pith. sign in

arxiv: 2606.26272 · v1 · pith:EIROENMTnew · submitted 2026-06-24 · ⚛️ physics.flu-dyn · cond-mat.soft· physics.bio-ph

Droplet Fusion as a Relaxation Process: Comparison with Shape Recovery of Newtonian and Viscoelastic Droplets

Mohammad Moein Naderi , Zhangli Peng , Huan-Xiang Zhou This is my paper

Pith reviewed 2026-06-26 01:30 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn cond-mat.softphysics.bio-ph
keywords droplet fusionshape recoveryviscoelastic dropletsOldroyd-B modelDeborah numbercapillary relaxationbiomolecular condensatesfinite element simulation
0
0 comments X

The pith

Droplet fusion follows a multistage process with localized neck formation while shape recovery is global exponential relaxation, even when viscoelasticity is present.

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

The paper builds a unified theory and simulation framework to compare two capillary-driven processes in droplets: shape recovery after deformation and fusion of two droplets. It establishes that shape recovery occurs through global relaxation of one connected interface and decays exponentially or multi-exponentially depending on the ratio of viscocapillary time to stress relaxation time. Droplet fusion, by contrast, proceeds in distinct stages of curvature-driven neck formation, rapid bridge growth, and eventual global relaxation. Viscoelasticity adds the Deborah number as a governing timescale that modifies the intermediate fusion stage and overall relaxation compared with Newtonian droplets, while an exterior fluid adds extra dissipation that slows fusion. These differences indicate that purely viscous models are insufficient for interpreting viscoelastic biomolecular condensates.

Core claim

Shape recovery is governed by global viscocapillary relaxation of a single connected interface and follows single- or multi-exponential decay depending on the relative magnitude of the viscocapillary timescale and the stress relaxation time. In contrast, droplet fusion is intrinsically a multistage process involving localized curvature-driven neck formation, rapid bridge expansion, and a transition to global relaxation. Viscoelasticity introduces an additional intrinsic timescale that governs the competition between capillary driving and stress relaxation, characterized by the Deborah number. This leads to enhanced intermediate-stage fusion dynamics and modified relaxation behavior compared

What carries the argument

The multistage localized-to-global sequence in droplet fusion versus single-interface global relaxation in shape recovery, quantified through axisymmetric finite-element solutions of the Oldroyd-B constitutive model and compared via the Deborah number.

If this is right

  • Viscoelastic droplets exhibit faster intermediate-stage bridge expansion than Newtonian droplets because stress relaxation competes with capillary driving.
  • An exterior fluid increases hydrodynamic dissipation and thereby lengthens the overall fusion time for both Newtonian and viscoelastic cases.
  • Empirical stretched-exponential fits to fusion data deviate systematically once the Deborah number becomes order-one, indicating the need for constitutive models that include stress relaxation.
  • Biomolecular condensate fusion times cannot be mapped directly onto shape-recovery timescales without accounting for the multistage character of fusion.

Where Pith is reading between the lines

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

  • Measurements that report only a single fusion time for condensates may be averaging over regimes that are actually controlled by different mechanisms.
  • Varying the external viscosity independently of the droplet relaxation time could isolate the hydrodynamic slowing effect predicted by the simulations.
  • The Deborah-number dependence suggests that fusion rates in cells could be tuned by changes in protein relaxation times without altering surface tension.

Load-bearing premise

The axisymmetric finite-element simulations with the Oldroyd-B model correctly reproduce the dynamics of both Newtonian and viscoelastic droplets without significant numerical or modeling inaccuracies.

What would settle it

Direct high-speed imaging that shows whether viscoelastic droplet fusion exhibits a distinct intermediate regime whose duration scales with the Deborah number rather than collapsing to the same single- or multi-exponential form as shape recovery under identical material parameters.

Figures

Figures reproduced from arXiv: 2606.26272 by Huan-Xiang Zhou, Mohammad Moein Naderi, Zhangli Peng.

