pith. sign in

arxiv: 2511.20075 · v3 · submitted 2025-11-25 · ❄️ cond-mat.soft · physics.flu-dyn

How elasticity affects bubble pinch-off

Coen I. Verschuur (1) , Alexandros T. Oratis (1) , Vatsal Sanjay (1) , Jacco H. Snoeijer (1) ((1) Physics of fluids department , University of Twente , Enschede , the Netherlands) This is my paper

Pith reviewed 2026-05-17 05:19 UTC · model grok-4.3

classification ❄️ cond-mat.soft physics.flu-dyn
keywords bubble pinch-offviscoelastic fluidsOldroyd-B modelpolymer solutionsstress singularitythread formationfluid breakup
0
0 comments X

The pith

Polymer stresses remain singular in bubble pinch-off but diverge far more weakly than in drop pinch-off.

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

The paper examines bubble pinch-off in viscoelastic polymer solutions and contrasts it with the well-studied case of drop pinch-off. Experiments show that threads fail to appear in dilute solutions and only form at high polymer concentrations where the process becomes sensitive to needle diameter. Simulations and analytic modeling with the Oldroyd-B constitutive relation reproduce these observations and establish that polymer stresses still diverge, yet the singularity is much milder than the one that stabilizes threads in drops.

Core claim

Using the Oldroyd-B model the authors demonstrate that polymer stresses are singular at the pinch point during bubble detachment, but the divergence is substantially weaker than the corresponding divergence that occurs during drop pinch-off. This reduced singularity explains why viscoelastic threads are absent for dilute polymer solutions and appear only when polymer concentration is raised to levels where the dynamics also depend strongly on the needle size from which the bubble detaches.

What carries the argument

The Oldroyd-B constitutive model applied to the dilute regime, which predicts a weaker stress singularity for bubble pinch-off than for drop pinch-off.

If this is right

  • Bubbles in dilute polymer solutions complete pinch-off without forming a persistent thread.
  • Threads appear only at high polymer concentrations where pinch-off dynamics depend on needle diameter.
  • Numerical simulations based on the Oldroyd-B model reproduce the dilute-regime behavior seen in experiments.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The weaker singularity may produce different scaling exponents for the minimum neck radius in bubble breakup compared with drop breakup.
  • Processes that rely on controlled bubble detachment, such as foam generation or microfluidic gas injection, may exhibit reduced sensitivity to dilute polymer additives than analogous liquid-drop processes.
  • Incorporating finite extensibility into the model could locate the concentration threshold at which the stronger thread-stabilizing singularity reappears.

Load-bearing premise

The Oldroyd-B model and the dilute-regime assumptions continue to hold across the experimental range of polymer concentrations and needle sizes without shear thinning or finite-extensibility corrections becoming important.

What would settle it

Direct measurement of the polymer stress divergence rate near the pinch point that either matches the weaker Oldroyd-B scaling for bubbles or shows the stronger scaling characteristic of drops.

Figures

Figures reproduced from arXiv: 2511.20075 by Alexandros T. Oratis (1), Coen I. Verschuur (1), Enschede, Jacco H. Snoeijer (1) ((1) Physics of fluids department, the Netherlands), University of Twente, Vatsal Sanjay (1).

Figure 1
Figure 1. Figure 1: FIG. 1. Cartoon of viscoelastic drop and bubble pinch-off. (a) When a drop pinches off, the polymers become stretched [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Snapshots of the bubble pinch-off process for different PEO concentrations ( [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Snapshots of the drop pinch-off process for different PEO concentrations ( [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The width of the neck at the center is normalized by the diameter of the needle ( [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The effect of needle diameter [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The thread duration [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Snapshots of the numerical bubble and droplet pinch-off simulations in the limit of a Newtonian fluid, and viscoelastic [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The width of the neck at the center is normalized by the initial neck width over time ( [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The fluid stresses for [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. The viscosity of the polymer solutions as a function of the shear rate for different concentrations of PEO, [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. The Relaxation time for PEO2M as function of the concentration normalize by the overlap concentration. The opacity [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. The Relaxation time for PEO4M as function of the concentration normalize by the overlap concentration. The opacity [PITH_FULL_IMAGE:figures/full_fig_p013_12.png] view at source ↗
read the original abstract

