3.1. Kinematic model

Let us consider a bubble resting against a rigid flat substrate, as depicted in the side and top views in figure 4(a,b). Given the small Bond number, $Bo = {(\rho _{ { l}} - \rho _{ { g}}) g R_0^2}/{\sigma } \sim O(10^{-3})$ , where $(\rho _{ { l}} - \rho _{ { g}})$ is the density difference between water and air and $g$ is the gravitational acceleration, the influence of gravitational effects is negligible for the problem at hand. Consequently, under equilibrium conditions, the bubble can be approximated as a spherical cap with a static contact angle $\alpha _0$ . We define the origin of the reference frame at the geometric centre of the uncut sphere. The surface of the bubble at equilibrium can be parametrically described as ${\boldsymbol{R}}_0(\theta ,\phi )$ , where $\theta \in [-\alpha _0,\alpha _0]$ is the polar angle and $\phi\,{\in}\,[0, 2\pi ]$ is the azimuthal angle. By normalising the bubble equilibrium surface ( $\|{\boldsymbol{R}}_0\| = 1$ ), the arc angle and arc length values are equal. The surface deformation is defined as $\delta {\boldsymbol{R}}(\theta ,\phi ,t)$ . This deformation may displace the contact line, denoted as $\boldsymbol{\Gamma }(\theta )$ , along the substrate. The unit vector normal to ${\boldsymbol{R}}_0$ is denoted by $\boldsymbol{n}$ . The unit vector tangential to ${\boldsymbol{R}}_0$ and normal to $\boldsymbol{\Gamma }$ is denoted by $\boldsymbol{t}$ . The unit vector tangential to ${\boldsymbol{R}}_0$ and to $\boldsymbol{\Gamma }$ is denoted by ${\boldsymbol{b}} = {\boldsymbol{n}} \times {\boldsymbol{t}}$ . The unit vector normal to the substrate is denoted as ${\boldsymbol{n}}_{\textrm {s}}$ . The bubble deformation can be decomposed into the component normal to the bubble equilibrium surface, defined as

(3.1) \begin{equation} \delta _{\perp } {\boldsymbol{R}} = \left (\delta {\boldsymbol{R}} \,{\boldsymbol\cdot}\, {\boldsymbol{n}} \right ){\boldsymbol{n}} = \eta {\boldsymbol{n}}, \end{equation}

and the component tangential to the bubble equilibrium surface, represented by

(3.2) \begin{equation} \delta _{\parallel } {\boldsymbol{R}} = \left (\delta {\boldsymbol{R}} \,{\boldsymbol\cdot}\, {\boldsymbol{t}} \right ){\boldsymbol{t}} = \tau {\boldsymbol{t}}, \end{equation}

as $\delta {\boldsymbol{R}} \perp {\boldsymbol{b}}$ . An example of normal deformation $\eta$ in the plane perpendicular to the substrate defined by the projection line $\text{A}{-}\text{A}$ , and thus identified as $\eta (\theta ,0,t)$ , is shown in figure 4(c). In contrast, an example of normal deformation $\eta$ in the plane parallel to the substrate defined by the projection line $\text{B}{-}\text{B}$ , and thus identified as $\eta (\pi /2,\phi ,t)$ , is shown in figure 4(d).

The equilibrium contact angle $\alpha _0$ is related to the surface tensions of the solid–liquid $\sigma _{{ sl}}$ , solid–gas $\sigma _{\textit{sg}}$ and liquid–gas $\sigma _{{ lg}}$ through the Young–Dupré equation:

(3.3) \begin{equation} ({\boldsymbol{n}} \,{\boldsymbol\cdot}\, {\boldsymbol{n}}_{\textrm { s}})|_{\theta = \alpha _0} \equiv \cos (\alpha _0) = \frac {\sigma _{{ sg}} - \sigma _{{ sl}}}{\sigma _{{ lg}}}. \end{equation}

A perturbation in the bubble surface $\delta {\boldsymbol{R}}$ results in a change in the wetting condition (Tyuptsov Reference Tyuptsov1966; Myshkis et al. Reference Myshkis, Babskii, Kopachevskii, Slobozhanin and Tyuptsov1987):

(3.4) \begin{equation} \partial _{\theta }\eta |_{\theta =\alpha _0} - \cot (\alpha _0)\eta |_{\theta =\alpha _0} = -\delta \alpha . \end{equation}

A full derivation of this relation is provided in Appendix A. We use this kinematic boundary condition to determine the spectrum of shape modes for the deformation $\delta {\boldsymbol{R}}$ under various wetting conditions.

3.2. Shape modes for wall-attached bubbles

The tangential deformation $\tau (\theta ,\phi ,t)$ does not affect the curvature of the bubble, and thus is irrelevant for analysing the dynamic behaviour of the interface. To perform a spectral analysis on the normal deformation $\eta (\theta , \phi , t)$ , it is customary to expand it in a series of real spherical harmonic functions $Y_{l}^{m}(\theta ,\phi )$ of degree $l$ and order $m$ , with $0 \leqslant m \leqslant l$ :

(3.5) \begin{equation} \eta (\theta ,\phi ,t) = \sum _m a_l^{m}(t) Y_{l}^{m}(\theta ,\phi ), \end{equation}

where $a_l^{m}(t)$ represents the time-dependent amplitude of each spherical harmonic $Y_{l}^{m}(\theta ,\phi ) = N_{l}^{m} P_{l}^{m} (\cos (\theta )) \cos (m \phi )$ . Here $P_{l}^{m} (\cos (\theta ))$ is a Legendre function and the normalisation factor $N_{l}^{m} = (\max (P_{l}^{m} (\cos (\theta )) \cos (m \phi )))^{-1}$ ensures that the spherical harmonic is constrained to a maximum magnitude of unity. For a free bubble, which constitutes a closed surface, the following periodic boundary conditions must hold:

(3.6) \begin{align} &\eta (\theta ,\phi ,t) = \eta (\theta +2\pi ,\phi ,t), \quad \partial _{\theta }\eta (\theta ,\phi ,t) = \partial _{\theta }\eta (\theta +2\pi ,\phi ,t), \quad \forall \theta , \phi , t, \end{align}

(3.7) \begin{align} &\eta (\theta ,\phi ,t) = \eta (\theta ,\phi +2\pi ,t), \quad \partial _{\phi }\eta (\theta ,\phi ,t) = \partial _{\phi }\eta (\theta ,\phi +2\pi ,t), \quad \forall \theta , \phi , t. \end{align}

This implies that the surface deformation can only be represented by Legendre functions with integer degrees $l$ and orders $m$ , commonly known as associated Legendre polynomials. From the linear framework used to derive (1.1), it follows that the driving frequency and bubble radius determine the degree $l$ of the shape mode, but not its order $m$ . Consequently, for a given $l$ , there are $l+1$ possible values of $m$ , each occurring with equal probability (see figure 1). Modes that share the same resonance frequency, and thus the same $l$ , are referred to as degenerate. Nonlinear interactions dictate the combination of degenerate modes that ultimately emerges for each $l$ . The first seven modes with integer degree $l$ for $m=0$ are depicted in figure 20(a,b). The mode spectrum of a free bubble for the first seven degrees is depicted in figure 5(a). The degree $l$ can be interpreted as the ‘energy level’ of the mode, while the order $m$ can be seen as the number of ‘states’ that become available with increasing $l$ . The ‘energy level’ $l$ can be augmented by increasing either the driving frequency or the bubble radius.

