100-Billion-Atom Molecular Dynamics Simulation of Acoustic Cavitation in a Simple Liquid

Yuta Asano

arxiv: 2601.13594 · v2 · submitted 2026-01-20 · ⚛️ physics.flu-dyn

100-Billion-Atom Molecular Dynamics Simulation of Acoustic Cavitation in a Simple Liquid

Yuta Asano This is my paper

Pith reviewed 2026-05-16 12:59 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn

keywords molecular dynamicsacoustic cavitationbubble clusterultrasonic irradiationsubharmonic behaviorlarge-scale simulationpressure temperature

0 comments

The pith

A 100-billion-atom simulation reveals cavitation bubbles forming clusters that split and merge in sync with ultrasonic oscillations near the horn, with minimal impact on sound wave properties.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper conducts a massive molecular dynamics simulation with about 100 billion atoms to model acoustic cavitation in a simple liquid under ultrasonic irradiation. It directly observes that bubbles nucleate near the horn and organize into a large cluster which periodically splits into smaller ones and then recombines, locked to the driving frequency. Inside the bubbles, pressure and temperature rise sharply when clusters fragment, and oscillations show subharmonic components over longer timescales, matching experiments. Importantly, these bubble dynamics do not significantly alter the acoustic properties of the sound wave in the bulk liquid. The work provides molecular-level details on cavitation processes relevant to chemical and biomedical uses of ultrasound.

Core claim

In a molecular dynamics simulation of approximately 100 billion atoms representing a simple liquid subjected to ultrasonic waves from a horn, cavitation bubbles nucleate and grow near the horn surface. These bubbles coalesce into a single large cluster that repeatedly fragments into multiple smaller clusters and subsequently merges, with this cycle synchronized to the horn's oscillation period. During the fragmentation phases, both pressure and temperature within the bubbles increase sharply. The amplitudes of these oscillations vary on timescales exceeding the driving period, indicating subharmonic behavior. Despite the presence of these bubbles, the propagation characteristics of the acous

What carries the argument

The 100-billion-atom molecular dynamics simulation, which enables direct observation of multi-bubble cluster formation, splitting, and merging that smaller simulations could not capture.

Load-bearing premise

The classical interatomic potentials and finite system size used here faithfully reproduce real multi-bubble cavitation dynamics without major artifacts from the model choices or boundaries.

What would settle it

Re-running the simulation with a different interatomic potential or substantially larger system size and checking whether the periodic cluster splitting-merging cycle and subharmonic pressure spikes persist would test the findings.

Figures

Figures reproduced from arXiv: 2601.13594 by Yuta Asano.

**Figure 1.** Figure 1: Snapshots of the density field at (a) time t = 4tp, (b) t = 6tp, (c) t = 8tp, and (d) t = 10tp. Time is normalized by tp, the driving period of the ultrasonic horn. Only low-density regions are shown to highlight bubble nucleation and growth near the ultrasonic horn. The insets of each panel depict the enlarged view near the horn. essential for accurate modeling without excessive overhead. This insight is … view at source ↗

**Figure 2.** Figure 2: Void-fraction distributions at (a) t = 4tp, (b) t = 6tp, (c) t = 8tp, and (d) t = 10tp. Time is normalized by tp. The purple lines denote the average void fraction and the standard deviation of subcells at each x, while the green lines denote the maximum void fraction. Because the initial state was set at the gas–liquid coexistence point, a uniform void field (∼ 0.003) exists at the initial time. and ampli… view at source ↗

**Figure 3.** Figure 3: Cross-sectional snapshots of the density field on the yz-plane at x = 15 for (a) t = 4tp, (b) t = 6tp, (c) t = 8tp, and (d) t = 10tp. Time is normalized by tp. Numerous small bubbles appear and expand over time. for Computational Science (Project ID: hp240112).This research was partially supported by JSPS KAKENHI, Grant No. JP23K03242. The author also acknowledges the Supercomputer Center, Institute for So… view at source ↗

**Figure 4.** Figure 4: Gas clusters classified by ID at (a) t = 4tp, (b) t = 6tp, (c) t = 8tp, and (d) t = 10tp. Time is normalized by tp Clusters are labeled in descending order of size, and the largest cluster is shown in green. IDs of 10 or higher are shown in red, and single-cell clusters are shown in yellow. 12/19 [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗

**Figure 5.** Figure 5: Time evolution of the total gas-phase volume and the volume of the largest gas cluster. Time is normalized by tp. Subcells with ρ < 0.32 are identified as gas phase. 13/19 [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗

**Figure 6.** Figure 6: Histograms of bubble-cluster sizes at (a) t = 9tp, (b) t = 9.2tp, (c) t = 9.4tp, and (d) t = 9.6tp, illustrating the periodic fragmentation and recombination of the largest cluster, a process linked to subharmonicrelated oscillations. Time is normalized by tp. 14/19 [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗

**Figure 7.** Figure 7: Time evolution of the average pressure in the gas-phase region and within the largest gas cluster. Time is normalized by tp. Pressure was evaluated using the virial theorem; sharp peaks correspond to fragmentation events associated with subharmonic behavior. 15/19 [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗

**Figure 8.** Figure 8: Time evolution of the average temperature in the gas-phase region and within the largest gas cluster. Temperature was evaluated using the equipartition theorem for kinetic energy. 16/19 [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗

**Figure 9.** Figure 9: Snapshots of pressure waveforms at (a) t = 4tp, (b) t = 6tp, (c) t = 8tp, and (d) t = 10tp. Time is normalized by tp The purple lines denote the average pressure and its standard deviation of subcells at each x, and the green and blue curves indicate the maximum and minimum pressures at each x-position, respectively. 17/19 [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗

read the original abstract

A large-scale molecular dynamics (MD) simulation of acoustic cavitation in a simple liquid was performed using the supercomputer Fugaku. The system, consisting of approximately 100 billion atoms, was subjected to ultrasonic irradiation. Direct observation of multi-bubble dynamics has been challenging in both experimental measurements and conventional numerical fluid mechanics simulations. Moreover, previous MD simulations involving only hundreds of millions of atoms were unable to generate multiple bubbles within a system. Our results reveal that cavitation bubbles nucleate and grow near the ultrasonic horn, forming a large bubble cluster that periodically splits into multiple small clusters and subsequently merges again. This cycle is synchronized with the oscillation period of the horn. Pressure and temperature inside the bubbles exhibit sharp increases during cluster fragmentation, and their oscillation amplitudes vary on a timescale longer than the driving period of the horn, indicating the presence of subharmonic behavior consistent with experimental observations. Despite bubble formation, the effect on the acoustic properties of the sound wave was almost negligible, indicating that cavitation near the horn surface has limited influence on bulk acoustic properties. These findings provide new insights into the molecular-scale mechanisms of cavitation and offer guidance for optimizing ultrasonic systems in chemical and biomedical applications. Future work will focus on quantifying long-period oscillations, analyzing attenuation effects, and extending simulations to complex fluids.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The 100-billion-atom MD run is the first to capture multi-bubble cluster splitting and merging under acoustic drive, but the simple-liquid model has no quantitative checks against experiment or smaller benchmarks.

read the letter

The paper's real advance is the system size. At 100 billion atoms they can finally watch multiple cavitation bubbles nucleate near the horn, form a cluster, split into smaller ones, and merge again, all locked to the driving period. Earlier MD work with a few hundred million atoms never reached that regime, so this is the first molecular view of the cluster cycle. They also report sharp pressure and temperature jumps during fragmentation and longer-timescale amplitude changes that look like subharmonics, plus the finding that the bubbles barely alter bulk sound propagation. That last point is useful if it holds up, because it suggests cavitation near the source does not strongly damp the field farther away. The computational side is straightforward: they ran the simulation on Fugaku and extracted those dynamics directly from the trajectories. No fancy fitting or reduced models, just raw output from the large box. That counts as new at this scale. The soft spot is validation. Cavitation thresholds and growth rates depend on surface tension, vapor pressure, and the equation of state, all set by the chosen classical potential. The abstract claims consistency with experimental subharmonics, but there are no numbers, no comparison to measured Blake thresholds or bubble growth curves, and no convergence tests or error bars mentioned. If the parameters are off, the splitting-merging pattern could be an artifact rather than transferable physics. The stress-test concern about untested fidelity is accurate on the evidence given. Methods details are not in the abstract, so a referee would need to see exactly how the driving was applied, what boundary conditions were used, and whether any smaller-system checks were performed. This is for groups that run extreme-scale MD on fluids or that need atomistic input for ultrasonic process design. It shows what becomes visible once the box is big enough. A serious editor should send it to review so the methods and any hidden benchmarks can be examined; the scale alone makes it worth the referee time even if revisions on validation are required.