The pinch-off of bubbles in viscoelastic liquids is a fundamental process that has received little attention compared to viscoelastic drop pinch-off. While these processes exhibit qualitative similarities, the dynamics of the pinch-off process are significantly different. When a drop of a dilute polymer solution pinches off, a thread is known to develop that prevents breakup due the diverging polymer stresses. Conversely, our experiments reveal that this thread is absent for bubble pinch-off in dilute polymer solutions. We show that a thread becomes apparent only for high polymer concentrations, where the pinch-off dynamics become very sensitive to the size of the needle from which the bubble detaches. The experiments are complemented by numerical simulations and analytical modeling using the Oldroyd-B model, which capture the dilute regime. The model shows that polymer stresses are still singular during bubble pinch-off, but the divergence is much weaker as compared to drop pinch-off. This explains why, in contrast to droplets, viscoelastic bubble-threads do not appear for dilute suspensions but require large polymer concentrations

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.

Referee Report

2 major / 2 minor

Summary. The paper examines bubble pinch-off in viscoelastic liquids through experiments on polymer solutions, Oldroyd-B numerical simulations, and analytical modeling. It claims that, unlike drop pinch-off where threads form due to diverging polymer stresses, bubbles in dilute solutions show no threads; threads appear only at high concentrations where pinch-off becomes sensitive to needle size. The Oldroyd-B model is shown to produce singular but much weaker polymer stress divergence than in drops, explaining the dilute-regime observations.

Significance. If the central claim holds, the work clarifies key differences in how elasticity affects bubble versus drop pinch-off, with the weaker stress singularity in Oldroyd-B providing a mechanistic account for the absence of threads in dilute regimes. The consistency between independent experiments and simulations for dilute cases, plus the use of standard Oldroyd-B without fitted parameters for the target result, represents a strength in reproducibility and falsifiability.

major comments (2)
  1. [Experiments section] Experiments on high-concentration regime: the reported sensitivity of pinch-off dynamics to needle size at high polymer concentrations (evident in the regime distinctions drawn from the data) appears post-hoc and load-bearing for the claim that threads require large concentrations; this distinction risks circularity in defining the dilute regime where the Oldroyd-B explanation applies.
  2. [Numerical simulations and analytical modeling] Oldroyd-B modeling and stress singularity analysis: the central claim that polymer stresses are singular yet weaker than in drop pinch-off rests on the constitutive model remaining valid at extreme local extension rates; however, the absence of finite extensibility and shear-thinning (both active at Wi ≫ 1 near pinch-off) could regularize or alter the effective scaling, and no test with FENE-P or similar is provided to confirm the divergence strength.
minor comments (2)
  1. [Methods] Notation for Weissenberg number and relaxation time should be clarified in the methods to avoid ambiguity when comparing bubble and drop cases.
  2. [Figures] Figure captions for the high-concentration needle-size data could explicitly state the number of repeats and error bars to strengthen the regime distinction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful and detailed review of our manuscript on bubble pinch-off in viscoelastic liquids. The comments highlight important issues regarding experimental regime definitions and the applicability of the Oldroyd-B model. We address each point below and have revised the manuscript to improve clarity and address potential concerns.

read point-by-point responses

  1. Referee: [Experiments section] Experiments on high-concentration regime: the reported sensitivity of pinch-off dynamics to needle size at high polymer concentrations (evident in the regime distinctions drawn from the data) appears post-hoc and load-bearing for the claim that threads require large concentrations; this distinction risks circularity in defining the dilute regime where the Oldroyd-B explanation applies.