Figure 5. Spectrum of allowed shape modes for the first seven $k$ values, from 0 to 6, for (a) a free bubble, (b) a pinned wall-attached bubble, (c) a fixed-contact-angle wall-attached bubble and (d) an unconstrained wall-attached bubble. The static contact angle considered for the wall-attached bubbles is $\alpha _0 = 3\pi /4$ . Parameters $l$ and $m$ are the degree and order, respectively, of the spherical harmonic $Y_{l_{k}}^{m}$ , and $k$ is an index that sequentially orders shape modes of a specific order $m$ . For a free bubble, both $l$ and $m$ are integers, resulting in a degenerate spectrum where different shape modes share the same $l$ . In contrast, for a pinned or fixed-contact-angle wall-attached bubble, $l$ is generally a non-integer, leading to a non-degenerate spectrum. For an unconstrained wall-attached bubble, the spectrum is no longer quantised in $l$ but is continuous and remains degenerate.

In general, spherical harmonics with an integer degree are not suitable for describing the surface deformation of a bubble in contact with a substrate, as they may not comply with the variational wetting condition (3.4). For a pinned bubble, the deformation basis functions must satisfy the following boundary condition:

(3.8) \begin{equation} \eta |_{\theta =\alpha _0} = 0. \end{equation}

This condition is fulfilled by Legendre functions $P_{l}^{m} (\cos (\theta ))$ of real, though not necessarily integer, degree $l$ that are zero at $\theta =\alpha _0$ . Since the distinct real values of $l$ for which this condition holds depend separately on $m$ , they will be denoted by $l_{k}(m)$ , where $k$ is an integer index that sequentially orders the various roots $l$ at a given $m$ . The index $k$ is chosen to start at $m$ , analogous to the case of integer $l$ . For a free bubble, $k$ corresponds to $l$ . In the azimuthal direction, periodic boundary conditions must hold:

(3.9) \begin{align} &\eta (\theta ,\phi ,t) = \eta (\theta ,\phi +2\pi ,t), \quad \partial _{\phi }\eta (\theta ,\phi ,t) = \partial _{\phi }\eta (\theta ,\phi +2\pi ,t), \quad \forall \theta , \phi , t, \end{align}

which restrict the value of $m$ to an integer. The first seven modes with real degrees $l$ for $m=0$ that comply with the condition (3.8) for $\alpha _0 = 3\pi /4$ are depicted in figure 20(c,d). The contact angle $\alpha _0 = 3\pi /4$ is chosen to align with the experimental observations discussed in the following sections. The mode spectrum of a pinned bubble with $\alpha _0 = 3\pi /4$ for the first seven $k$ values is reported in table 1(a) and also depicted in figure 5(b). In this case, shape modes are non-degenerate as each mode corresponds to a unique $l$ value. This leads to a richer, more granular spectrum where all permissible modes can distinctly manifest. It is important to recognise that a superposition of modes sharing the same degree $l$ but differing in order $m$ cannot satisfy the boundary condition along the entire contact line, since $\phi$ -dependent deformation components cannot cancel out everywhere. Only individual modes that produce uniform deformation along the contact line can meet this constraint. Compared with a free bubble, the $l$ value for a given $k$ value is higher, as the same number of shape-mode features must be compressed into a shorter arc of the circumference. Consequently, for a given radius of curvature, the value of $l$ increases as the contact angle decreases. Finally, it can be observed that the deviation of $l$ from $k$ increases as $k$ increases and $m$ decreases.

If the contact line is free to move but the contact angle is fixed ( $\delta \alpha =0$ ), the deformation basis functions must satisfy the following boundary condition:

(3.10) \begin{equation} ({\partial _{\theta }\eta }/{\eta })|_{\theta =\alpha _0} = \cot (\alpha _0). \end{equation}

Again, this condition is met by Legendre functions $P_{l}^{m} (\cos (\theta ))$ of real, though not necessarily integer, degree $l$ and integer order $m$ . The first seven modes with real degrees $l$ for $m=0$ that comply with the condition (3.10) for $\alpha _0 = 3\pi /4$ are depicted in figure 20(e, f). The mode spectrum of a fixed-contact-angle bubble with $\alpha _0 = 3\pi /4$ for the first seven $k$ values is reported in table 1(b) and also depicted in figure 5(c). The $l$ value for a given $k$ and $m$ values is lower compared with the pinned-bubble case. These findings indicate that the pinning conditions can markedly influence the resonance frequency of shape modes, including the breathing mode.

In the family of Legendre functions describing the deformation of a free bubble, characterised by integer indices $l$ and $m$ , orthogonality is satisfied for both fixed $l$ and fixed $m$ :

(3.11) \begin{align} \big \langle P_{l}^m, P_{l}^{m'}\big\rangle &= \int _0^{\pi } \frac {P_{l}^m (\cos (\theta )) P_{l}^{m'} (\cos (\theta ))}{1-\cos (\theta )^2} \sin (\theta ) {\textrm d}\theta = 0, \quad \text{for}\; m \neq m', \end{align}

(3.12) \begin{align} \left \langle P_{l}^m, P_{l'}^{m}\right \rangle &= \int _0^{\pi } P_{l}^m (\cos (\theta )) P_{l'}^{m} (\cos (\theta )) \sin (\theta ) {\textrm d}\theta = 0, \quad \text{for}\; l \neq l'. \end{align}

The corresponding spherical harmonics $Y_{l}^m = N_{l}^m P_{l}^m (\cos (\theta )) \cos (m \phi )$ are thus orthogonal to each other:

(3.13) \begin{equation} \big \langle Y_{l}^m, Y_{l'}^{m'}\big \rangle = \int _{\theta =0}^{\pi } \int _{\phi =0}^{2\pi } Y_{l}^m Y_{l'}^{m'} \sin (\theta ) {\textrm d}\phi \,{\textrm d}\theta = 0, \quad \text{for}\; l \neq l'\; \text{and/or}\; m \neq m'. \end{equation}

Orthogonality is also satisfied in the families of Legendre functions describing the deformation of a pinned bubble and a fixed-contact-angle bubble. However, since no function shares the same $l$ , orthogonality holds only for fixed $m$ :

(3.14) \begin{equation} \big \langle P_{l_k}^m, P_{l_{k'}}^{m}\big \rangle = \int _0^{\alpha _0} P_{l_k}^m (\cos (\theta )) P_{l_{k'}}^{m} (\cos (\theta )) \sin (\theta ) {\textrm d}\theta = 0, \quad \text{for}\; k \neq k'. \end{equation}

In these families as well, the corresponding spherical harmonics are therefore orthogonal to each other:

(3.15) \begin{equation} \big \langle Y_{l_k}^m, Y_{l_{k'}}^{m'}\big \rangle = \int _{\theta =0}^{\alpha _0} \int _{\phi =0}^{2\pi } Y_{l_k}^m Y_{l_{k'}}^{m'} \sin (\theta ) {\textrm d}\phi \,{\textrm d}\theta = 0, \quad \text{for}\; k \neq k'\; \text{and/or}\; m \neq m'. \end{equation}

Spherical harmonics belonging to different families are in general not orthogonal, and the inner product between them is non-zero. Consequently, the spherical harmonics in these two families are not volume-preserving as they are not orthogonal to $Y_0^0=1$ , and therefore their 1-norm is non-zero.

