the type of jet formation as a function of the normalised stand-off distance 
$\gamma$ and the normalised channel length 
$l^*={l}/{R_{max}}$ and is collected from simulations and experiments. The reverse jet occurs once a part of the expanding bubble reaches the lower end of the channel. A rough criterion for that to occur is 
$R_{max}\gtrsim d+l\Leftrightarrow l^*\lesssim 1-\gamma$, the dashed diagonal line shows 
$l^*=1-\gamma$. Thus, as this line is approached from the standard jet regime, an increasingly stronger backflow occurs. The transition region between the standard jet and the reverse jet spans nearly the entire regime map shown in figure 16. Only for the case 
$l^*=2.11$, 
$\gamma =1.58$ (data point at the top right of figure 16), the bubble is seeded so far away from the boundary that we cannot detect an effect of the channel on the bubble shape, i.e. the indentation in the lower bubble wall as seen in figure 4(e–h) is not formed. The needle jet is only found for 
$\gamma \lesssim 0.25$ and sufficiently long channels. This region is indicated with the horizontal line in figure 16). For very short channels, the reverse jet and not the needle jet occurs. This is caused by the fact that the reverse jet develops earlier than the main collapse and therefore prevents the annular flow focusing necessary for the needle jet formation. Thus for small 
$l^*$, the reverse jet region takes over even if 
$\gamma \lesssim 0.25$. The simulations presented in this work are limited by the assumption of axisymmetry. This causes noticeable differences compared to the experiments once fragmentation of bubbles or droplets occur. In the simulation, this yields torus-like structures that are unstable in reality (see Reuter & Mettin Reference Reuter and Mettin2016). Another important limitation is the lack of a phase transition model, which prevents an accurate prediction of the condensation of water vapour present in real cavitation bubbles. Instead, the simulated bubble contains an non-condensable gas that cushions further bubble oscillations. This affects the rebound but also the development of non-axisymmetric instabilities that grow with time. Therefore, we only show the first 200 
$\mathrm {\mu }{\rm s}$ of bubble dynamics, which includes the first and second bubble collapse only. However, this may have important contributions during the later bubble dynamics. The axisymmetric simulations reveal that not only the first but also successive collapses transport considerable amounts of liquid through the channel. With asymmetries and instabilities as well as energy losses due to condensation influencing the fluid dynamics in reality, we expect less liquid transport than predicted in the simulations.
Regime map showing the parameter space of the standard jet, the reverse jet and the needle jet with a dependence on the normalised stand-off distance 
$\gamma$ and the normalised channel length 
$l^*=l/R_{max}$. The dashed lines show 
$\gamma =1-l^*$ and 
$\gamma =0.25$.
While understanding the pumping mechanism of viscous liquids was the initial goal, the work could be extended to use the flow below the boundary. This flow affects the strength of the bubble collapse as some of the kinetic energy is used for the transport in the micro-channels. For example, a perforated plate with through-holes that are filled with an immiscible and sufficiently viscous liquid may dissipate some of the kinetic energy and therefore reduce the energy available that is focused during the near boundary bubble collapse. This may reduce cavitation erosion, as was discussed already for perforated plates containing gas (see Gonzalez-Avila et al. Reference Gonzalez-Avila, Nguyen, Arunachalam, Domingues, Mishra and Ohl2020). Also, we speculate that emulsification of liquids in an ultrasonic bath where two liquids are separated by a perforated plate could benefit from the here found reverse jet. Another point of interest is the jet and spray produced by the bubble if the domain on the opposite side of the perforated wall is filled with a gas instead of a liquid. Potential applications include needleless micro-injections via a fast liquid jet in air and jet-based printing.
Supplementary material and movies
Videos of the numerical and experimental results as well as the code of the numerical solver CavBubbleFoam and an example simulation case can be found as supplementary material to this article.
Supplementary material and movies are available at https://doi.org/10.1017/jfm.2022.480.
Acknowledgements
We are thankful for the discussions with M. Koch, C. Lechner and Q. Zeng on details of the solver in OpenFOAM. Also the help from F. Denner is gratefully acknowledged. We thank R. Mettin for lending some of the laser equipment.