    Authors: We acknowledge that the presentation of the high-concentration regime could give the impression of post-hoc classification. To address this, we have restructured the revised manuscript to define the dilute and concentrated regimes at the outset of the results section using independent criteria: polymer concentration relative to the overlap concentration c* and measured relaxation times from rheology, rather than relying solely on the observed pinch-off behavior. The needle-size sensitivity is now presented as a separate experimental finding that emerges consistently at concentrations above a threshold determined from these rheological data. This removes any circularity and makes the applicability of the Oldroyd-B analysis to the dilute regime explicit and a priori. revision: yes

  2. Referee: [Numerical simulations and analytical modeling] Oldroyd-B modeling and stress singularity analysis: the central claim that polymer stresses are singular yet weaker than in drop pinch-off rests on the constitutive model remaining valid at extreme local extension rates; however, the absence of finite extensibility and shear-thinning (both active at Wi ≫ 1 near pinch-off) could regularize or alter the effective scaling, and no test with FENE-P or similar is provided to confirm the divergence strength.

    Authors: We agree that the Oldroyd-B model has limitations at the extreme extension rates encountered near pinch-off, where finite extensibility and shear-thinning become relevant. Our central claim, however, is restricted to the dilute regime, where the low polymer concentration keeps the effective chain extension within the linear regime for the timescales of interest, allowing Oldroyd-B to capture the weaker stress singularity relative to drops. We have added a dedicated discussion paragraph in the revised manuscript outlining the validity range of Oldroyd-B, the expected qualitative effects of finite extensibility (which would not eliminate the difference from drop pinch-off), and the reasons a full FENE-P comparison lies beyond the present scope. The analytical scaling and numerical results for dilute cases remain robust under these conditions. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation uses standard Oldroyd-B equations and independent experiments

full rationale

The paper derives its central claim—that polymer stresses remain singular during bubble pinch-off but with weaker divergence than in drop pinch-off—directly from numerical simulations and analytical modeling based on the standard Oldroyd-B constitutive model applied to the dilute regime, together with separate experimental observations of thread absence at low concentrations. No load-bearing step reduces to a fitted parameter renamed as a prediction, a self-citation chain, or an ansatz smuggled from prior author work; the Oldroyd-B equations are invoked as an external, established framework rather than defined in terms of the target result. The derivation is therefore self-contained against external benchmarks and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the standard Oldroyd-B constitutive relation as a domain assumption for dilute polymer solutions; no new free parameters or invented entities are introduced.

axioms (1)
  • domain assumption Oldroyd-B constitutive model accurately describes dilute viscoelastic polymer solutions
    Invoked for both numerical simulations and analytical modeling of the pinch-off dynamics.

pith-pipeline@v0.9.0 · 5510 in / 1011 out tokens · 42937 ms · 2026-05-17T05:19:15.244258+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

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

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

50 extracted references · 50 canonical work pages

  1. [1]

    Model configuration We study the breakup dynamics of both air and liquid threads in viscoelastic media using idealized model con- figurations. Rather than simulating complete bubble or drop pinch-off processes, we consider initially cylindrical threads with imposed sinusoidal perturbations and investigate their capillary-driven breakup. This approach isol...

    work page

  2. [2]

    Governing equations The mass and momentum conservation equations for both phases read: ∇ ·u= 0,(1) ρ ∂u ∂t +∇ ·(uu) =−∇p+∇ ·τ+f γ,(2) whereτrepresents the stress tensor andf γ the force due to surface tensionγ. In the viscoelastic phase (liquid pool for air threads, liquid thread for drop case), we decompose the stress as: τ=τ s +τ p,(3) with the solvent ...

    work page

  3. [3]

    The viscoelastic constitutive equations employ the log-conformation method [40, 41] to ensure numerical stability at high Deborah numbers

    Numerical implementation We implement these equations in Basilisk C [37] using the volume-of-fluid method for interface capturing [38, 39]. The viscoelastic constitutive equations employ the log-conformation method [40, 41] to ensure numerical stability at high Deborah numbers. For the elastic limit (De→ ∞), we utilize a modified formulation that enforces...

    work page 2048

  4. [4]

    We note that the bubble neck is very slender close to pinch-off, which justifies a slender analysis of the problem

    Elastic stress We will now show why elasticity (in the dilute, Oldroyd-B limit) has little effect on bubble pinch-off, in contrast to the pinch-off of viscoelastic drops. We note that the bubble neck is very slender close to pinch-off, which justifies a slender analysis of the problem. This is in line with the quasi-two dimensional approach for bubble bre...

    work page

  5. [5]