If both the contact line and contact angle are free to move, the deformation can adopt the shape of a Legendre function with any real $l$ value. This is because there are no boundary conditions imposed along the $\theta$ direction, including the periodicity condition required for a free bubble. Consequently, as shown in figure 5(d), the mode spectrum for a bubble with a free contact line and contact angle is no longer quantised in $l$ , but, rather, it becomes continuous. However, it remains quantised in $m$ due to the $2\pi$ -periodicity in the $\phi$ direction that still holds. In this case, the spectrum exhibits degeneracy, as shape modes can share the same degree $l$ , a scenario analogous to that of a free bubble.

5.1. Breathing mode

The bubble deformation $\eta (\theta ,\phi , t)$ can be decomposed into a volume-variant component $\eta _{ { volume}}(\theta ,\phi , t)$ and a volume-invariant component $\eta _{ { shape}}(\theta ,\phi , t)$ :

(5.1) \begin{equation} \eta (\theta ,\phi , t) = \eta _{{ volume}}(\theta ,\phi , t) + \eta _{ { shape}}(\theta ,\phi , t). \end{equation}

For a free bubble, $\eta _{{ volume}}(\theta ,\phi , t)=a_0^0(t)Y_0^0(\theta ,\phi )$ , which corresponds to the breathing mode. The shape deformation, on the other hand, is described by $\eta _{{ shape}}(\theta ,\phi , t)=\sum a_l^m(t) Y_l^m(\theta ,\phi )$ with $l\neq 0$ . For a wall-attached bubble, performing such a decomposition in terms of spherical harmonics is not straightforward. As mentioned previously, there are no spherical harmonics that simultaneously satisfy the wetting condition and are volume-invariant. Nevertheless, we can still experimentally track the temporal evolution of the volumetric change, define an equivalent breathing mode amplitude and compare it with classical theoretical models for spherical bubbles. This proves useful to elucidate the accuracy of these models in the context of wall-attached bubbles and serves as a valuable means to validate the value of the acoustic driving pressure measured by the hydrophone. For this purpose, we define an average deformation of the bubble surface, analogous to an increase $\delta R(t)$ in the radius of the spherical cap, $ \delta R(t) = ({1}/{(2\pi (1-\cos (\alpha _0)))})\int _0^{2\pi } \int _0^{\alpha _0} \eta (\theta ,\phi ,t)\sin {\theta }\,{\textrm d}\theta \,{\textrm d}\phi .$ The volume of this equivalent spherical cap is consequently given by $ V(t) = (({2}/{3}) - \cos (\alpha _0) + ({1}/{3})\cos ^3(\alpha _0) )\pi (R_0 + \delta R(t) )^3.$ The radius $R_{\textit{eq}}(t)$ of a spherical bubble, which has the same volume as the equivalent spherical cap, is given by $ R_{\textit{eq}}(t) = (({3V(t)}/{4 \pi }) ) ^{{1}/{3}} = ( ({1}/{2}) - ({3}/{4})\cos (\alpha _0) + ({1}/{4})\cos ^3(\alpha _0) )^{{1}/{3}} (R_0+ \delta R(t) ).$ For a contact angle $\alpha _0 \approx 140^\circ$ , $R_{\textit{eq}}(t) \approx 0.99 (R_0+ \delta R(t) )$ . Therefore, from now on, the difference between the two will be regarded as negligible, and the term $R(t)$ will be used.

To experimentally measure $R(t)$ we employ an image-processing algorithm to extract the bubble contour from the side-view video recordings. As long as the bubble maintains axisymmetry, i.e. before developing sectoral shape modes, its volume $V(t)$ can be determined by integrating the area of disks stacked along the axis of symmetry $z$ , defined by the bubble contour $R(z,t)$ , as $ V(t) =\pi \int _{z^{-}}^{z^{+}} R(z,t)^2 {\textrm d}z,$ where $z^{-}$ and $z^{+}$ represent the limiting values within which the contour of the bubble $R(z,t)$ is defined. The radius $R(t)$ of a volume-equivalent spherical bubble can then be determined as

(5.2) \begin{equation} R(t) = \left (\frac {3V(t)}{4 \pi } \right ) ^{1/3}. \end{equation}

The radial motion of this equivalent spherical bubble of radius $R(t)$ in proximity to a solid boundary and subjected to a spatially uniform acoustic field can be approximated by the Rayleigh–Plesset equation for mildly compressible Newtonian fluids. The influence of the nearby solid boundary is incorporated using the method of images (Strasberg Reference Strasberg1953), yielding the following formulation:

(5.3) \begin{equation} \rho _{ { l}} \left (R \ddot R + \frac {3}{2} \dot R^2 \right )= \left ( 1 + \frac {R}{c_{ { l}}} \frac {d}{{\textrm d}t} \right ) p_{ { g}} -2\frac {\sigma _{{ lg}}}{R} - p_{\infty } - p_{ { d}}\left (t\right ) - \rho _{ { l}} \frac {2R\dot {R}^2 + R^2 \ddot {R}}{2d} - 4{\,{\,\rm \mu}} _{ { l}} \frac {\dot R}{R}, \end{equation}

where overdots denote time differentiation, $\rho _{ { l}}$ the liquid density, $c_{ { l}}$ the speed of sound in the medium, $p_{ { g}}$ the gas pressure within the bubble, $\sigma _{{ lg}}$ the liquid–gas surface tension, $p_{\infty }$ the undisturbed ambient pressure, $p_{{ d}}(t)$ the acoustic driving pressure, ${\,{\,\rm \mu}} _{ { l}}$ the dynamic viscosity of the medium and $d=R_0$ the distance from the centre of the bubble to the wall. A full derivation of the pressure term accounting for the rigid boundary is provided in Appendix D. The method of images is typically valid when the distance between the bubble and the wall is much larger than the bubble radius. In our case, where the bubble is attached to the wall, finite-size and multipole effects may become significant. More advanced models have been developed to describe bubble dynamics near boundaries (Doinikov, Aired & Bouakaz Reference Doinikov, Aired and Bouakaz2011; Hay et al. Reference Hay, Ilinskii, Zabolotskaya and Hamilton2013), which capture more complex interactions than the simple image method. Nevertheless, despite its simplicity, the method of images accurately predicts the experimentally observed resonance size and damping of wall-attached bubbles, showing strong agreement with both radial dynamics (figure 7) and shape-mode pressure inception (figure 11), which we examine in detail later. To solve equation (5.3), it is necessary to determine the gas pressure $p_{ { g}}$ within the bubble. Given its simplicity, the assumption of a polytropic process is widely employed: $p_{ { g}} = p_{ { g,0}} ({R_0}/{R} )^{3n}, \ p_{ { g,0}} = p_{\infty } + ({2\sigma _{{ lg}}}/{R_0}),$ with $p_{ { g,0}}$ denoting the bubble internal pressure at rest, $R_0$ the equilibrium bubble radius and $n$ the polytropic index.

