1. Introduction
Applications of ultrasound are ubiquitous in engineering and science, including imaging (Oelze et al. Reference Oelze, O'Brien, Blue and Zachary2004), chemo-mechanical activations, i.e. sonochemistry (Potisek et al. Reference Potisek, Davis, Sottos, White and Moore2007; Li, Nagamani & Moore Reference Li, Nagamani and Moore2015), as well as non-destructive evaluation and structural health monitoring (Naik, Malhotra & Popovics Reference Naik, Malhotra and Popovics2003; Kim et al. Reference Kim, In, Kim, Kurtis and Jacobs2014), owing to its non-destructive nature. Distinct advantages of ultrasound include the flexibility to control the mechanical and thermal energy in a spatiotemporal way. It has a successful transition into clinical or biomedical applications, e.g. necrosis of cancer cells as a new therapeutic modality (Kennedy Reference Kennedy2005). The need to deliver spatiotemporally controlled moderated acoustic energy to a target area is a critical ability of ultrasound in medicine (Kim et al. Reference Kim, Lau, Halmes, Oelze, Moore and Li2019). The moderate use of the acoustic energy offers the solutions for clinical challenges, e.g. blood-brain barrier (BBB) opening (Hynynen et al. Reference Hynynen, McDannold, Vykhodtseva and Jolesz2001), local drug delivery (Zhang et al. Reference Zhang, Yu, Bomba, Zhu and Gu2016) and neurosurgery (Leinenga et al. Reference Leinenga, Langton, Nisbet and Götz2016).
Of particular importance in various applications is the consideration of the cavitation effect at an air–fluid interface (Cinbis, Mansour & Khuri-Yakub Reference Cinbis, Mansour and Khuri-Yakub1993). When ultrasonic energy is focused at the free surface, acoustic radiation force induces a deformation of the surface, resulting in acoustic fountain (Li, Li & Li Reference Li, Li and Li1997) and ultrasonic atomization (Woods & Loomis Reference Woods and Loomis1927; Simon et al. Reference Simon, Sapozhnikov, Khokhlova, Crum and Bailey2015); other phenomena include medical nebulizers and gas diffusion (Simon et al. Reference Simon, Sapozhnikov, Khokhlova, Crum and Bailey2015). Numerous experimental and numerical studies have explored the formation of an acoustic fountain and noted that the fountain shape is influenced by several factors, including acoustic input power, ultrasonic wavelength, frequency and sonication time. For instance, Simon et al. (Reference Simon, Sapozhnikov, Khokhlova, Crum and Bailey2015) investigated the effect of frequency, temperature, type of media and acoustic radiation force, showing that the bubble plays a crucial role in the atomization in drop-chain fountains. Xu, Yasuda & Liu (Reference Xu, Yasuda and Liu2016) showed distinct features of ultrasonic fountains in comparison with pump-induced fountains; in particular, they noted that the velocity field from the pump is much greater than that by ultrasound for a given fountain height. Despite the efforts to understand acoustic fountains, wave phenomena under the free surface have not been quantified. Uncovering such acoustic streaming is needed to characterize the fountain phenomenon, e.g. shape.
Herein, we first investigate acoustic fountain and acoustic streaming simultaneously. This is achieved by means of particle image velocimetry. Integrated with focused ultrasound (FUS), acoustic fountain (on the free surface) and acoustic streaming (beneath the free surface) are characterized over time. To better understand these phenomena, we consider three distinct fountain shapes, namely, weak, intermediate (stable) and highly forced fountains (explosive), by varying the acoustic pressure from 0.5 to 1.4 MPa (spatial-peak temporal average intensity (
$I_{\textrm {SPTA}}$), 
$8.4\ \textrm {W}\ \textrm {cm}^{-2}$ to 
$66.1\ \textrm {W}\ \textrm {cm}^{-2}$). Besides, we explored a different selection of the centre frequency of ultrasound (550 kHz and 1.1 MHz) to get additional insight into the spatial control of acoustic fountains. By tracking the changes in particle motion and the fountain at the free surface, we provide new quantitative information on the time-averaged transverse velocity distribution and geometry of fountains.