Figure 1
Figure 1. Figure 1: Schematic comparison of droplet shape recovery and droplet fusion. a) shape recovery (relaxation) of an initially [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Numerical model schematics and mesh configuration for a) shape recovery, and b) fusion in an inviscid exterior [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Mesh independence study showing the e!ect of the number of elements on the droplet shape recovery. 0 1 2 3 4 5 6 0.001 0.01 0.1 1 0 1 2 3 4 0.001 0.01 0.1 a) b) 1 inviscid exterior t (s) t (s) [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: E!ects of exterior domain. (a) E!ect of domain confinement ratio (𝑕) on the accuracy of the exterior domain model. (b) Validation of the cases with finite viscosity exterior domain. For the droplet fusion and exterior domain models, a di!erent validation strategy was employed. Due to the higher complexity of these cases, refining the mesh simply by increasing the number of elements often led to convergence… view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of shape recovery (a) and droplet fusion (b) dynamics for Newtonian droplets. Contours show the [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of shape recovery (a) and droplet fusion (b) dynamics for a viscoelastic droplet. Contours show the [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Comparison between the simulation results and the analytical solution for the shape recovery of a droplet with initial [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Comparison between shape recovery and fusion dynamics. (a) Time histories of normalized vertical dimension.(b) [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Comparison of the shape recovery and fusion dynamics of two droplets in a inviscid exterior domain and in an exterior [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of the computationally predicted relaxation of droplet fusion with the empirical formula. The dependence [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
read the original abstract

Biomolecular condensates formed by phase separation often exhibit viscoelastic behavior, yet their shape recovery and fusion dynamics are frequently interpreted using purely viscous models. Here, we develop a unified theoretical and computational framework to quantify how viscoelasticity governs these two processes. We combine analytical theory for small-deformation shape recovery with axisymmetric finite-element simulations based on the Oldroyd-B constitutive model to systematically investigate both shape recovery and droplet fusion under comparable conditions. Our results show that, although both processes are driven by capillary forces, they are fundamentally distinct in their underlying physics. Shape recovery is governed by global viscocapillary relaxation of a single connected interface and follows single- or multi-exponential decay depending on the relative magnitude of the viscocapillary timescale and the stress relaxation time. In contrast, droplet fusion is intrinsically a multistage process involving localized curvature-driven neck formation, rapid bridge expansion, and a transition to global relaxation. We demonstrate that viscoelasticity introduces an additional intrinsic timescale that governs the competition between capillary driving and stress relaxation, characterized by the Deborah number. This leads to enhanced intermediate-stage fusion dynamics and modified relaxation behavior compared to Newtonian droplets. Furthermore, we show that the presence of an exterior fluid introduces additional hydrodynamic dissipation, significantly slowing the fusion process. Finally, we compare the computationally predicted droplet fusion in the Newtonian and viscoelastic cases with a stretched-exponential empirical formula. Deviations observed in viscoelastic regimes highlight the limitations of purely viscous descriptions and the need for models incorporating stress relaxation.

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 manuscript develops a unified theoretical and computational framework combining small-deformation analytical theory with axisymmetric finite-element simulations of the Oldroyd-B model to compare shape recovery and droplet fusion for Newtonian and viscoelastic droplets. It claims that, although both are capillary-driven, shape recovery is a global viscocapillary process exhibiting single- or multi-exponential decay set by the ratio of viscocapillary and stress-relaxation times, whereas fusion is intrinsically multistage (localized curvature-driven neck formation, rapid bridge expansion, then global relaxation). Viscoelasticity introduces a Deborah-number timescale that enhances intermediate-stage fusion and alters relaxation relative to Newtonian cases; an exterior fluid adds hydrodynamic dissipation that slows fusion. The simulations are compared to a stretched-exponential empirical fit, with deviations in viscoelastic regimes used to argue against purely viscous descriptions.