Figure 7. Experimental and predicted time evolution of the equivalent radius of a bubble in contact with a glass substrate subjected to an ultrasound driving at a frequency $f_{ { d}} = {30}\,\textrm {kHz}$ . In (a), the bubble has an an equilibrium radius $R_0 = {89.0}\;\unicode{x03BC} {\textrm{m}}$ and the ultrasound pressure amplitude is $p_{ { a}} = {2.1}\,{\textrm{kPa}}$ . In (b), the bubble has an an equilibrium radius $R_0 = {102.6}\;\unicode{x03BC} {\textrm{m}}$ and the ultrasound pressure amplitude is $p_{ { a}} = {4.6}\,{\textrm{kPa}}$ . Theoretical predictions are obtained by solving the Rayleigh–Plesset equation using the following physical parameters: $p_{\infty } \,{=}\,{101.0}\,{\textrm{kPa}}$ , $\sigma _{{ lg}} \,{=}\, {72}\,\textrm {mN}\,{\textrm {m}^{-1}}$ , $c_{{ l}} \,{=}\, {1481}\,\textrm {m}\,{\textrm {s}^{-1}}$ , ${\,{\,\rm \mu}} _{ { l}} \,{=}\, {9.54}\times {10^{-4}}\,{\textrm{Pa}}\,{\textrm{s}}$ and $\rho _{ { l}} \,{=}\, {997.8}\,\textrm {kg}\,{\textrm {m}^{-3}}$ . The prediction obtained using the polytropic gas model, with $n = \gamma = 1.4$ , is shown by the dotted grey line. The prediction obtained using the Zhou gas model, with parameters $\gamma =1.4$ , $K_{{ g}} = {0.026}\,\textrm {W}\,{\textrm {m}^{-1}}{\textrm {K}^{-1}}$ , ${\mathcal{R}} = {287}\,\textrm {J}\,{\textrm {kg}^{-1}}\,{\textrm {K}^{-1}}$ and $T_{ { g}}|_{R} = {295}\,\textrm {K}$ , is represented by the continuous black line.

In figure 7, the predictions of the theoretical model are compared with the experimental radial time evolution of bubbles subjected to ultrasound forcing at a frequency of $f_{ { d}} = {30}\,\textrm {kHz}$ . We assume for the polytropic gas model that the gas within the bubble undergoes an adiabatic process, characterised by an exponent $n = \gamma$ , where $\gamma = 1.4$ denotes the specific heat ratio for the gas. This assumption is justified by the high Péclet number, $ Pe = {R_0^2 \omega _{ { d}}}/{D_{ { g}}} \sim 100$ , where $\omega _{ { d}}$ is the driving angular frequency and $D_{ { g}}$ represents the thermal diffusivity of the gas. Figure 7(a) illustrates the case of a resonant-sized bubble with an equilibrium radius $R_0 = {89.0}\;\unicode{x03BC} {\textrm{m}}$ and ultrasound pressure amplitude $p_{{ a}} = {2.1}\,{\textrm{kPa}}$ . Here, the theoretical prediction based on the polytropic gas model significantly overestimates the experimental results and manifests a continuous growth behaviour. Conversely, figure 7(b) shows results for an over-resonant bubble with an equilibrium radius $R_0 = {102.6}{\;\rm \unicode{x03BC}} {\textrm{m}}$ and ultrasound pressure amplitude $p_{ { a}} = {4.6}\,{\textrm{kPa}}$ . In this case, the theoretical curve obtained using the polytropic gas model exhibits a pronounced beating pattern, which is absent in the experimental data. A beating phenomenon arises from the interference between two signals with different frequencies, $f_1$ and $f_2$ , where the beat frequency is given by $f_{ { beat}} = |f_2 - f_1|$ . The observed beat frequency of approximately ${4}\,\textrm {kHz}$ suggests that the second signal, which overlaps with the forced response of the bubble at ${30}\,\textrm {kHz}$ , corresponds to the free response at the resonance frequency, which for this bubble size is approximately ${26}\,\textrm {kHz}$ . From this, we deduce that the strong overestimation seen in figure 7(a) results from constructive interference between the forced and free responses, both occurring at approximately the same frequency for this bubble size. We attribute the anomalous, underdamped free response, absent in the experimental observations, to the lack of thermal damping in the polytropic gas model. Thermal damping, as discussed by Chapman & Plesset (Reference Chapman and Plesset1971), is the dominant damping mechanism for bubbles of the sizes considered in this study.

To address the thermal interaction, we employ the Zhou model (Zhou Reference Zhou2021) for the bubble gas pressure $p_{ { g}}$ . A detailed description of the model is provided in Appendix E. Figure 7 illustrates that the theoretical predictions derived from this model align remarkably well with the experimental results, showing a much milder beating pattern.

5.2. Meniscus waves

Upon bubble oscillations, travelling surface waves are emitted from the pinning line. These waves move towards the bubble symmetry axis, $z$ , and equilibrate to form an axisymmetric harmonic standing wave that oscillates at the same frequency as the ultrasound applied. Figure 8 displays the initial emission of the wave, its propagation and the formation of a steady-state pattern. This phenomenon has previously been observed and investigated on flat liquid surfaces within containers, where a meniscus forms along the walls (Douady Reference Douady1990; Torres et al. Reference Torres, Pastor, Jiménez and De Espinosa1995; Batson, Zoueshtiagh & Narayanan Reference Batson, Zoueshtiagh and Narayanan2013; Ward, Zoueshtiagh & Narayanan Reference Ward, Zoueshtiagh and Narayanan2019; Shao et al. Reference Shao, Wilson, Saylor and Bostwick2021). When the container is subjected to harmonic vertical acceleration, travelling edge waves are emitted from the walls. These waves are not generated by a parametric instability but rather by synchronous adjustments of the meniscus profile in response to the harmonic modulation of the acceleration field, resulting in the harmonic emission of an axisymmetric wave from the wall in order to conserve fluid mass. Their occurrence requires no minimum threshold in the magnitude of the driving acceleration, and their amplitude is linearly proportional to this magnitude. The meniscus forms when the contact angle deviates from $90^\circ$ , and meniscus waves occur regardless of whether the contact line is pinned or free. These waves can be suppressed by filling the liquid container to the brim, thereby pinning the contact line at the brim with a contact angle of $90^\circ$ . To our knowledge, our study is the first to recognise the presence of meniscus waves on ultrasound-driven bubbles in contact with a solid surface. In this case, the presence of a wall mandates that the acceleration of the bubble interface near the wall must be parallel to the wall, as dictated by the non-penetration condition. As a result, when the contact angle deviates from $90^\circ$ , the acceleration along the pinning line of the bubble is not orthogonal to the bubble surface and meniscus waves are emitted. Conversely, when a bubble in contact with a wall exhibits a contact angle of $90^\circ$ meniscus waves are not expected to occur, as the direction of acceleration is orthogonal to the bubble surface at all points.