The study is organized as follows. We describe the FUS particle image velocimetry (FUS-PIV) set-up in § 2. The induced flow and general fountain shapes generated by two different frequency transducers and the ability of the FUS set-up to control the fountain shape spatiotemporally are discussed in § 3. A first-order model for the fountain shape is provided in § 3.4. The main conclusions and future directions for the proposed FUS-PIV technique are elaborated in § 4.
2. Experimental set-up
Acoustic fountains were induced at the free surface of a quiescent water body using FUS. A sinusoidal voltage signal was generated with a function generator (33500B Keysight Technologies) and amplified through a radiofrequency (RF) power amplifier (NP Technologies Inc.). The amplified signal was sent to FUS transducers (FUS Instruments) through a 10 MHz low-pass filter and an impedance matching network. As illustrated in figure 1(a), the FUS set-up was synchronized with the computer to control the sonication time and the input parameters in the function generator, also enabling spatial control of acoustic energy. A pair of FUS transducers were used; one for 550 kHz and the other for 1.1 MHz nominal centre frequencies. The focal distance and aperture of each transducer were 60 mm and 75 mm, respectively (
$f$-number of 0.8). The spatial pressure distribution at the focal spot was also estimated using the hydrophone – the obtained full width at half-maximum (FWHM) beamwidth and depth of field of each transducer were approximately 2.2 mm and 12.5 mm for 550 kHz and 1.2 mm and 5.9 mm for the 1.1 MHz. The height of the water surface was fixed to 60 mm to guarantee that the transmitted acoustic energy was focused on the free surface. In the water, the output voltage signal was detected at the focal spot using a fibreoptic hydrophone (Precision Acoustics) and converted to peak acoustic pressure by applying hydrophone sensitivity (
$96.00\ \textrm {mV}\ \textrm {MPa}^{-1}$ for 550 kHz and 
$112.04\ \textrm {mV}\ \textrm {MPa}^{-1}$ for 1.1 MHz) (Kim et al. Reference Kim, Lau, Halmes, Oelze, Moore and Li2019). The peak acoustic pressure was varied in the range from 0.5 to 1.4 MPa and focused on the free surface. The spatial-peak temporal average intensity, 
$I_{\text {SPTA}}$, was estimated using (2.1), ranging from 8.4 to 
$66.1\ \textrm {W}\ \textrm {cm}^{-2}$:
\begin{equation} {I}_{\text{SPTA}} = p^{2} (2 \rho c)^{-1}, \end{equation}
where p is the peak acoustic pressure amplitude, c the speed of sound, and 
$\rho$ the density of the medium. It is hypothesized that acoustic radiation force occurs at the focal spot as the water absorbs the acoustic energy – this is a result of a momentum transfer from the ultrasonic wave to the target media (Rudenko, Sarvazyan & Emelianov Reference Rudenko, Sarvazyan and Emelianov1996). A continuous wave (CW) with the sonication time of 60 s was considered to keep the consistent fountain shape for each case. Note that the level of pressure and intensity guarantees negligible temperature changes during the fountain formation. The temperature increase 
$T_{r} = {2 \alpha {I}}/{\gamma }<0.5\,^{\circ }\textrm {C}$ (Tabaru et al. Reference Tabaru, Yoshikawa, Azuma, Asami and Hashiba2012), where 
$\alpha =7.65\times 10^{-3}\ \textrm {m}^{-1}$ and 
$\gamma =4.5\times 10^4\ \textrm {J}/^{\circ }\textrm {C}$ are the absorption coefficient and the specific heat at constant volume, and I is the average intensity created by the ultrasonic device.
Figure 1. (a) Basic schematic of the experimental set-up illustrating the focused ultrasound transducer connection, PIV and field of view (FOV). (b) Details around the fountain and forces acting on the surface; here, 
$p_{L}$, 
$\sigma$ and 
$\tau$ are the Langevin pressure, surface tension and shear stress at the interface, whereas 
$\rho$ and 
$\rho _{a}$ are the density of water and the density of air.