Funding
This work was funded by the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) under contract OH 75/4-1.
Declaration of interests
The authors report no conflict of interest.
Appendix A. Verification of the simulations
Figure 17(a) shows a comparison between the Volume-of-Fluid simulation of a freely oscillating (unbound) bubble and the prediction of the Keller–Miksis equation with equal maximum bubble radius 
$R_{max}=470$ 
$\mathrm {\mu }{\rm m}$. It was done for a geometry of dimensions 
$100\,{\rm mm} \times 50\,{\rm mm}$, which is tenfold larger than what was used for the rest of this work to avoid shock wave reflections at the outer boundaries of the simulation geometry. It shows excellent agreement for the first oscillation period. Since the collapse of the bubble in the VoF simulation is not perfectly spherical due to asphericities in the geometry and the mesh grid, its second oscillation period shows a weaker expansion than the Keller–Miksis equation predicts.
(a) Comparison between a VoF simulation of a cavitation bubble far from any boundaries in a large geometry (
$100\,{\rm mm} \times 50\,{\rm mm}$) and a solution of the Keller–Miksis equation for a radially oscillating bubble with equal maximum bubble radius 
$R_{max}$ and (b) grid dependence study for the case of a bubble near a perforated boundary for 
$l=1000$ 
$\mathrm {\mu }{\rm m}$, 
$r_{c}=50$ 
$\mathrm {\mu }{\rm m}$, 
$d=500$ 
$\mathrm {\mu }{\rm m}$.
A mesh resolution dependence study is shown in figure 17(b), using the case of a bubble near a perforated plate with the same geometric parameters as in figure 4(e–h). The grid resolution at the outer boundary of the geometry was kept constant, while the region including the bubble and the channel were discretised using square cells with a width of 10, 5 and 2.5 
$\mathrm {\mu }{\rm m}$. A cell width of 2.5 
$\mathrm {\mu }{\rm m}$ was used in the rest of this work. For a better direct comparison between different cell sizes, a cylindrical bubble is seeded without interface smearing, and the seed size is kept constant at a radius of 10 
$\mathrm {\mu }{\rm m}$ and a height of 20 
$\mathrm {\mu }{\rm m}$. For finer meshes, the bubble reaches a slightly larger maximum radius and exhibits a sharper liquid–gas interface. The interface of the thin liquid film that forms between the bubble and the wall after the first collapse is sharper with a finer mesh.
Appendix B. Estimation of the pumped volume
Here we aim to estimate the pumped volume in dependence of the liquid viscosity 
$\mu$ and the channel radius 
$r_{c}$ in general form including later times than the simulation covers, i.e. 
$t>200\,\mathrm {\mu }$s. We neglect the driving force associated with the bubble dynamics. The laminar volumetric flow rate 
$\dot V$ in a cylindrical channel is given via the Hagen–Poiseuille equation (Pfitzner Reference Pfitzner1976):
\begin{equation} \dot V=\frac{{\rm \pi} r_{c}^4\Delta p}{8\mu l}. \end{equation}
The pressure drop 
$\Delta p$ is in equilibrium with the viscous forces acting upon the fluid in the channel. If the flow driving force taken away, viscosity decelerates the flowing liquid via
\begin{equation} \Delta p={-}\frac{F}{A}={-}\frac{\rho Al\dot u}{A}={-}\frac{\rho l\ddot V}{{\rm \pi} r_{c}^2}, \end{equation}
turning (B1) into a differential equation for the flow rate 
$\dot V$:
\begin{equation} \dot V={-}\frac{r_{c}^2\rho}{8\mu}\ddot V. \end{equation}Equation (B3) is solved by
\begin{equation} \dot V(t)=\exp\left({-\frac{8\mu}{r_{c}^2\rho}t}\right)\dot V_0. \end{equation}
Integrating (B4) from 
$t=0$ to 
$t=\infty$ gives the total pumped volume:
\begin{equation} V=\frac{r_{c}^2\rho}{8\mu}\dot V_0=\frac{{\rm \pi} r_{c}^4\rho}{8\mu}u_0. \end{equation}
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