Figure 8. Time evolution of harmonic meniscus waves produced by the harmonic acceleration of the bubble surface close to the wall. The wave pattern is highlighted by performing a subtraction operation between the X-ray image capturing the wave at its peak and that with the wave at its trough. (a) Initial emission of the travelling meniscus wave. (b) Propagation of the travelling meniscus wave. (c) Steady state of meniscus waves forming a standing-wave pattern.

5.3. Faraday waves

When the amplitude of the ultrasound pressure exceeds a critical threshold, a half-harmonic parametric instability known as Faraday instability emerges, supplanting the harmonic pattern of meniscus waves. The resulting standing-wave pattern, also termed shape mode, is the result of the nonlinear interaction between the external periodic forcing and the natural frequencies of the bubble interface. This interaction leads to the amplification of certain modes while damping others, resulting in the observable patterns. Initially, the Faraday wave pattern exhibits axisymmetry, akin to meniscus waves, resulting in a purely zonal shape mode. The wavelength of Faraday waves is approximately twice that of meniscus waves, and its amplitude is substantially higher, as evidenced in figure 9(a,b). As time proceeds, Faraday waves progressively lose their axisymmetry, ultimately stabilising into a non-axisymmetric shape mode. This finding supports the theoretical prediction for spherical entities as stated by Chossat et al. (Reference Chossat, Lauterbach and Melbourne1991), which asserts that no axisymmetric mode with $l \geqslant 2$ is inherently stable. The ultimate shape mode of the bubble results from the superposition of a sectoral mode onto the pre-existing zonal mode, as depicted in figure 9(c). This transition is evidenced by the development of lobes in the azimuthal direction, which overlap with the existing axisymmetric lobes in the polar direction. The absence of layer-symmetric patterns indicates that tesseral shape modes are not present. This contrasts with the findings of Fauconnier et al. (Reference Fauconnier, Béra and Inserra2020), who reported the presence of tesseral modes. However, their study employed a different experimental set-up: the substrate was made of PMMA, the ultrasound was oriented parallel to the substrate and shape modes were identified solely from a top-view perspective.

Figure 9. Comparison between meniscus waves and Faraday waves on the same bubble as that of figure 8. (a) Meniscus waves are harmonic waves whose emergence is not contingent upon surpassing a minimum amplitude threshold in the ultrasound driving. (b) Faraday waves are half-harmonic waves characterised by a wavelength approximately double that of meniscus waves and a significantly higher amplitude. Their occurrence is contingent upon surpassing a critical threshold in the ultrasound driving amplitude. Initially, Faraday waves are axisymmetric, i.e. the shape mode is purely zonal. (c) Over time, Faraday waves gradually lose their axisymmetry, manifesting a sectoral mode that overlays the pre-existing zonal mode. This transition is marked by the emergence of lobes in both the polar and azimuthal directions. The persistence of the bottom lobe indicates the absence of tesseral modes.

Given the truncated geometry of a wall-attached bubble, the degree $l$ of the observed zonal shape mode, generally a real number, can be estimated only by measuring the spacing between the ridges of the shape mode as seen from the side view and comparing it with the circumference of the bubble. The length of the arc $a$ subtended by the chord of length $c$ joining two ridges of the shape mode is calculated as

(5.4) \begin{equation} a = 2R_0\sin ^{-1}\left (\frac {c}{2 R_0}\right )\!. \end{equation}

Thus, $l$ is given by the number of times the arc length can divide the bubble circumference:

(5.5) \begin{equation} l = 2\pi \frac {R_0}{a}. \end{equation}

Figure 10. Experimentally measured shape mode degree (a) and order (b) of bubbles in contact with a glass substrate, subjected to an ultrasound driving frequency of $f_{ { d}} = {30}\,\textrm {kHz}$ , as a function of their equilibrium radius. The degree varies continuously with the bubble radius, whereas the order is quantised and only assumes integer values.

Conversely, the order $m$ of the observed sectoral shape mode can be readily determined by counting the number of lobes along the azimuthal direction when the bubble is viewed from above. Since the bubble contour remains closed and periodic in this view, $m$ must be an integer. At a fixed ultrasound driving frequency $f_{ { d}}={30}\,\textrm {kHz}$ , the measured values of $l$ vary approximately linearly with the bubble equilibrium radius $R_0$ , while the values of $m$ show a stepwise progression, as illustrated in figure 10. The fact that $l$ spans a practically continuous range – unlike the quantised behaviour expected under strict boundary conditions – suggests that the contour endpoints of the bubble are effectively free. This interpretation is supported by the observed motion of the contact line and changes in contact angle during bubble oscillation. Thus, the current configuration can be considered as corresponding to the idealised scenario of a wall-attached bubble with free boundary conditions, exhibiting a spectrum of shape modes that is continuous in $l$ and discrete in $m$ , as discussed in § 3.2 and shown in figure 5(d). It can be determined from figure 10 that, for a given radius $R_0$ , the value of $m$ of the sectoral mode that manifests is approximately the integer closest to the observed value of $l$ for that radius. Therefore, the bubble deformation $\eta (\theta ,\phi ,t)$ can be decomposed as

(5.6) \begin{equation} \eta (\theta ,\phi ,t) = \underbrace {a_0^0(t)Y_0^0(\theta ,\phi )}_{\text{Breathing mode}} + \underbrace {a_l^0(t)Y_l^0(\theta ,\phi )}_{\text{Zonal mode}} + \underbrace {a_l^{\lfloor l \rceil }(t)Y_l^{\lfloor l \rceil }(\theta ,\phi )}_{\text{Sectoral mode}}, \end{equation}

where $\lfloor \rceil$ indicates the rounding operator. It is important to note that, for each value of $l$ , only a specific pair of values of $m$ manifest from the set of possible values, specifically the zeroth value and the highest possible value of $m$ for the given $l$ . Consequently, under the present wetting conditions, the spectrum of shape modes is degenerate, similar to that of a free bubble (Cattaneo et al. Reference Cattaneo, Guerriero, Shakya, Krattiger, G. Paganella, Narciso and Supponen2025), with the difference that the values of $l$ are real and continuous rather than integer.

Figure 11. Shape-mode order $m$ of bubbles in contact with a glass substrate, subjected to an ultrasound driving frequency $f_{ { d}} = {30}\,\textrm {kHz}$ , as a function of the equilibrium radius and the ultrasound pressure amplitude. Experimental data are depicted with symbols, while the theoretical threshold for the onset of shape modes is based on the model from Francescutto & Nabergoj (Reference Francescutto and Nabergoj1978), as detailed in the text. The solid lines correspond to a surface tension value $\sigma _{{ lg}} = {65}\,\textrm {mN}\,{\textrm {m}^{-1}}$ , and the dotted lines represent $\sigma _{{ lg}} = {72}\,\textrm {mN}\,{\textrm {m}^{-1}}$ . The other parameters used in the model are: $\rho _{ { l}} = {997.8}\,\textrm {kg}\,{\textrm {m}^{-3}}$ , ${\,\rm \mu} _{{ l}} = {9.54}\times {10^{-4}}\,{\textrm{Pa}}\,{\textrm{s}}$ and ${\,\rm \mu} _{ { e}}$ extracted from the theoretical plots in Chapman & Plesset (Reference Chapman and Plesset1971).

