Shape modes and jet formation on ultrasound-driven wall-attached bubbles

TL;DR

This work investigates how micrometer-scale wall-attached bubbles respond to ultrasound, revealing a four-stage progression of interfacial instabilities: spherical breathing, harmonic axisymmetric meniscus waves, half-harmonic axisymmetric Faraday waves (zonals), and their superposition with sectoral Faraday waves. A dual-view imaging approach (top-view visible and side-view X-ray) captures the full three-dimensional shape evolution, uncovering a continuous spectrum in shape-mode degree l on a wall, with discrete azimuthal order m, and a threshold-driven transition to jetting from the non-substrate side. The theoretical framework combines a kinematic wetting model and Legendre-function-based shape modes under different boundary conditions, while predictions for Faraday thresholds are reconciled with experiments using a thermal-damping gas model (Zhou). Three-dimensional boundary-element simulations corroborate the observed dynamics and quantify jet formation at low acoustic pressures, highlighting a jetting mechanism that concentrates energy at a surface lobe and yields high impact pressures, with direct relevance to ultrasound-assisted cleaning and drug delivery. Together, the study advances understanding of wall-confined bubble dynamics, bridging theory, high-speed imaging, and numerical simulations to illuminate shape-mode selection and jetting behavior near solid boundaries.

Abstract

Understanding how substrate-attached bubbles respond to ultrasound is important for applications from industrial cleaning to biomedical therapy. Under ultrasonic excitation, bubbles can deform through Faraday instability and periodically emit high-speed jets. Although this behavior is increasingly well understood for free bubbles, the dynamics of wall-attached bubbles remain largely unexplored. In particular, the three-dimensional selection and evolution of non-spherical modes and their relation to jetting have not been resolved. We investigate micrometric air bubbles in contact with a rigid substrate and driven by ultrasound, using a dual-view imaging setup combining top-view bright-field microscopy with side-view phase-contrast X-ray imaging. This approach reveals a stepwise evolution of bubble shape through four regimes: spherical oscillations, harmonic axisymmetric meniscus waves, half-harmonic axisymmetric Faraday waves, and the superposition of half-harmonic sectoral Faraday waves. This contrasts with free bubbles, which jump directly to their final Faraday pattern at instability onset. For the chosen substrate, the observed shape-mode spectrum is degenerate and spans a continuous range of mode degrees, consistent with theoretical predictions based on kinematic arguments. Free bubbles, although also degenerate, remain limited to discrete spherical harmonics. Measured ultrasound pressure thresholds for Faraday instability agree with classical interface-stability theory modified for a rigid boundary. Complementary 3D boundary-element simulations reproduce the observed shape evolution. Finally, we identify the acceleration threshold for cyclic jetting: unlike free bubbles, wall-attached bubbles always jet from the side not constrained by the substrate.

Figures (20)

• Figure 1: Shape modes of a spherical interface ordered by degree $l$ and order $m$, up to the sixth degree. Zonal modes occupy the column marked in red ($m=0$), sectoral modes are located along the highlighted diagonal in blue ($l=m$), and tesseral modes are distributed across the remaining yellow area ($l\neq m$). The $l=0$ mode corresponds to a uniform spherical deformation and is commonly referred to as breathing mode.
• Figure 2: (a) Sessile water droplet with a 5mm radius subjected to vertical vibrations at a frequency of 1080Hz. Reproduced with permission from the work by Vukasinovic2000Vibration-InducedAtomization. (b) Wall-attached air bubble with a 68µm radius subjected to ultrasound driving at a frequency of 100kHz. Reproduced with permission from the work by Cattaneo2023JetSubstrate.
• Figure 3: (a) Schematics of the experimental setup. (C1-C2) Cameras, (GC) Glass capillary, (LI) LED illuminator, (MC) Microfluidic chip, (OL1-OL2) Objective lenses, (S) Scintillator, (SH) Sample holder, (SR) Sound reflector, (TL1-TL2) Tube lenses, (TW1-TW2) Telescopic windows, (US) Ultrasound transducer. (b) Geometry of the microfluidic bubble-generator chip. A single bubble is diverted from the upward bubble stream and propelled towards the bottom of the glass capillary using a manually operated syringe. (c) Testing conditions with a single microbubble positioned underneath a glass capillary. (d) Acoustic driving pressure produced by the ultrasound transducer and measured by a needle hydrophone, normalised to the steady-state amplitude value.
• Figure 4: Definition sketch of a bubble resting against a rigid flat substrate with a static contact angle $\alpha_0$. (a) and (b) depict the cross-sections along the planes defined by projection lines $\text{A}-\text{A}$ and $\text{B}-\text{B}$, respectively. The equilibrium surface, represented by a dashed line, is defined as ${\boldsymbol{R}}_0(\theta,\phi)$, where polar angle $\theta \in [-\alpha_0,\alpha_0]$ and the azimuthal angle $\phi \in [0, 2\pi]$ serve as surface coordinates. The time-dependent surface deformation is denoted as $\delta{\boldsymbol{R}}(\theta,\phi,t)$. The deformed surface ${\boldsymbol{R}}_0(\theta,\phi) + \delta{\boldsymbol{R}}(\theta,\phi,t)$ is depicted by a solid line. $\boldsymbol{n}$, $\boldsymbol{t}$ and $\boldsymbol{b}$ denote the normal, tangential and binormal unit vectors to the equilibrium surface, respectively. The contact line $\boldsymbol{\Gamma}$ is shown as a dotted line. (c) and (d) represent the surface deformation normal to the equilibrium surface $\delta{\boldsymbol{R}}(\theta,\phi,t) \cdot \boldsymbol{n} = \eta$ in the $\text{A}-\text{A}$ and $\text{B}-\text{B}$ plane, respectively.
• 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 $\alpha_0 = 3\pi/4$. $l$ and $m$ are the degree and order, respectively, of the spherical harmonic $Y_{l_{k}}^{m}$. $k$ 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.