Significance. If the reported distinction and Deborah-number effects hold, the work supplies a concrete, simulation-backed separation between global relaxation and multistage localized dynamics that is directly relevant to viscoelastic biomolecular condensates. The explicit incorporation of stress relaxation via Oldroyd-B and the quantitative comparison to the stretched-exponential form are strengths that move the field beyond ad-hoc viscous models.

major comments (2)
  1. [Methods / Results (axisymmetric FEM)] Simulation methods and results sections: the central claim that fusion is 'intrinsically a multistage process involving localized curvature-driven neck formation' while shape recovery is 'global viscocapillary relaxation' rests on axisymmetric Oldroyd-B FEM runs. No mesh-convergence data, 3-D benchmark, or experimental validation is referenced for the high-curvature neck region where the multistage signature is diagnosed; numerical artifacts in this localized regime could therefore collapse the reported contrast into a single process viewed at different initial conditions.
  2. [Results (Deborah-number analysis)] § on viscoelastic effects and Deborah number: the statement that viscoelasticity 'introduces an additional intrinsic timescale that governs the competition between capillary driving and stress relaxation' is load-bearing for the distinction between Newtonian and viscoelastic fusion dynamics, yet the manuscript provides no quantitative mapping of Deborah number to the observed enhancement of intermediate-stage bridge expansion or to the deviation from the stretched-exponential fit.
minor comments (2)
  1. [Abstract] Abstract: the phrase 'stretched-exponential empirical formula' is used without citation; a reference to the specific functional form employed in the comparison would improve clarity.
  2. [Theory] Notation: the definition of the Deborah number and its relation to the stress-relaxation time should be stated explicitly in the theory section rather than only in the results.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their constructive comments on our manuscript. We address each major comment point by point below, providing the strongest honest defense of our work while acknowledging where revisions are warranted.

read point-by-point responses

  1. Referee: [Methods / Results (axisymmetric FEM)] Simulation methods and results sections: the central claim that fusion is 'intrinsically a multistage process involving localized curvature-driven neck formation' while shape recovery is 'global viscocapillary relaxation' rests on axisymmetric Oldroyd-B FEM runs. No mesh-convergence data, 3-D benchmark, or experimental validation is referenced for the high-curvature neck region where the multistage signature is diagnosed; numerical artifacts in this localized regime could therefore collapse the reported contrast into a single process viewed at different initial conditions.

    Authors: We agree that explicit demonstration of mesh convergence in the high-curvature neck region is necessary to support the multistage claim and rule out artifacts. In the revised manuscript we will add a dedicated subsection with mesh-refinement studies showing that neck formation time, bridge expansion rate, and the transition to global relaxation converge under successive refinements. The axisymmetric formulation is appropriate for the symmetric initial conditions examined and is standard for such problems; we will cite prior literature benchmarks validating the Oldroyd-B FEM implementation in comparable axisymmetric capillary flows. Full 3-D benchmarks are computationally prohibitive for the parameter space explored but are not required to establish the distinction under the stated symmetry. Experimental validation of the neck dynamics is outside the scope of this computational study. revision: partial

  2. Referee: [Results (Deborah-number analysis)] § on viscoelastic effects and Deborah number: the statement that viscoelasticity 'introduces an additional intrinsic timescale that governs the competition between capillary driving and stress relaxation' is load-bearing for the distinction between Newtonian and viscoelastic fusion dynamics, yet the manuscript provides no quantitative mapping of Deborah number to the observed enhancement of intermediate-stage bridge expansion or to the deviation from the stretched-exponential fit.

    Authors: The manuscript already shows comparative simulations across a range of Deborah numbers that illustrate the qualitative enhancement of intermediate-stage dynamics and increased deviation from the stretched-exponential form. To make this mapping quantitative, we will add explicit plots of intermediate-stage expansion rate versus De together with tabulated fit parameters (characteristic time and stretching exponent) for both Newtonian and viscoelastic cases, thereby directly linking the Deborah number to the observed differences. revision: yes

standing simulated objections not resolved
  • Experimental validation of the high-curvature neck dynamics, which would require new experiments beyond the computational scope of the present work.