Velocity fields and ultrasonic streaming processes were acquired in a vertical plane using a low-speed planar PIV system from TSI. A field of view (FOV) of 
$50\ \textrm {mm} \times 36 \ \textrm {mm}$ crossing the high-intensity ultrasonic focus spot was illuminated with a 1 mm thick laser sheet produced by a 
$250\ \textrm {mJ}\ \textrm {pulse}^{-1}$ double-pulsed laser from Quantel. The FOV was located at the centre of the container covering a symmetric region of 
${\pm }20$ mm in the horizontal direction from the fountain centre (ultrasonic focused spot, which was set as the origin); the FOV covers the 10 mm above the water–air interface to capture the shape of the fountain. The working fluid was seeded with 
$11\ \mathrm {\mu }\textrm {m}$, silver-coated hollow glass spheres. Ten sets of 80 image pairs (800 vector fields), covering a sonication period of 30 s, were collected for the 550 kHz FUS, whereas five sets of 160 image pairs were obtained for the 1.1 MHz FUS covering a 60 s sonication both at an acquisition frequency of 2.4 Hz with an 8 MP CCD PowerView camera (
$3320\ \textrm {pixels}\times 2496\ \textrm {pixels}$). A Nikon AF Micro-Nikkor lens with a focal length of 105 mm and a focal ratio of 
$f/2.8$ was used to maximize the focus. The image pairs were interrogated using a recursive cross-correlation method via the TSI Insight 4G software. The final interrogation window had a size of 
$24\ \textrm {pixels} \times 24\ \textrm {pixels}$ with 50 % overlap, resulting in a vector grid spacing 
${\rm \Delta} x = {\rm \Delta} y = 190\ \mathrm {\mu }\textrm {m}$.
3. Results
3.1. Fountain shape characteristics
Snapshots illustrating bulk shape features of the weak, intermediate and highly forced fountains induced by different peak acoustic pressures are shown in figure 2. Noticeable shape changes occurred throughout the process of increasing the transducer output voltage for the tested transducers of 550 kHz and 1.1 MHz, which produced the three ultrasonic fountain regimes. A slow increase of the acoustic pressure from 0 MPa to 0.4 MPa for the 550 kHz transducer produced no change in water-surface shape. The force 
$F = ({2\alpha }/{{\rho }{c}^2}){|p|}^2$, where 
$\alpha$ is the attenuation coefficient and 
$p$ is the peak pressure as defined in (2.1) (Xu et al. Reference Xu, Yasuda and Liu2016), created by the ultrasonic transducer, is insufficient to sustain the fountain formation. A weak fountain is formed for acoustic pressure over a threshold of 0.5 MPa for the 550 kHz transducer, as shown in figure 2(a). The generated acoustic radiation force is sufficient to accelerate the fluid at the focal spot, creating an upward flow in the close vicinity of the water–air interface. Further increase in the acoustic pressure of over 0.7 MPa for the 550 kHz transducer resulted in a step-change in fountain height leading to the intermediate fountain regime, as shown in figure 2(b). Here, the radiation force is sufficient to support a larger fountain surface area as well as the stronger gravitational potential generated by the higher fountain. An even larger acoustic pressure results in a highly forced fountain, where the comparatively high radiation pressure causes a breakup at the fountain surface, i.e. explosive fountain; ultrasonic atomization occurs in this regime (figure 2c). We point out that the process is not steady in this highly forced regime, and the atomization is essentially an unsteady process.
Figure 2. Snapshots of various fountain shape created by a 550 kHz transducer focused at the water–air interface. (a) Weak, (b) intermediate and (c) highly forced fountains. Scale bar: 5 mm.