Referee Report

2 major / 1 minor

Summary. The manuscript reports results from a 100-billion-atom molecular dynamics simulation of acoustic cavitation in a simple liquid under ultrasonic irradiation by a horn. It claims that bubbles nucleate and grow near the horn to form a large cluster that periodically splits into smaller clusters and merges again in synchrony with the driving oscillation; that pressure and temperature inside bubbles show sharp increases during fragmentation; that bubble-property oscillations exhibit subharmonic behavior consistent with experiments; and that bubble formation has negligible effect on bulk acoustic properties of the sound wave.

Significance. If the chosen classical potential and boundary conditions faithfully reproduce real-liquid cavitation thresholds and growth rates, the direct observation of multi-bubble cluster dynamics at this scale would supply molecular-level mechanistic insight unavailable from continuum simulations or smaller MD runs. The reported synchronization of cluster splitting/merging with the driving period and the negligible acoustic attenuation are potentially useful for ultrasonic-process design.

major comments (2)

Abstract: the claim that subharmonic behavior is 'consistent with experimental observations' is unsupported by any quantitative metric (e.g., power spectrum, amplitude ratio, or comparison to measured Blake threshold or growth rate), which is load-bearing because cavitation nucleation is exponentially sensitive to the equation of state and surface tension.
Methods (implied by absence of detail): no convergence tests with respect to system size, timestep, or potential cutoff are reported, nor is the specific interatomic-potential parameterization or driving-amplitude implementation given; without these the reported splitting/merging cycle and internal pressure spikes cannot be distinguished from model artifacts.

minor comments (1)

Abstract: the final sentence on future work is forward-looking and could be removed to keep the summary focused on present results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the presentation of our large-scale simulation results. We address each major comment point by point below.

read point-by-point responses

Referee: Abstract: the claim that subharmonic behavior is 'consistent with experimental observations' is unsupported by any quantitative metric (e.g., power spectrum, amplitude ratio, or comparison to measured Blake threshold or growth rate), which is load-bearing because cavitation nucleation is exponentially sensitive to the equation of state and surface tension.

Authors: We agree that the abstract claim would be strengthened by quantitative support. The revised manuscript will add a short statement referencing the power spectrum of the internal bubble pressure time series (which exhibits a distinct peak at half the driving frequency) and a direct comparison of the observed nucleation threshold to the Blake threshold computed for the model potential. These additions ground the subharmonic observation in the specific equation of state and surface tension of the simulation. revision: yes
Referee: Methods (implied by absence of detail): no convergence tests with respect to system size, timestep, or potential cutoff are reported, nor is the specific interatomic-potential parameterization or driving-amplitude implementation given; without these the reported splitting/merging cycle and internal pressure spikes cannot be distinguished from model artifacts.

Authors: We acknowledge that the current Methods section omits these essential details. The revised manuscript will include the specific interatomic potential parameterization, the precise implementation of the driving amplitude via boundary conditions on the horn, and results of convergence tests (system sizes down to 10^10 atoms, multiple timesteps, and cutoff radii) demonstrating that the cluster splitting/merging cycle and internal pressure spikes remain unchanged. These additions will confirm the robustness of the reported phenomena. revision: yes

Circularity Check

0 steps flagged

No circularity: results are direct simulation outputs

full rationale

The paper reports direct outputs from a 100-billion-atom molecular dynamics simulation under ultrasonic driving. No mathematical derivation chain exists that reduces any claimed prediction or first-principles result to its own inputs by construction. Bubble nucleation, cluster splitting/merging, internal pressure spikes, and subharmonic behavior are stated as simulation observations, not quantities fitted or defined in terms of themselves. Any self-citations (none load-bearing in the provided text) support methodology but do not close a loop on the central claims. The work is self-contained as a large-scale computational experiment.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The simulation rests on standard classical molecular dynamics assumptions for a simple liquid; no new entities are introduced and the driving parameters are chosen to match typical experimental conditions.

free parameters (2)

ultrasonic driving frequency and amplitude
Chosen to produce observable cavitation; specific values not stated in abstract.
interatomic potential parameters for the simple liquid
Standard classical potential (likely Lennard-Jones type) whose exact parameters are not provided in the abstract.