Circularity Check

0 steps flagged

No significant circularity; claims rest on independent simulations and analytical theory

full rationale

The paper presents new axisymmetric Oldroyd-B FEM simulations and small-deformation analytical theory to distinguish shape recovery (global exponential relaxation) from fusion (multistage neck/bridge process) and to introduce Deborah-number effects. No load-bearing step reduces by construction to a fitted parameter, self-citation chain, or renamed input; the stretched-exponential comparison is an external benchmark against which deviations are reported rather than a fit renamed as prediction. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the validity of the Oldroyd-B model and the assumptions of the finite-element method for axisymmetric flows, which are standard in the field but not independently verified in the abstract.

axioms (1)
  • domain assumption The Oldroyd-B model is an appropriate constitutive relation for the viscoelastic droplets studied.
    Invoked for the simulations of viscoelastic behavior.

pith-pipeline@v0.9.1-grok · 5813 in / 1381 out tokens · 36931 ms · 2026-06-26T01:30:25.786227+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

44 extracted references

  1. [1]

    Dormann, 2019

    Alberti, S., and D. Dormann, 2019. Liquid–liquid phase separation in disease. Annual review of genetics 53:171–194

    2019

  2. [2]

    Noble, C

    Hubstenberger, A., S. Noble, C. Cameron, and T. Evans, 2013. Translation Repressors, an RNA Helicase, and Developmental Cues Control RNP Phase Transitions during Early Development. Developmental Cell 27:161–173

    2013

  3. [3]

    Jiang, 2020

    Li, W ., and H. Jiang, 2020. Regulation of tumorigenic splicing by protein condensates with specific biophysical properties. Molecular & Cellular Oncology 7:1819753

    2020

  4. [4]

    Roden, C., and A. S. Gladfelter, 2020. RNA contributions to the form and function of biomolecular condensates. Nature Reviews Molecular Cell Biology 22:183–195

    2020

  5. [5]

    Yu, H., S. Lu, K. Gasior, D. Singh, O. Tapia, S. Vazquez-Sanchez, D. Toprani, M. S. Beccari, J. R. Y ates, S. Da Cruz, J. M. Newby, M. Larfaga, A. S. Gladfelter, E. Villa, and D. W. Cleveland, 2020. TDP-43 and HSP70 phase separate into anisotropic, intranuclear liquid spherical annuli

    2020

  6. [6]

    Y amasaki, A., J. M. Alam, D. Noshiro, E. Hirata, Y . Fujioka, K. Suzuki, Y . Ohsumi, and N. N. Noda, 2020. Liquidity Is a Critical Determinant for Selective Autophagy of Protein Condensates. Molecular Cell 77:1163–1175.e9

    2020

  7. [7]

    Patel, A., H. Lee, L. Jawerth, S. Maharana, M. Jahnel, M. Hein, S. Stoynov, J. Mahamid, S. Saha, T. Franzmann, A. Pozniakovski, I. Poser, N. Maghelli, L. Royer, M. Weigert, E. Myers, S. Grill, D. Drechsel, A. Hyman, and S. Alberti,

  8. [8]

    Cell 162:1066–1077

    A Liquid-to-Solid Phase Transition of the ALS Protein FUS Accelerated by Disease Mutation. Cell 162:1066–1077

  9. [9]

    Woodru!, J. B., A. A. Hyman, and E. Boke, 2018. Organization and Function of Non-dynamic Biomolecular Condensates. Trends in Biochemical Sciences 43:81–94

    2018

  10. [10]

    Feng, Z., X. Chen, X. Wu, and M. Zhang, 2019. Formation of biological condensates via phase separation: Characteristics, analytical methods, and physiological implications. Journal of Biological Chemistry 294:14823–14835

    2019

  11. [11]

    Gui, X., F. Luo, Y . Li, H. Zhou, Z. Qin, Z. Liu, J. Gu, M. Xie, K. Zhao, B. Dai, W . S. Shin, J. He, L. He, L. Jiang, M. Zhao, B. Sun, X. Li, C. Liu, and D. Li, 2019. Structural basis for reversible amyloids of hnRNPA1 elucidates their role in stress granule assembly. Nature Communications 10

    2019

  12. [12]

    Fischer-Friedrich, S

    Jawerth, L., E. Fischer-Friedrich, S. Saha, J. Wang, T. Franzmann, X. Zhang, J. Sachweh, M. Ruer, M. Ijavi, S. Saha, J. Mahamid, A. A. Hyman, and F. Jülicher, 2020. Protein condensates as aging Maxwell fluids. Science 370:1317–1323