The step-change produced in the steady, i.e. the weak and intermediate, fountain height for acoustic pressures of 0.5 MPa to 0.9 MPa every 0.1 MPa is shown in figure 3(a) with time-averaged shapes. A similar step-change of the steady fountain height was also noted by Xu et al. (Reference Xu, Yasuda and Liu2016). The characteristic of the intermediate fountain following the definition for forced jet fountains (Mehaddi et al. Reference Mehaddi, Vaux, Candelier and Vauquelin2015; Hunt & Debugne Reference Hunt and Debugne2016), including the fountain top height 
$z_{t}$, width 
$b_{t}$ and the steady fountain height 
$z_{ss}$ are given in figure 3(b).
Figure 3. Time-averaged fountain shapes induced by the 550 kHz transducer. (a) Weak and intermediate fountain shapes at various input pressures, and (b) features of the intermediate fountain. The transducer focal spot is set at the origin of the coordinate system; see figure 1(a).
3.2. Time-averaged flow field
Features of the streaming and flow induced by the focused ultrasound are presented in figure 4 for the three fountain regimes. The induced flow was different between the fountain regimes; however, it exhibited similar flow patterns within a given regime. The weak fountain (0.5 MPa) displayed a very weak lateral flow near the ultrasonic focal spot; see figure 4(a). A stronger lateral flow pattern close to the water–air interface and symmetric about the focal spot occurred with intermediate, stationary fountains, as shown in figure 4(b–d) for 0.7, 0.8 and 0.9 MPa acoustic pressure. The flow field in the highly forced regime was characterized by a not symmetric, energetic toroidal vortex; it influenced a comparatively broader region, as evidenced in figure 4(e,f). A similar pattern is also noted in the axial flow, as shown in figure 5. Weak fountains induced incipient vortex formation and low flow, locally stronger about the focal spot of the transducer. In contrast, evident recirculation vortices were formed in the fountains under the highly forced regime; flow patterns extended relatively far from the transducer focal spot.
Figure 4. Streamlines superimposed with time-averaged transverse velocity distributions u below the (a) weak (0.5 MPa), (b–d) intermediate (0.7, 0.8 and 0.9 MPa) and (e,f) highly forced fountains (1.0 and 1.2 MPa).
Figure 5. Streamlines superimposed with time-averaged axial flow velocity distributions below the (a) weak (0.5 MPa), (b) intermediate (0.7 MPa) and (c) highly forced fountains (1.0 MPa).
Additional insight on the acoustic-induced streaming flow is obtained with the swirling strength 
$\varLambda _{ci}d/U_{ref}$ in figure 6. Here, 
$\varLambda _{ci}$ is the magnitude of the imaginary part of the velocity gradient tensor complex eigenvalues (Zhou et al. Reference Zhou, Adrian, Balachandar and Kendall1999) with its sign indicating the directionality of the swirl (Wu & Christensen Reference Wu and Christensen2006). This quantity is normalized by 
$U_{ref}/d$ where 
$d$ is the FUS transducer aperture and 
$U_{ref}= 0.01\ \textrm {m}\ \textrm {s}^{-1}$, the maximum velocity magnitude noted in the intermediate fountains. For low acoustic pressure settings, i.e. weak fountains, small-scale vortical structures were induced due to the acoustic streaming effect. Sufficiently high acoustic pressure produced a comparatively larger axisymmetric vortex structure forming a toroidal vortex ring; faster induced axial flow encountered the quasi-steady fluid at the water–air interface. Also, the small-scale coherent motions extended deeper in the highly forced regime (figure 6c; see also Slama et al. Reference Slama, Gilles, Chiekh and Bera2019).
Figure 6. Instantaneous swirling strength 
$\varLambda _{ci}d/U_{ref}$ induced under the (a) weak (0.5 MPa), (b) intermediate (0.7 MPa) and (c) highly forced fountains (1.0 MPa).
3.3. Distinct modulation of transducer frequency
The transducer frequency produced particular effects on the fountains. It also induced shared features, namely, three distinct regime behaviours (weak, intermediate and highly forced) that occurred with 550 kHz and 1.1 MHz, as shown in figure 7. Streaming flow field characteristics and the forming process as the ultrasound pressure increased followed a similar process as described in §§ 3.1 and 3.2, not reported for brevity.