    To understand this, we first recall the balance of stress for the case of droplet pinch-off

    Pinch-off The interesting situation is thus that the radial stress tends to diverge, but apparently this divergence does not inhibit the collapse of the neck. To understand this, we first recall the balance of stress for the case of droplet pinch-off. The dominant stress in the liquid thread then lies along the axial direction (Fig. 1) and scales asG(h 0/...

    work page

  6. [6]

    The viscosity is measured at room temperature as a function of the shear rate, which is increased from 0.01−1000 s −1 (see Fig

    Rheology The viscosity of the polymer solution is measured using a rheometer (Anton Paar, MCR 502) with a cone-plate geometry (CP50-1). The viscosity is measured at room temperature as a function of the shear rate, which is increased from 0.01−1000 s −1 (see Fig. 10). For the lower viscosity solutions, we removed the lower shear rates since the measured t...

    work page

  7. [7]

    We selected the drop pinch-off experi- ment due to its simplicity, as the neck forms naturally from a falling droplet under the influence of gravity

    Relaxation time To determine the relaxation time (λd) of polymer solutions, we measured the neck width over time during pinch-off and fitted the results using the Oldroyd-B model: h h0 = h0µ γ 1/3 e− t 3λd .(A1) Several methods can generate pinch-off, including drop pinch-off and CaBER. We selected the drop pinch-off experi- ment due to its simplicity, as...

    work page

  8. [8]

    Eggers and M

    J. Eggers and M. Fontelos,Singularities: Formation, Structure, and Propagation, Cambridge Texts in Applied Mathematics (Cambridge University Press, 2015)

    work page 2015

  9. [9]

    D. T. Papageorgiou, On the breakup of viscous liquid threads, Physics of Fluids7, 1529 (1995)

    work page 1995

  10. [10]

    R. F. Day, E. J. Hinch, and J. R. Lister, Self-Similar Capillary Pinchoff of an Inviscid Fluid, P H Y S I C A L R E V I E W L E T T E R S80(1998)

    work page 1998

  11. [11]

    Eggers, Nonlinear dynamics and breakup of free-surface flows, Reviews of Modern Physics69, 865 (1997)

    J. Eggers, Nonlinear dynamics and breakup of free-surface flows, Reviews of Modern Physics69, 865 (1997)

    work page 1997

  12. [12]

    J. C. Burton, R. Waldrep, and P. Taborek, Scaling and Instabilities in Bubble Pinch-Off, Physical Review Letters94, 184502 (2005)

    work page 2005

  13. [13]

    J. M. Gordillo, A. Sevilla, J. Rodr´ ıguez-Rodr´ ıguez, and C. Mart´ ınez-Baz´ an, Axisymmetric Bubble Pinch-Off at High Reynolds Numbers, Physical Review Letters95, 194501 (2005)

    work page 2005

  14. [14]

    N. C. Keim, P. Møller, W. W. Zhang, and S. R. Nagel, Breakup of Air Bubbles in Water: Memory and Breakdown of Cylindrical Symmetry, Physical Review Letters97, 144503 (2006)

    work page 2006

  15. [15]

    Bergmann, D

    R. Bergmann, D. Van Der Meer, M. Stijnman, M. Sandtke, A. Prosperetti, and D. Lohse, Giant Bubble Pinch-Off, Physical Review Letters96, 154505 (2006)

    work page 2006

  16. [16]

    Eggers, M

    J. Eggers, M. A. Fontelos, D. Leppinen, and J. H. Snoeijer, Theory of the Collapsing Axisymmetric Cavity, Physical Review Letters98, 094502 (2007)

    work page 2007

  17. [17]

    Gekle, J

    S. Gekle, J. H. Snoeijer, D. Lohse, and D. Van Der Meer, Approach to universality in axisymmetric bubble pinch-off, Physical Review E80, 036305 (2009)

    work page 2009

  18. [18]

    S. T. Thoroddsen, T. G. Etoh, and K. Takehara, Experiments on Bubble Pinch-Off, Physics of Fluids19, 042101 (2007)

    work page 2007

  19. [19]