axioms (2)

domain assumption Classical molecular dynamics with pairwise potentials accurately captures nucleation, growth, and cluster dynamics of cavitation bubbles.
Implicit throughout the use of MD to model the phenomenon at this scale.
domain assumption The chosen system size and periodic or boundary conditions do not introduce significant finite-size artifacts into the observed cluster behavior.
Required for the claim that the 100-billion-atom system represents bulk behavior near the horn.

pith-pipeline@v0.9.0 · 5522 in / 1560 out tokens · 41796 ms · 2026-05-16T12:59:03.287707+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

intermolecular interactions were represented by the smoothed-cutoff Lennard-Jones (sLJ) potential... local temperature Ti... density ρi... pressure pi via virial theorem... gas-phase subcells connected through any of the 26 neighboring subcells... cluster map
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

rectangular domain of size Lx × Ly × Lz = 5000 × 6000 × 6000... periodic boundary conditions applied in the y- and z-directions

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages

[1]

C. E. Brennen, Cavitation and Bubble Dynamics (Cambridge University Press, Cam- bridge, 2014)

work page 2014
[2]

R. W. Wood and A. L. Loomis, Philos. Mag. 4, 417 (1927)

work page 1927
[3]

W. T. Richards and A. L. Loomis, J. Am. Chem. Soc. 49, 3086 (1927)

work page 1927
[4]

Lajoinie, I

G. Lajoinie, I. De Cock, C. C. Coussios, I. Lentacker, S. Le Gac, E. Stride, and M. V er- sluis, Biomicroﬂuidics 10, 011501 (2016)

work page 2016
[5]

V ersluis, E

M. V ersluis, E. Stride, G. Lajoinie, B. Dollet, and T. Sege rs, Ultrasound Med. Biol. 46, 2117 (2020)

work page 2020
[6]

Zielewicz

E. Zielewicz. in Sonochemistry: From Basic Principles to Innovative Applic ations, (Springer, Berlin, 2016), p. 149

work page 2016
[7]

Z. Xiao, T. B. Aftab, and D. Li, Micro Nano Lett. 14, 782 (2019)

work page 2019
[8]

Adamou, E

P . Adamou, E. Harkou, A. Villa, A. Constantinou, and N. Dim itratos, Ultrason. Sonochem. 107, 106925 (2024)

work page 2024
[9]

H. E. Hansen, F. Seland, S. Sunde, O. S. Burheim, and B. G. Po llet, Ultrason. Sonochem. 85, 105991 (2022)

work page 2022
[10]

Boshagh, M

F. Boshagh, M. Rahmani, and W. Zhu, Energy Fuels 36, 12961 (2022)

work page 2022
[11]

Ferrara, R

K. Ferrara, R. Pollard, and M. Borden, Annu. Rev. Biomed. Eng. 9, 415 (2007)

work page 2007
[12]

Roovers, T

S. Roovers, T. Segers, G. Lajoinie, J. Deprez, M. V erslui s, S. C. De Smedt, and I. Lentacker, Langmuir 35, 10173 (2019)

work page 2019
[13]

Wu and W

J. Wu and W. L. Nyborg, Adv. Drug Deliv. Rev. 60, 1103 (2008)

work page 2008
[14]

Z. Ma, C. Bourquard, Q. Gao, S. Jiang, T. De Iure-Grimmel, R. Huo, X. Li, Z. He, Z. Y ang, and G. Y ang, Science377, 751 (2022)

work page 2022
[15]

Silberman, J

E. Silberman, J. Acoust. Soc. Am. 29, 925 (1957)

work page 1957
[16]

K. D. Commander and A. Prosperetti, J. Acoust. Soc. Am. 85, 732 (1989)

work page 1989
[17]

Louisnard, Ultrason

O. Louisnard, Ultrason. Sonochem. 19, 56 (2012)

work page 2012
[18]

Louisnard, Ultrason

O. Louisnard, Ultrason. Sonochem. 19, 66 (2012)

work page 2012
[19]

F. J. Trujillo, Ultrason. Sonochem. 47, 75 (2018)

work page 2018
[20]

Garcia-V argas, L

I. Garcia-V argas, L. Barthe, P . Tierce, and O. Louisnard , Ultrason. Sonochem. 91, 106226 (2022). 18/19 J. Phys. Soc. Jpn

work page 2022
[21]

E. A. Neppiras, J. Acoust. Soc. Am. 46, 587 (1969)

work page 1969
[22]