Figure 7. Snapshots of various fountain shape created by a 1.1 MHz transducer focused at the water–air interface. (a) Weak, (b) intermediate and (c) highly forced fountains. Scale bar: 5 mm.
Distinct shape characteristics of the intermediate fountains induced by the 550 kHz and 1.1 MHz frequencies are shown in figure 8(a). The 1.1 MHz frequency produced a steady fountain height (
$z_{ss}$) and a width of nearly one half of that formed with the 550 kHz, indicating that the acoustic radiation force created from the transducer may play a central role in modulating the fountain shape. This is supported by Xu et al. (Reference Xu, Yasuda and Liu2016) and Lim, Kim & Kim (Reference Lim, Kim and Kim2019), which indicated that radiation pressure contributes more to the formation of acoustic fountains compared with the kinetic energy of the liquid medium; this is also verified at the end of § 3.3. Here, Langevin radiation pressure as the time-averaged pressure experience by the planar target (Hasegawa et al. Reference Hasegawa, Kido, Iizuka and Matsuoka2000) is chosen for the acoustic radiation pressure formulation (Chu & Apfel Reference Chu and Apfel1982; Lee & Wang Reference Lee and Wang1993). Pressure intensity for a focused type transducer from Kino (Reference Kino1987) is given by
\begin{equation} p_{{L}}(r)=2\left(I_{0} / c\right) {jinc}^{2}\left(\frac{r a}{z_{0} \lambda}\right), \end{equation}
where 
$jinc(x) \equiv J_{1}(x)/x$ and 
$J_{1}(x)$ is the Bessel function of the first kind. Also, 
$I_{0}$ is the peak intensity at the centreline of the transducer, 
$a$ is half of the transducer aperture (
$a = 37.5$ mm for both 550 kHz and 1.1 MHz transducers), 
$\lambda$ is the wavelength defined as 
$c/f$ where 
$c$ is the sound speed and 
$f$ is the ultrasonic frequency, and 
$z_{0}$ is the focal distance (
$f = 75$ and 60 mm for the 550 kHz and 1.1 MHz transducers, respectively). For spherical-focused-type ultrasonic transducers, peak intensity 
$I_{0} = 0.0506 {\rm \pi}(\eta V_{p}^{2}/{c R})(a / \lambda z_{0})^{2}$ with 
$\eta$ being the power conversion coefficient, 
$V_p$ the peak voltage applied at the transducer terminals and 
$R$ the electrical impedance of the transducer (Cinbis et al. Reference Cinbis, Mansour and Khuri-Yakub1993); additional details on the transducer can be found in Kim et al. (Reference Kim, Lau, Halmes, Oelze, Moore and Li2019).
Figure 8. (a) Shape characteristics of intermediate fountains for the 550 kHz (0.7 MPa) and 1.1 MHz (1.3 MPa) transducers. (b) Theoretical Langevin radiation pressure profile at the weak and intermediate fountain threshold pressure for the 550 kHz (solid lines) and 1.1 MHz (solid-dashed lines) transducers.
Computed Langevin pressure profiles and comparison with the experimentally obtained fountain-shape profiles are shown in figure 8(b). The Langevin pressure was 
$O(4)$ larger than the dynamic pressure associated with the flow, e.g. peak 
$p_L/q \approx 30/0.001 \, (q=\frac{1}{2}\rho u^{2})$ from the 550 kHz (0.7 MPa) intermediate fountain, demonstrating the dominant role of acoustic pressure. Indeed, the general fountain-shape trend matches well with the pressure profile, as the formulated profiles exhibit lower height for weak fountains as well as a nearly one-half height and width for the 1.1 MHz transducer.