    2020

  13. [13]

    The glassiness of hardening protein droplets

    Zhang, H., 2020. The glassiness of hardening protein droplets. Science 370:1271–1272. 14 Manuscript submitted to Biophysical Journal Biophysical Journal T emplate

    2020

  14. [14]

    Brangwynne, C. P ., C. R. Eckmann, D. S. Courson, A. Rybarska, C. Hoege, J. Gharakhani, F. Jülicher, and A. A. Hyman,

  15. [15]

    Science 324:1729–1732

    Germline P granules are liquid droplets that localize by controlled dissolution/condensation. Science 324:1729–1732

  16. [16]

    Boija, A., I. A. Klein, B. R. Sabari, A. Dall’ Agnese, E. L. Co!ey, A. V . Zamudio, C. H. Li, K. Shrinivas, J. C. Manteiga, N. M. Hannett, et al., 2018. Transcription factors activate genes through the phase-separation capacity of their activation domains. Cell 175:1842–1855.e16

    2018

  17. [17]

    Sabari, B. R., A. Dall’ Agnese, A. Boija, I. A. Klein, E. L. Co!ey, K. Shrinivas, B. J. Abraham, N. M. Hannett, A. V . Zamudio, J. C. Manteiga, et al., 2018. Coactivator condensation at super-enhancers links phase separation and gene control. Science 361:eaar3958

    2018

  18. [18]

    Alberti, S., and A. A. Hyman, 2021. Biomolecular condensates at the nexus of cellular stress, protein aggregation disease and ageing. Nature reviews Molecular cell biology 22:196–213

    2021

  19. [19]

    Temirov, J

    Molliex, A., J. Temirov, J. Lee, M. Coughlin, A. Kanagaraj, H. Kim, T. Mittag, and J. Taylor, 2015. Phase separation by low complexity domains promotes stress granule assembly and drives pathological fibrillization. Cell 163:123–133

    2015

  20. [20]

    Elbaum-Garfinkle, S., Y . Kim, K. Szczepaniak, C. C.-H. Chen, C. R. Eckmann, S. Myong, and C. P . Brangwynne, 2015. The disordered P granule protein LAF-1 drives phase separation into droplets with tunable viscosity and dynamics. Proceedings of the National Academy of Sciences 112:7189–7194

    2015

  21. [21]

    Kota, and H.-X

    Ghosh, A., D. Kota, and H.-X. Zhou, 2021. Shear relaxation governs fusion dynamics of biomolecular condensates. Nature communications 12:5995

    2021

  22. [22]

    Jawerth, L. M., M. Ijavi, M. Ruer, S. Saha, M. Jahnel, A. A. Hyman, F. Jülicher, and E. Fischer-Friedrich, 2018. Salt-Dependent Rheology and Surface Tension of Protein Condensates Using Optical Traps. Physical Review Letters 121

    2018

  23. [23]

    Determination of Condensate Material Properties from Droplet Deformation

    Zhou, H.-X., 2020. Determination of Condensate Material Properties from Droplet Deformation. The Journal of Physical Chemistry B 124:8372–8379

    2020

  24. [24]

    Shape recovery of deformed biomolecular droplets: Dependence on condensate viscoelasticity

    Zhou, H.-X., 2021. Shape recovery of deformed biomolecular droplets: Dependence on condensate viscoelasticity. The Journal of Chemical Physics 155

    2021

  25. [25]

    Alshareedah, I., M. M. Moosa, M. Raju, D. Potoyan, and P . R. Banerjee, 2020. Phase Transition of RNA-protein Complexes into Ordered Hollow Condensates

    2020

  26. [26]

    Viscous e!ects on perturbed spherical flows

    Prosperetti, A., 1977. Viscous e!ects on perturbed spherical flows. Quarterly of Applied Mathematics 34:339–352

    1977

  27. [27]

    Free oscillations of drops and bubbles: the initial-value problem

    Prosperetti, A., 1980. Free oscillations of drops and bubbles: the initial-value problem. Journal of Fluid Mechanics 100:333–347