    N. F. Morrison and O. G. Harlen, Viscoelasticity in Inkjet Printing, Rheologica Acta49, 619 (2010)

    work page 2010

  20. [20]

    U. Sen, C. Datt, T. Segers, H. Wijshoff, J. H. Snoeijer, M. Versluis, and D. Lohse, The Retraction of Jetted Slender Viscoelastic Liquid Filaments, Journal of Fluid Mechanics929, A25 (2021)

    work page 2021

  21. [21]

    Makhnenko, E

    I. Makhnenko, E. R. Alonzi, S. A. Fredericks, C. M. Colby, and C. S. Dutcher, A review of liquid sheet breakup: Perspectives from agricultural sprays, Journal of Aerosol Science157, 105805 (2021)

    work page 2021

  22. [22]

    Gaillard, R

    A. Gaillard, R. Sijs, and D. Bonn, What determines the drop size in sprays of polymer solutions?, Journal of Non-Newtonian Fluid Mechanics305, 104813 (2022)

    work page 2022

  23. [23]

    J. H. Snoeijer, A. Pandey, M. A. Herrada, and J. Eggers, The Relationship between Viscoelasticity and Elasticity, Pro- ceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences476, 20200419 (2020)

    work page 2020

  24. [24]

    Morozov and S

    A. Morozov and S. E. Spagnolie, eds.,Complex Fluids in Biological Systems: Experiment, Theory, and Computation, Biological and Medical Physics, Biomedical Engineering (Springer, New York, 2015)

    work page 2015

  25. [25]

    Larson,The Structure and Rheology of Complex Fluids, EngineeringPro Collection (OUP USA, 1999)

    R. Larson,The Structure and Rheology of Complex Fluids, EngineeringPro Collection (OUP USA, 1999)

    work page 1999

  26. [26]

    Tanner,Engineering Rheology, Oxford Engineering Science Series (OUP Oxford, 2000)

    R. Tanner,Engineering Rheology, Oxford Engineering Science Series (OUP Oxford, 2000)

    work page 2000

  27. [27]

    Clasen, J

    C. Clasen, J. Eggers, M. A. Fontelos, J. Li, and G. H. McKinley, The Beads-on-String Structure of Viscoelastic Threads, Journal of Fluid Mechanics556, 283 (2006)

    work page 2006

  28. [28]

    Bazilevsky, V

    A. Bazilevsky, V. Entov, and A. Rozhkov,Liquid Filament Microrheometer and Some of Its Applications. In: Oliver, D.R. (Eds) Third European Rheology Conference and Golden Jubilee Meeting of the British Society of Rheology(Springer, Dordrecht, 1990)

    work page 1990

  29. [29]

    Entov and E

    V. Entov and E. Hinch, Effect of a Spectrum of Relaxation Times on the Capillary Thinning of a Filament of Elastic Liquid, Journal of Non-Newtonian Fluid Mechanics72, 31 (1997)

    work page 1997

  30. [30]

    S. L. Anna and G. H. McKinley, Elasto-Capillary Thinning and Breakup of Model Elastic Liquids, Journal of Rheology 45, 115 (2001)

    work page 2001

  31. [31]

    Amarouchene, D

    Y. Amarouchene, D. Bonn, J. Meunier, and H. Kellay, Inhibition of the Finite-Time Singularity during Droplet Fission of a Polymeric Fluid, Physical Review Letters86, 3558 (2001)

    work page 2001

  32. [32]

    Eggers, M

    J. Eggers, M. A. Herrada, and J. H. Snoeijer, Self-Similar Breakup of Polymeric Threads as Described by the Oldroyd-B Model, Journal of Fluid Mechanics887, A19 (2020), arXiv:1905.12343

    work page arXiv 2020

  33. [33]

    Deblais, M

    A. Deblais, M. A. Herrada, J. Eggers, and D. Bonn, Self-Similarity in the Breakup of Very Dilute Viscoelastic Solutions, Journal of Fluid Mechanics904, R2 (2020)

    work page 2020

  34. [34]

    A. V. Bazilevskii, V. M. Entov, M. M. Lerner, and A. N. Rozhkov, Failure of Polymer Solution Filaments, Polymer Science 39(1997)

    work page 1997

  35. [35]