3.4. Theoretical formulation
Here, we develop a simple model to predict the basic shape characteristic of ultrasonic fountains, which can provide insight towards controlling acoustic fountains. Let us consider the force diagram illustrated in figure 1(b). There, 
$p_{L}$, 
$\sigma$ and 
$\tau$ are the Langevin pressure (3.1), surface tension and shear stress at the interface, respectively, whereas 
$\rho$ and 
$\rho _{a}$ represent the density of water and the density of air, respectively. Assuming an immiscible water–air interface and neglecting potential variations in fluid properties due to the ultrasound heating, a momentum balance with a zero net vertical momentum at the steady-state fountain surface is given by
\begin{align} &2{\rm \pi} r p_{L} \cos \theta \,\mathrm{d} l + 2{\rm \pi} r \rho {(v|_{r})}^2\,\mathrm{d} r\nonumber\\ &\quad =2{\rm \pi} r \left(\rho-\rho_{a}\right) g h \,\mathrm{d} r + 2 {\rm \pi}r \tau \sin \theta \,\textrm{d}l + 2 {\rm \pi}\sigma(r \sin \theta|_{r} - r \sin \theta|_{r+\mathrm{d} r}), \end{align}
where θ is the angle between fountain surface and the horizontal direction, v is the streamwise velocity and g is the gravitational acceleration (figure 1b). Substituting 
$\mathrm {d} l = \mathrm {d} r/ \cos \theta$, dividing by 
$2 {\rm \pi}r \,\textrm {d}r$ and applying Taylor series expansion on (3.2) gives
\begin{equation} \left[p_{L} + \rho v^{2} - \tau \tan\theta -\left(\rho-\rho_{a}\right) g h\right] =\frac{\sigma}{r} \frac{ \mathrm{d}(r \sin \theta)}{\mathrm{d} r}. \end{equation}
Using 
$\sin \theta ={\mathrm {d} h}/{\mathrm {d} r}[1+({\mathrm {d} h}/{\mathrm {d} r})^{2}]^{-1 / 2}$ and 
$\tan \theta = {\mathrm {d} h}/{\mathrm {d} r}$, (3.3) can be written as
\begin{equation} \frac{1}{\sigma}\left\{p_{L} -(\rho-\rho_{a}) g h + \rho v^{2}\right\}= \zeta^{-{3}/{2}} \left(\frac{\mathrm{d}^{2} h}{\mathrm{d} r^{2}}+\frac{\zeta}{r} \frac{\mathrm{d} h}{\mathrm{d} r}+ \tau \frac{\mathrm{d} h}{\mathrm{d} r}\right). \end{equation}
Here, 
$\zeta \equiv 1+(\mathrm {d}h / \mathrm {d} r)^{2}$. Once the distribution of 
$p_{L}$, 
$\rho v^{2}$ and 
$\tau$ are obtained, (3.4) can be solved numerically with two prescribed boundary conditions, namely, 
$\mathrm {d}h / \mathrm {d} r=0$ at 
$r=0$ from symmetry and 
$h_{0}=({p_{L}(0)}/({(\rho -\rho _{a}) g}))$ from the force balance at 
$r=0$, to model the shape of steady, axisymmetric acoustic fountains.
3.5. Contributions of forces to fountain formation
Assessment of the contributing forces from (3.3) can be obtained by expressing it in a non-dimensional form. By considering the radius of curvature of the acoustic fountains 
$R$, the maximum velocity magnitude 
$U$ (
${=}U_{ref}$, at 
$z\approx 0$) and the peak Langevin pressure 
$p_{0} = p_{L}(0)$, the variables can be expressed as
\begin{equation} r'=\frac{r}{R}, \quad h'=\frac{h}{R}, \quad v'=\frac{v}{U}, \quad p'=\frac{p_{L}}{p_{0}}, \end{equation}
and dividing by 
$\sigma /R$, (3.6) can be written as follows:
\begin{equation} {Pr}\,p' + {We}\,v'^{2} - {Ca}\,\boldsymbol{\nabla}' v'\times \tan\theta - {Bo}\,h' = \frac{1}{r'} \frac{ \mathrm{d}(r' \sin \theta)}{\mathrm{d} r'}, \end{equation}
where 
${Pr} = {p_{0}R}/{\sigma }$ is the reduced acoustic pressure, and 
$We={\rho U^{2} R}/{\sigma }$, 
${Ca}={\mu U}/{\sigma }, \mu$ is the dynamic viscosity of water, and 
${Bo}=({(\rho -\rho _{a})gR^{2}})/{\sigma }$ are the Weber, Capillary and Bond numbers. For instance, the 550 kHz (0.5 MPa) case had 
${Pr} \sim 10^{0}$, 
${We}\sim 10^{-4}$, 
${Ca}\sim 10^{-3}$ and 
${Bo}\sim 10^{0}$.