The occurrence of the Faraday instability, as detected experimentally in relation to bubble size and ultrasound pressure amplitude, is summarised in figure 11. The instability region appears to be arranged into distinct tongues, each corresponding to the specific order $m$ observed. This finding is notable as the presence of distinct tongues indicates that the shape modes exhibit discrete resonance sizes for a fixed acoustic driving frequency. Given that the accessible range of shape-mode degrees $l$ is continuous, one would expect the associated resonant bubble sizes to be similarly continuous. However, the quantisation of the azimuthal wavenumber $m$ introduces a selectivity on the bubble size, resulting in a discrete number of resonance sizes. The experimentally observed instability thresholds agree well with those predicted by the theory from Francescutto & Nabergoj (Reference Francescutto and Nabergoj1978) and Nabergoj & Francescutto (Reference Nabergoj and Francescutto1978) developed for shape oscillations of free bubbles. Full details of our implementation of the Francescutto & Nabergoj model for wall-attached bubbles are provided in Appendix F. The minimum inception threshold for shape modes occurs at the bubble radius that corresponds to the resonant radius, which is also close to the resonance size for the shape mode with $m=4$ , making it the most easily excited shape mode. We note that to achieve alignment with the experimental data, we found it necessary to use a surface tension value of ${65}\,\textrm {mN}\,{\textrm {m}^{-1}}$ . This is lower than the surface tension of a clean air–water interface, which is around ${72}\,\textrm {mN}\,{\textrm {m}^{-1}}$ . The reduction in surface tension could be attributed to the presence of surfactants in the liquid medium that have adsorbed onto the bubble interface and thus decreased its surface tension. This value closely aligns with that reported in Versluis et al. (Reference Versluis, Goertz, Palanchon, Heitman, Van Der Meer, Dollet, De Jong and Lohse2010), where the authors investigated the shape modes of free air bubbles in an aqueous medium driven by ultrasound. The theory predicted by Francescutto & Nabergoj (Reference Francescutto and Nabergoj1978) achieves strong agreement with experiments even without accounting for meniscus waves, suggesting that these waves are unlikely to be a significant factor in triggering shape modes. This observation aligns with the findings of Batson et al. (Reference Batson, Zoueshtiagh and Narayanan2013), who determined in the context of flat water–air interfaces that significant influence of meniscus waves is more likely to occur for harmonic axisymmetric Faraday waves, as they share the same frequency and pattern with meniscus waves.

Figure 12. Time evolution of the experimental amplitude of the breathing, zonal and sectoral modes for the case $l \approx m=3$ ( $l =3.22$ ). The red dotted line marks the threshold amplitude of the breathing mode for the onset of shape modes. The amplitude of the breathing mode is shown as dashed during the occurrence of the sectoral mode, indicating that accurate measurement is not possible due to the loss of bubble axisymmetry. Below is a sequence of top-down optical and side-view X-ray images illustrating the dynamics of the bubble. The images have been denoised and their backgrounds have been removed for visual clarity. The equilibrium bubble radius is $R_0 = {72} {\;\rm \unicode{x03BC}} {\textrm{m}}$ . The ultrasound driving frequency is $f_{{ d}} = {30}\,\textrm {kHz}$ and its amplitude is $p_{ { a}} = {5.3} \,{\textrm{kPa}}$ .

Figure 13. Time evolution of the experimental amplitude of the breathing, zonal and sectoral modes for the case $l \approx m=4$ ( $l = 4.21$ ). The red dotted line marks the threshold amplitude of the breathing mode for the onset of shape modes. The amplitude of the breathing mode is shown as dashed during the occurrence of the sectoral mode, indicating that accurate measurement is not possible due to the loss of bubble axisymmetry. Below is a sequence of top-down optical and side-view X-ray images illustrating the dynamics of the bubble. The images have been denoised and their backgrounds have been removed for visual clarity. The equilibrium bubble radius is $R_0 = {89} {\;\rm \unicode{x03BC}} {\textrm{m}}$ . The ultrasound driving frequency is $f_{ { d}} = {30}\,\textrm {kHz}$ and its amplitude is $p_{ { a}} = {2.1} \,{\textrm{kPa}}$ .

Figure 14. Time evolution of the experimental amplitude of the breathing, zonal and sectoral modes for the case $l \approx m=5$ ( $l =4.85$ ). The red dotted line marks the threshold amplitude of the breathing mode for the onset of shape modes. The amplitude of the breathing mode is shown as dashed during the occurrence of the sectoral mode, indicating that accurate measurement is not possible due to the loss of bubble axisymmetry. Below is a sequence of top-down optical and side-view X-ray images illustrating the dynamics of the bubble. The images have been denoised and their backgrounds have been removed for visual clarity. The equilibrium bubble radius is $R_0 = {103} {\;\rm \unicode{x03BC}} {\textrm{m}}$ . The ultrasound driving frequency is $f_{{ d}} = {30}\,\textrm {kHz}$ and its amplitude is $p_{ { a}} = {4.4} \,{\textrm{kPa}}$ .

5.4. Time evolution of shape modes

The time evolution of the amplitude of bubble deformation components, specifically the breathing, zonal and sectoral modes, is presented in figures 12–15 for cases with $l \approx m = 3$ , $l \approx m = 4$ , $l \approx m = 5$ and $l \approx m = 6$ , respectively. The amplitude of the breathing mode $a_0^0(t)$ is approximately measured using the method outlined in § 5.1. This approach remains accurate as long as the bubble maintains an approximately axisymmetric shape. The amplitude $a_l^0(t)$ of a zonal shape mode $Y_{l}^{0}(\theta ,\phi )=P_{l}^{0}(\cos (\theta ))$ can be determined by calculating the inner product between the cross-section of the bubble deformation, $\eta (\theta ,0,t)$ , along the $z$ axis, and the shape mode itself:

(5.7) \begin{equation} a_l^0(t) = \frac {2l+1}{2} \langle \eta (\theta ,0,t), P_{l}^{0}(\cos (\theta )) \rangle = \frac {2l+1}{2}\ \int _{0}^{\alpha _0} \eta (\theta ,0,t) P_{l}^{0} (\cos (\theta )) \sin (\theta ) {\textrm d}\theta , \end{equation}

where $ {(2l+1)}/{2}$ is a normalisation factor that follows from Rodrigues’ formula. The amplitude $a_m^m(t)$ of a sectoral shape mode $Y_{m}^{m}(\theta ,\phi )= N_{m}^{m} P_{m}^{m} (\cos (\theta )) \cos (m \phi )$ can be computed in a similar way by computing the inner product between the cross-sections along the $x$ axis and $y$ axis of the bubble deformation $\eta (\pi /2,\phi ,t)$ and of the shape mode $Y_{m}^{m}(\pi /2,\phi )=\cos (m\phi )$ , respectively, as follows:

(5.8) \begin{equation} a_m^m(t) = \frac {1}{\pi } \left \langle \eta (\pi /2,\phi ,t), \cos (m\phi )\right \rangle = \frac {1}{\pi }\ \int _{0}^{2\pi } \eta (\pi /2,\phi ,t) \cos (m\phi ) {\textrm d}\phi . \end{equation}

However, under the present conditions, quantifying the amplitude of the zonal mode using (5.7) is unfeasible. The non-quantised nature of the shape mode precludes precise measurement of its degree. Moreover, the bubble deformation, constrained within the range $\theta = \pm \alpha _0$ , renders the calculation inherently ill-posed. Finally, the concurrent presence of a sectoral shape mode complicates the isolation of its specific contribution to the cross-sectional deformation. To address these challenges, we estimate the amplitude of the shape mode by measuring the amplitude of the lower-lobe oscillation, at $\theta = 0$ . By definition, it directly corresponds to the amplitude of the shape mode, and it remains unaffected by sectoral modes, as these modes exhibit zero deformation at $\theta = 0$ . These issues do not arise in estimating the sectoral mode amplitude due to the quantised nature of the shape-mode order, the periodicity of the deformation and the fact that the zonal mode merely introduces a uniform offset to the deformation in the examined cross-section.

Figure 15. Time evolution of the experimental amplitude of the breathing, zonal and sectoral modes for the case $l \approx m=6$ ( $l =6.02$ ). The red dotted line marks the threshold amplitude of the breathing mode for the onset of shape modes. The amplitude of the breathing mode is shown as dashed during the occurrence of the sectoral mode, indicating that accurate measurement is not possible due to the loss of bubble axisymmetry. Below is a sequence of top-down optical and side-view X-ray images illustrating the dynamics of the bubble. The images have been denoised and their backgrounds have been removed for visual clarity. The equilibrium bubble radius is $R_0 = {129}{\;\rm \unicode{x03BC}} {\textrm{m}}$ . The ultrasound driving frequency is $f_{{ d}} = {30}\,\textrm {kHz}$ and its amplitude is $p_{ { a}} = {10.5} \,{\textrm{kPa}}$ .

In all examined cases, the amplitude of the breathing mode ranges between 1.5 and 2.4 times the threshold required to trigger shape modes for the specific bubble size under investigation. The temporal evolution of shape modes exhibits consistent patterns across all cases. The zonal mode invariably emerges first, followed by the sectoral mode. At the breathing mode amplitudes relative to the shape-mode threshold employed, the zonal mode typically occurs after 10–14 ultrasound cycles. A notable exception is observed in the case of $l \approx m = 3$ , where the zonal mode initiates after just 4 cycles. Following the onset of the zonal mode, the sectoral mode typically manifests after 6–10 ultrasound cycles. The amplitude of the zonal mode increases approximately linearly with time until the inception of the sectoral mode. At its peak, the zonal mode is 6–8 times more intense than the breathing mode. This factor appears to increase as the interval between the onset of the zonal mode and the initiation of the sectoral mode prolongs. The pronounced amplification of bubble deformation produced by shape modes arises from their intrinsic capacity to concentrate kinetic energy at specific regions on the bubble surface, the shape-mode lobes. This contrasts with the breathing mode, where kinetic energy and deformation are distributed uniformly across the entire bubble surface. The concentration of energy at specific points produced by shape modes is the key factor that enables jet formation at low-amplitude ultrasound pressures, as shown in this study. Upon the onset of the sectoral mode, the amplitude of the zonal mode ceases to increase and begins to decline, stabilising at approximately 0.5–0.7 times its prior peak value. Simultaneously, the intensity of the sectoral mode grows to a level comparable to that of the zonal mode, indicating a possible energy transfer from the zonal to the sectoral mode. This decrease in the overall shape-mode intensity suggests that the most favourable conditions for jet formation occur when only the zonal mode is active, as this configuration results in the highest deformation amplitude. For the remainder of the recording period, the zonal and sectoral modes exhibit minimal variation, maintaining a largely stable pattern. We do not observe significant changes in the amplitude of the breathing mode when the zonal mode develops. Once the sectoral mode emerges, extracting volume becomes troublesome, as the bubble shape loses axisymmetry. Therefore, we can only assume that the amplitude remains constant during this regime as well.

The delayed emergence of the ultimate shape-mode pattern observed in wall-bounded bubbles contrasts with the behaviour of free bubbles, where the final Faraday pattern appears directly, without transitioning through an intermediate stage of purely axisymmetric shape modes (Cattaneo et al. Reference Cattaneo, Guerriero, Shakya, Krattiger, G. Paganella, Narciso and Supponen2025). One possible explanation is the friction at the contact line, which may restrict the bubble oscillatory motion along the azimuthal direction, thus hindering the onset of surface destabilisation in that direction. In contrast, motion along the polar direction is not similarly constrained, making zonal modes easier to develop initially. These zonal modes, superimposed on the breathing-mode spherical oscillation, may then generate sufficient surface deformation to eventually trigger the onset of sectoral modes. Interestingly, this progression is not unprecedented – it has been observed in sessile droplets on vibrating surfaces, where zonal modes appear first. Subsequently, the radial oscillation of these axisymmetric waves induces azimuthal waves at the contact line, leading to the formation of sectoral patterns (Vukasinovic et al. Reference Vukasinovic, Smith and Glezer2007; Panda et al. Reference Panda, Kahouadji, Tuckerman, Shin, Chergui, Juric and Matar2023).

5.5. Instability threshold for repeated jets

When the amplitude of the shape mode exceeds a critical threshold, the inward folding of the bottom shape-mode lobe generates a high-velocity jet directed towards the substrate. After onset, jetting occurs at each cycle of the shape mode. Given that the shape mode exhibits a half-harmonic behaviour, the jet frequency is half that of the applied ultrasound driving. Shape-mode-induced cyclic jets on bubbles have likely been observed in previous studies that documented repeated jets resulting from surface deformations at driving frequencies spanning hertz (Crum Reference Crum1979), kilohertz (Prabowo & Ohl Reference Prabowo and Ohl2011) and megahertz (Vos et al. Reference Vos, Dollet, Versluis and De Jong2011) ranges.

Figure 16. (a) X-ray image sequence of shape-mode-induced jetting. The bottom lobe undergoes maximum excursion, and when it rapidly folds inwards, a singularity forms at the fluid interface, where both surface curvature and velocity diverge, leading to the formation of a high-speed jet directed towards the substrate, driven by the self-focusing of kinetic energy at the surface singularity. The equilibrium radius of the bubble is ${121}{\;\rm \unicode{x03BC}} {\textrm{m}}$ , the driving pressure is $p_{ { a}} = {9.4}\,{\textrm{kPa}}$ and the frequency is $f_{ { d}} = {30}\,\textrm {kHz}$ . (b) X-ray image sequence of ultrasound-driven inertial jetting. The bubble forms a jet as soon as the ultrasound amplitude reaches its steady state, leaving no time for shape modes to develop. This produces a single, quick jet followed by rapid bubble fragmentation. The equilibrium radius of the bubble is ${91}{\;\rm \unicode{x03BC}} {\textrm{m}}$ , the driving pressure is $p_{ { a}} = {17}\,{\textrm{kPa}}$ and the frequency is $f_{ { d}} = {30}\,\textrm {kHz}$ . The background of the images has been removed for visual clarity.

