The Critical Plastocapillary Number for a Newtonian Liquid Filament Embedded into a Viscoplastic Fluid

Authors

Mohammad Tanver Hossain, University of Illinois at Urbana-Champaign
Wonsik Eom, University of Illinois at Urbana-Champaign
Arjun Shah, University of Illinois at Urbana-Champaign
Andrew Lowe, Chapman UniversityFollow
Douglas Fudge, Chapman UniversityFollow
Sameh H. Tawfick, University of Illinois at Urbana-Champaign
Randy Ewoldt, University of Illinois at Urbana-ChampaignFollow

Document Type

Article

Publication Date

6-11-2025

Abstract

The yield stress of a viscoplastic material can stabilize an embedded fluid tunnel against capillarity-induced breakup, enabling remarkable technologies such as embedded 3D printing of intricate, freeform, and small components. However, there is persistent disagreement in the published literature between the observed minimum stable diameter, 𝑑min, and the theoretical plastocapillary length 𝑝𝑐 = 2𝛤∕𝜎𝑦, with interfacial tension 𝛤 and bath yield stress 𝜎𝑦, leading to a prior hypothesis that the apparent surface tension 𝛤 is much smaller to enforce 𝑑min = 𝑝𝑐 . Here we introduce and experimentally test a new hypothesis that the critical diameter is set by the dimensionless plastocapaillary number, 𝑌𝛤 = 𝜎𝑦𝑑∕2𝛤, having a non-trivial critical value different than one, 𝑌𝛤𝑐 ≠ 1, and therefore the prior hypothesis of adjusting 𝛤 to enforce 𝑝𝑐 = 𝑑min is incorrect. We study several Newtonian inks (uncured polydimethylsiloxane (PDMS), highly refined mineral oil, silicone oil) extruded into a wide range of non-Newtonian viscoplastic bath materials (polyacrylic acid microgels, polysaccharide microgels, nanoclay gel, and micro-organogels). Across this wide parameter space, we observe a critical value of 𝑌𝛤𝑐 = 0.21±0.03. We explain this being less than one by analogy to other critical dimensionless groups with yield stress fluids, such as the gravitational stability of a suspended sphere or bubble, where the yield stress acts upon an effective area larger than the naïve estimate set only by embedded object diameter 𝑑. These results provide a new way to understand and predict the minimum stable diameter of embedded liquid filaments, as in embedded 3D printing, as 𝑑min = 𝑌𝛤𝑐 (2𝛤∕𝜎𝑦).

Comments

This article was originally published in Journal of Non-Newtonian Fluid Mechanics, volume 343, in 2025. https://doi.org/10.1016/j.jnnfm.2025.105440

Recommended Citation

M.T. Hossain, W. Eom, A. Shah, A. Lowe, D. Fudge, S.H. Tawfick, R.H. Ewoldt, The critical plastocapillary number for a Newtonian liquid filament embedded into a viscoplastic fluid, J. Non-Newton. Fluid Mech. 343 (2025) 105440, https://doi.org/10.1016/j.jnnfm.2025.105440

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The authors

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This work is licensed under a Creative Commons Attribution-Noncommercial 4.0 License