Equation (3.6) allows us to compare the significance of various forces contributing to the formation of acoustic fountain, namely, acoustic pressure, hydrostatic pressure caused by gravitation, and the Laplace pressure induced by surface tension. The acoustic radiation pressure is the main driving force balancing out gravitation and surface tension; whereas the axial momentum and viscous stress are orders smaller for determining the width and steady-state height of the fountain. An estimation of the interfacial viscous stress 
$\tau _{xz}$ in the axial direction is obtained from the derivative of the axial velocity 
$u(x)$ at 
$z=0$; see figure 9.
Figure 9. Axial velocity profile and derivative with respect to 
$x$, measured right below 
$z = 0$.
From the scaling, we can neglect the last term on both sides of (3.4), i.e. the axial momentum and viscous terms, to obtain a first-order approximation of the acoustic fountain shapes. Solutions for the weak and intermediate fountains are given in figure 10. Comparison with experiments shows a good agreement in the fountain heights 
$z_{ss}$ and width. Also, the fountain shape and curvature is well-captured in the weak fountain regime; however, this is not the case for the curvature of the intermediate fountain. The surface curvature departure indicates that 
${Ca} \tan (\theta )$ is not negligible in intermediate fountains, where 
$\tau \times \textrm {d}h/\textrm {d}r$ (3.4) may be comparable to the surface tension term as the surface slope, 
$\theta$, and the interfacial viscous stress increase. Note that a much smaller viscous stress and surface slope were in the weak fountain (figure 9), resulting in much better fountain shape prediction.
Figure 10. Estimated shapes of the (a,b) weak and (c,d) intermediate fountains generated by 550 kHz transducer. Panels (a,c) are calculated from theoretical model and (b,d) from the experiments. (e) Comparison of the profiles at the centre, i.e., 
$y=0$.
4. Conclusions
Specially designed experiments using a combination of focused ultrasound and PIV techniques were conducted to study acoustic fountains and streaming flows simultaneously. We characterized three acoustic regimes, namely, weak, intermediate and highly forced fountains. A basic model is proposed to estimate fountain shape. Good agreement is obtained on the fountain width and steady fountain height (
$z_{ss}$) utilizing the streaming flow data and theoretical Langevin pressure formulation. However, discrepancies occurred in the fountain curvature for intermediate fountains; the viscous stress term is found to be responsible for that. The agreement between our theoretical modelling and experiments demonstrates the dominating factors associated with the formation of the acoustic fountain, namely, the acoustic radiation force, gravity and surface tension. In contrast, kinetic energy from the streaming flow was not dominant (Nightingale et al. Reference Nightingale, Soo, Nightingale and Trahey2002). We also confirmed that viscous stress is not a central factor determining the height and width of acoustic fountains, but instead has a non-trivial impact on predicting the curvature. Given the results, we envision that the proposed FUS-PIV set-up can be exploited in a biological context. For instance, future work will focus on optimizing the FUS-PIV set-up to examine the wave distortion caused by the interactions with tissue-mimicking materials or bone tissue. This approach will further advance the understanding of not only fluid flow phenomena in biomaterials, but also stress or strain development in the gastrointestinal tract, such as the oesophagus, stomach and intestines.
Acknowledgements
G.K. and S.C. contributed equally to this paper.
Declaration of interests
The authors report no conflict of interest.