Figure 16(a) illustrates that, for a bubble with an equilibrium radius $R_0 = {121}{\;\rm \unicode{x03BC}} {\textrm{m}}$ , subjected to a driving pressure $p_{ { a}} = {9.4}\,{\textrm{kPa}}$ and frequency $f_{ { d}} = {30}\,\textrm {kHz}$ , just before jet emission occurs, the bottom shape-mode lobe folds inward, deforming into an approximate truncated cone. This creates a singularity point at the fluid interface, characterised by divergent surface curvature and velocity. The resulting self-focusing of kinetic energy at this singularity point initiates the jet formation. Jets originating from the bottom shape-mode lobe are predominantly observed due to its higher amplitude compared with other lobes, as it is not constrained by the substrate. This contrasts with free bubbles, where all lobes exhibit equal amplitude, allowing simultaneous jet formation when their amplitude is sufficiently high (Cattaneo et al. Reference Cattaneo, Guerriero, Shakya, Krattiger, G. Paganella, Narciso and Supponen2025). This jetting phenomenon can be considered the spherical analogue of the jets formed by Faraday waves on vertically vibrating liquid baths (Longuet-Higgins Reference Longuet-Higgins1983; Zeff et al. Reference Zeff, Kleber, Fineberg and Lathrop2000). Jets arising from a collapsing conical interface are also observed in other phenomena, including bubble bursting at fluid interfaces (Kientzler & Arons Reference Kientzler and Arons1954; Duchemin et al. Reference Duchemin, Popinet, Josserand and Zaleski2002), droplet impact on liquid pools (Worthington & Cole Reference Worthington and Cole1897; Thoroddsen et al. Reference Thoroddsen, Takehara, Nguyen and Etoh2018), cavitation bubble collapse in extremely close proximity to solid boundaries (Lechner et al. Reference Lechner, Lauterborn, Koch and Mettin2019; Reuter & Ohl Reference Reuter and Ohl2021), cavitation-bubble-pair interactions (Mishra et al. Reference Mishra, Bourquard, Roy, Lakkaraju, Ghosh and Supponen2022; Fan et al. Reference Fan, Bussmann, Reuter, Bao, Adami, Gordillo, Adams and Ohl2024) and bubble coalescence (Jiang et al. Reference Jiang, Rotily, Villermaux and Wang2024).

Shape-mode-induced jets are fundamentally different from classical inertial jets, which require considerably higher acoustic pressures to form – almost an order of magnitude more. Figure 16(b) illustrates an example of ultrasound-driven inertial jetting for a bubble with an equilibrium radius $R_0 = {91}{\;\rm \unicode{x03BC}} {\textrm{m}}$ , which is near its resonant size. The bubble is subjected to a driving pressure $p_{ { a}} = {17}\,{\textrm{kPa}}$ and frequency $f_{ { d}} = {30}\,\textrm {kHz}$ . Here, the bubble jets immediately upon reaching the steady-state ultrasound amplitude, leaving no time for shape modes to develop. The result is a single, swift jet followed by rapid fragmentation of the bubble. Unlike jets formed by shape modes, this jet arises from the intense pressure gradient applied to the bubble rather than from an instability at the interface. Thus, shape-mode-induced jets represent a unique mechanism for concentrating energy through relatively low acoustic excitation, marking a distinct contrast to the behaviour of classical inertial jets.

Figure 17. (a) Jetting occurrence as a function of bubble size, shape-mode order and driving ultrasound pressure. A minimum in driving pressure occurs at the resonant radius. (b) Jetting occurrence as a function of bubble size, shape-mode order and acceleration of the bottom shape-mode lobe. A distinct threshold in acceleration, regardless of bubble size, is identified.

In figure 17(a), we illustrate the relationship between bubble size and the ultrasound pressure needed to cause shape-mode-induced jet formation. The required pressure reaches a minimum at the resonant radius for the applied driving frequency. In figure 17(b), we identify a distinct threshold in the acceleration of the bottom shape-mode lobe of approximately $0.2$ μm μs⁻², irrespective of the bubble size, beyond which jetting occurs. The lobe acceleration $\ddot z_{ { lobe}}$ is estimated from the zonal shape mode amplitude as $\ddot z_{{ lobe}} \approx \omega _{0,l}^2 a_{l}^0$ , where $\omega _{0,l} = {\omega _{\textrm { {d}}}}/{2}$ is the angular frequency of the shape mode. This consistent acceleration threshold arises because Faraday instability-driven jetting depends solely on the motion of individual shape-mode lobes, which is independent of bubble size. While increasing the bubble size increases the number of lobes, their individual width – set by the driving frequency – remains unchanged. As a result, the critical lobe height and corresponding acceleration required for jetting remain constant across bubble sizes. This behaviour is fundamentally distinct from that of inertial jets, where jet strength scales with the pressure difference across the bubble, and therefore depends on both the magnitude of the pressure gradient and the bubble size (Supponen et al. Reference Supponen, Obreschkow, Tinguely, Kobel, Dorsaz and Farhat2016).

The speed of jets generated by shape modes is found to reach $u_{ { jet}} = {30}{\,\textrm {m s}^{-1}}$ . Using the fluid hammer pressure formula, $p_{ { wh}} = \rho _{ { l}} c_{ { l}} u_{{ jet}}$ , for a water flow impinging on a rigid material, the estimated impact pressure on the substrate is calculated to be up to 45 MPa. This suggests that even at relatively low ultrasound driving pressures, the repeated jet formations are sufficiently powerful to cause damage to biological and ductile materials. This mechanism may hold significant importance in ultrasound-based cleaning applications and, as demonstrated in our previous work, plays a pivotal role in sonoporation for targeted drug delivery (Cattaneo et al. Reference Cattaneo, Guerriero, Shakya, Krattiger, G. Paganella, Narciso and Supponen2025). As it displaces, the jet may eventually break into multiple droplets following the Rayleigh–Plateau instability. Jetting persists indefinitely unless the emergence of the sectoral shape mode dampens the lobe amplitude below the critical threshold, or the bubble shape becomes too chaotic as a result of the disruptive influence of jets on the shape itself.

Figure 18. Comparison of the time evolution of the bubble interface between experimental observations and numerical simulations using the BEM, shown from both top- and side-view perspectives. (a) Transition from a purely zonal shape mode to a combined zonal and sectoral mode. The bubble has an equilibrium radius $R_0 = {120}\,\rm {\;\rm \unicode{x03BC}} {\textrm{m}}$ and is driven by ultrasound with an amplitude $p_{{ a}} = {9.4}\,{\textrm{kPa}}$ at a frequency $f_{ { d}} = {30}\,\rm {kHz}$ . (b) Jetting formation during an almost purely zonal shape mode. The bubble has an equilibrium radius $R_0 = {129}\,\rm {\;\rm \unicode{x03BC}} {\textrm{m}}$ and is driven by ultrasound with an amplitude $p_{ { a}} = {10.5}\,{\textrm{kPa}}$ at a frequency $f_{{ d}} = {30}\,\textrm {kHz}$ .