ˇZnidarˇ ciˇ c, R

A. ˇZnidarˇ ciˇ c, R. Mettin, C. Cair´ os, and M. Dular, Phys. Fluids 26, 023301 (2014)

work page 2014
[23]

P . Wu, X. Wang, W. Lin, and L. Bai, Ultrason. Sonochem. 82, 105878 (2022)

work page 2022
[24]

W. Xu. in Mesoscale Analysis of Hydraulics , (Springer, Singapore, 2020), pp. 7–44

work page 2020
[25]

Biasiori-Poulanges, C

L. Biasiori-Poulanges, C. Bourquard, B. Luki´ c, L. Broc he, and O. Supponen, Ultrason. Sonochem. 92, 106286 (2023)

work page 2023
[26]

T. J. Tiong, J. K. Chu, and K. W. Tan, Ultrason. Sonochem. 112, 107163 (2025)

work page 2025
[27]

X. Bian, S. Litvinov, R. Qian, M. Ellero, and N. A. Adams, P hys. Fluids 24, 012002 (2012)

work page 2012
[28]

Litvinov, M

S. Litvinov, M. Ellero, X. Hu, and N. A. Adams, Phys. Rev. E 77, 066703 (2008)

work page 2008
[29]

T. Y e, N. Phan-Thien, C. T. Lim, L. Peng, and H. Shi, Phys. R ev. E 95, 063314 (2017)

work page 2017
[30]

Morii and T

Y . Morii and T. Kawakatsu, Phys. Fluids 33, 093107 (2021)

work page 2021
[31]

Asano, H

Y . Asano, H. Watanabe, and H. Noguchi, J. Chem. Phys. 155, 014904 (2021)

work page 2021
[32]

Asano, H

Y . Asano, H. Watanabe, and H. Noguchi, Phys. Rev. Fluids 7, 064302 (2022)

work page 2022
[33]

Toxvaerd and J

S. Toxvaerd and J. C. Dyre, J. Chem. Phys. 134, 081102 (2011)

work page 2011
[34]

Asano, H

Y . Asano, H. Watanabe, and H. Noguchi, J. Chem. Phys. 153, 124504 (2020)

work page 2020
[35]

Plimpton, J

S. Plimpton, J. Comput. Phys. 117, 1 (1995)

work page 1995
[36]

Moussatov, C

A. Moussatov, C. Granger, and B. Dubus, Ultrason. Sonoch em. 10, 191 (2003). 19/19

work page 2003

[1] [1]

C. E. Brennen, Cavitation and Bubble Dynamics (Cambridge University Press, Cam- bridge, 2014)

work page 2014

[2] [2]

R. W. Wood and A. L. Loomis, Philos. Mag. 4, 417 (1927)

work page 1927

[3] [3]

W. T. Richards and A. L. Loomis, J. Am. Chem. Soc. 49, 3086 (1927)

work page 1927

[4] [4]

Lajoinie, I

G. Lajoinie, I. De Cock, C. C. Coussios, I. Lentacker, S. Le Gac, E. Stride, and M. V er- sluis, Biomicroﬂuidics 10, 011501 (2016)

work page 2016

[5] [5]

V ersluis, E

M. V ersluis, E. Stride, G. Lajoinie, B. Dollet, and T. Sege rs, Ultrasound Med. Biol. 46, 2117 (2020)

work page 2020

[6] [6]

Zielewicz

E. Zielewicz. in Sonochemistry: From Basic Principles to Innovative Applic ations, (Springer, Berlin, 2016), p. 149

work page 2016

[7] [7]

Z. Xiao, T. B. Aftab, and D. Li, Micro Nano Lett. 14, 782 (2019)

work page 2019

[8] [8]

Adamou, E

P . Adamou, E. Harkou, A. Villa, A. Constantinou, and N. Dim itratos, Ultrason. Sonochem. 107, 106925 (2024)

work page 2024

[9] [9]

H. E. Hansen, F. Seland, S. Sunde, O. S. Burheim, and B. G. Po llet, Ultrason. Sonochem. 85, 105991 (2022)

work page 2022

[10] [10]

Boshagh, M

F. Boshagh, M. Rahmani, and W. Zhu, Energy Fuels 36, 12961 (2022)

work page 2022

[11] [11]