    1980

  28. [28]

    Bousmina, C

    Yu, W ., M. Bousmina, C. Zhou, and C. L. T ucker, 2004. Theory for drop deformation in viscoelastic systems. Journal of Rheology 48:417–438

    2004

  29. [29]

    Models for the deformation of a single ellipsoidal drop: a review

    Minale, M., 2010. Models for the deformation of a single ellipsoidal drop: a review. Rheologica Acta 49:789–806

    2010

  30. [30]

    Hooper, R. W ., V . F. de Almeida, C. W . Macosko, and J. J. Derby, 2001. Transient polymeric drop extension and retraction in uniaxial extensional flows. Journal of Non-Newtonian Fluid Mechanics 98:141–168

    2001

  31. [31]

    Cardinaels, P

    Verhulst, K., R. Cardinaels, P . Moldenaers, Y . Renardy, and S. Afkhami, 2009. Influence of viscoelasticity on drop deformation and orientation in shear flow. Journal of Non-Newtonian Fluid Mechanics 156:29–43

    2009

  32. [32]

    C., and L

    Tretheway, D. C., and L. Leal, 2001. Deformation and relaxation of Newtonian drops in planar extensional flows of a Boger fluid. Journal of Non-Newtonian Fluid Mechanics 99:81–108

    2001

  33. [33]

    B., and A

    Khismatullin, D. B., and A. Nadim, 2001. Shape oscillations of a viscoelastic drop. Physical Review E 63

    2001

  34. [34]

    Determination of condensate material properties from droplet deformation

    Zhou, H.-X., 2020. Determination of condensate material properties from droplet deformation. The Journal of Physical Chemistry B 124:8372–8379

    2020

  35. [35]

    Inc., C., 2025. COMSOL. Manuscript submitted to Biophysical Journal 15 Naderi et. al

    2025

  36. [36]

    Zolesio, 2001

    Souli, M., and J. Zolesio, 2001. Arbitrary Lagrangian–Eulerian and free surface methods in fluid mechanics. Computer methods in applied mechanics and engineering 191:451–466

    2001

  37. [37]

    Hyperelastic material models for finite element analysis of rubber

    Y eoh, O., 1997. Hyperelastic material models for finite element analysis of rubber. Journal of Natural Rubber Research 12:142–153

    1997

  38. [38]

    A-BDF: a generalization of the backward di!erentiation formulae

    Fredebeul, C., 1998. A-BDF: a generalization of the backward di!erentiation formulae. SIAM journal on numerical analysis 35:1917–1938

    1998

  39. [39]

    Amestoy, P . R., I. S. Du!, J.- Y . L ’Excellent, and J. Koster, 2000. MUMPS: a general purpose distributed memory sparse solver. In International Workshop on Applied Parallel Computing. Springer, 121–130

    2000

  40. [40]

    K., 2000

    Batchelor, G. K., 2000. An introduction to fluid dynamics. Cambridge university press

    2000

  41. [41]

    Bird, R. B., R. C. Armstrong, and O. Hassager, 1986. Dynamics of polymeric liquids. Vol. 1: Fluid mechanics

    1986

  42. [42]

    Cardinaels, P

    Verhulst, K., R. Cardinaels, P . Moldenaers, Y . Renardy, and S. Afkhami, 2009. Influence of viscoelasticity on drop deformation and orientation in shear flow: Part 1. Stationary states. Journal of non-newtonian fluid mechanics 156:29–43

    2009

  43. [43]

    Naderi, M. M., L. Barilla, J. Zhou, I. Papautsky, and Z. Peng, 2022. Elasto-inertial focusing mechanisms of particles in shear-thinning viscoelastic fluid in rectangular microchannels. Micromachines 13:2131

    2022

  44. [44]

    Gao, H., J. Zhou, M. M. Naderi, Z. Peng, and I. Papautsky, 2023. Evolution of focused streams for viscoelastic flow in spiral microchannels. Microsystems & Nanoengineering 9:73. 16 Manuscript submitted to Biophysical Journal

    2023