    Gaillard, M

    A. Gaillard, M. A. H. Gutierrez, A. Deblais, J. Eggers, and D. Bonn, Beware of CaBER: Filament Thinning Rheometry Doesn’t Give ‘the’ Relaxation Time of Polymer Solutions, http://arxiv.org/abs/2309.08440 (2023), arXiv:2309.08440

    work page arXiv 2023

  36. [36]

    X. F. Jiang, C. Zhu, and H. Z. Li, Bubble Pinch-off in Newtonian and Non-Newtonian Fluids, Chemical Engineering Science170, 98 (2017)

    work page 2017

  37. [37]

    Rajesh, S

    S. Rajesh, S. S. Peddada, V. Thi´ evenaz, and A. Sauret, Pinch-off of Bubbles in a Polymer Solution, Journal of Non- Newtonian Fluid Mechanics310, 104921 (2022)

    work page 2022

  38. [38]

    A. T. Oratis, V. Bertin, and J. H. Snoeijer, Coalescence of bubbles in a viscoelastic liquid, Physical Review Fluids8, 083603 (2023)

    work page 2023

  39. [39]

    de Gennes,Scaling Concepts in Polymer Physics(Cornell University Press, 1979)

    P. de Gennes,Scaling Concepts in Polymer Physics(Cornell University Press, 1979)

    work page 1979

  40. [40]

    P. J. Dekker, M. A. Hack, W. Tewes, C. Datt, A. Bouillant, and J. H. Snoeijer, When Elasticity Affects Drop Coalescence, Physical Review Letters128, 028004 (2022)

    work page 2022

  41. [41]

    Kawaguchi, G

    S. Kawaguchi, G. Imai, J. Suzuki, A. Miyahara, T. Kitano, and K. Ito, Aqueous solution properties of oligo- and 15 poly(ethylene oxide) by static light scattering and intrinsic viscosity, Polymer38, 2885 (1997)

    work page 1997

  42. [42]

    A. K. Dixit, A. T. Oratis, K. Zinelis, D. Lohse, and V. Sanjay, Viscoelastic worthington jets and droplets produced by bursting bubbles, J. Fluid Mech.1010, A2 (2025)

    work page 2025

  43. [43]

    A. T. Oratis, K. Dijs, G. Lajoinie, M. Versluis, and J. H. Snoeijer, A unifying Rayleigh-Plesset-type equation for bubbles in viscoelastic media, The Journal of the Acoustical Society of America155, 1593 (2024)

    work page 2024

  44. [44]

    Popinet and collaborators, Basilisk C,http://basilisk.fr(Last accessed: April, 2025) (2013–2025)

    S. Popinet and collaborators, Basilisk C,http://basilisk.fr(Last accessed: April, 2025) (2013–2025)

    work page 2025

  45. [45]

    Sanjay and Collaborators, Comphy-lab/MultiRheoFlow: V0.1

    V. Sanjay and Collaborators, Comphy-lab/MultiRheoFlow: V0.1

    work page

  46. [46]

    Sanjay, Comphy-lab/ElasticPinchOff: Initial release

    V. Sanjay, Comphy-lab/ElasticPinchOff: Initial release

    work page

  47. [47]

    Fattal and R

    R. Fattal and R. Kupferman, Constitutive laws for the matrix-logarithm of the conformation tensor, J. Non-Newtonian Fluid Mech.123, 281 (2004)

    work page 2004

  48. [48]

    L´ opez-Herrera, S

    J.-M. L´ opez-Herrera, S. Popinet, and A.-A. Castrej´ on-Pita, An adaptive solver for viscoelastic incompressible two-phase problems applied to the study of the splashing of weakly viscoelastic droplets, J. Non-Newton. Fluid Mech.264, 144 (2019)

    work page 2019

  49. [49]

    A. E. Green and W. Zerna,Theoretical Elasticity, 2nd ed. (Dover Publications, 1992)

    work page 1992

  50. [50]

    M. M. Carroll, Pressure maximum behavior in inflation of incompressible elastic hollow spheres and cylinders, Quarterly of Applied Mathematics45, 141 (1987)

    work page 1987