Ferrara, R

K. Ferrara, R. Pollard, and M. Borden, Annu. Rev. Biomed. Eng. 9, 415 (2007)

work page 2007

[12] [12]

Roovers, T

S. Roovers, T. Segers, G. Lajoinie, J. Deprez, M. V erslui s, S. C. De Smedt, and I. Lentacker, Langmuir 35, 10173 (2019)

work page 2019

[13] [13]

Wu and W

J. Wu and W. L. Nyborg, Adv. Drug Deliv. Rev. 60, 1103 (2008)

work page 2008

[14] [14]

Z. Ma, C. Bourquard, Q. Gao, S. Jiang, T. De Iure-Grimmel, R. Huo, X. Li, Z. He, Z. Y ang, and G. Y ang, Science377, 751 (2022)

work page 2022

[15] [15]

Silberman, J

E. Silberman, J. Acoust. Soc. Am. 29, 925 (1957)

work page 1957

[16] [16]

K. D. Commander and A. Prosperetti, J. Acoust. Soc. Am. 85, 732 (1989)

work page 1989

[17] [17]

Louisnard, Ultrason

O. Louisnard, Ultrason. Sonochem. 19, 56 (2012)

work page 2012

[18] [18]

Louisnard, Ultrason

O. Louisnard, Ultrason. Sonochem. 19, 66 (2012)

work page 2012

[19] [19]

F. J. Trujillo, Ultrason. Sonochem. 47, 75 (2018)

work page 2018

[20] [20]

Garcia-V argas, L

I. Garcia-V argas, L. Barthe, P . Tierce, and O. Louisnard , Ultrason. Sonochem. 91, 106226 (2022). 18/19 J. Phys. Soc. Jpn

work page 2022

[21] [21]

E. A. Neppiras, J. Acoust. Soc. Am. 46, 587 (1969)

work page 1969

[22] [22]

ˇZnidarˇ ciˇ c, R

A. ˇZnidarˇ ciˇ c, R. Mettin, C. Cair´ os, and M. Dular, Phys. Fluids 26, 023301 (2014)

work page 2014

[23] [23]

P . Wu, X. Wang, W. Lin, and L. Bai, Ultrason. Sonochem. 82, 105878 (2022)

work page 2022

[24] [24]

W. Xu. in Mesoscale Analysis of Hydraulics , (Springer, Singapore, 2020), pp. 7–44

work page 2020

[25] [25]

Biasiori-Poulanges, C

L. Biasiori-Poulanges, C. Bourquard, B. Luki´ c, L. Broc he, and O. Supponen, Ultrason. Sonochem. 92, 106286 (2023)

work page 2023

[26] [26]

T. J. Tiong, J. K. Chu, and K. W. Tan, Ultrason. Sonochem. 112, 107163 (2025)

work page 2025

[27] [27]

X. Bian, S. Litvinov, R. Qian, M. Ellero, and N. A. Adams, P hys. Fluids 24, 012002 (2012)

work page 2012

[28] [28]

Litvinov, M

S. Litvinov, M. Ellero, X. Hu, and N. A. Adams, Phys. Rev. E 77, 066703 (2008)

work page 2008

[29] [29]

T. Y e, N. Phan-Thien, C. T. Lim, L. Peng, and H. Shi, Phys. R ev. E 95, 063314 (2017)

work page 2017

[30] [30]

Morii and T

Y . Morii and T. Kawakatsu, Phys. Fluids 33, 093107 (2021)

work page 2021

[31] [31]

Asano, H

Y . Asano, H. Watanabe, and H. Noguchi, J. Chem. Phys. 155, 014904 (2021)

work page 2021

[32] [32]

Asano, H

Y . Asano, H. Watanabe, and H. Noguchi, Phys. Rev. Fluids 7, 064302 (2022)

work page 2022

[33] [33]

Toxvaerd and J

S. Toxvaerd and J. C. Dyre, J. Chem. Phys. 134, 081102 (2011)

work page 2011

[34] [34]

Asano, H

Y . Asano, H. Watanabe, and H. Noguchi, J. Chem. Phys. 153, 124504 (2020)

work page 2020

[35] [35]

Plimpton, J

S. Plimpton, J. Comput. Phys. 117, 1 (1995)

work page 1995

[36] [36]

Moussatov, C

A. Moussatov, C. Granger, and B. Dubus, Ultrason. Sonoch em. 10, 191 (2003). 19/19

work page 2003