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REVIEW 4 major objections 5 minor 58 references

Computing sessile droplet shapes on arbitrary surfaces with a new pairwise force smoothed particle hydrodynamics model

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A new polynomial pairwise-force profile for smoothed particle hydrodynamics makes simulated surface tension and contact angle scale-independent, so equilibrium sessile droplet shapes can be computed on arbitrary rough and chemically…

desk verdict New polynomial PF-SPH force profiles give a genuinely scale-independent surface tension calibration, but the contact-angle side leans on a suspect density isosurface and a partly self-referential fit—worth peer review all the same. read the letter →

arxiv 2412.03810 v2 pith:HNB6J2RY submitted 2024-12-05 physics.flu-dyn cs.NAmath.NA

Riley M Whebell , Timothy J Moroney , Ian W Turner , Ravindra Pethiyagoda , Scott W McCue This is my paper · ORCID
classification physics.flu-dyncs.NAmath.NA MSC 76M2876D45 PACS 47.55.D47.11.-j
keywords smoothedparticlehydrodynamicspairwiseforcemodelsurfacetensioncalibrationcontactanglesessiledropletwettingroughchemicallypatterned
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that a smoothed particle hydrodynamics (SPH) model with a new, physically motivated pairwise force profile can compute equilibrium sessile droplet shapes on arbitrary rough and chemically heterogeneous surfaces without ever tracking the liquid-gas interface or the contact line. The central difficulty it addresses is that once the surface is not flat and axisymmetric, both the shape of the contact line and the contact angle at each point are unknown, so grid-based methods that enforce a prescribed angle lose their footing. The paper's proposed fix is a scale-independent pairwise force: the fluid-fluid interaction strength $s_{\mathrm{ff}}$ sets the surface tension through $\sigma = 30.96 s_{\mathrm{ff}}$, and the ratio of fluid-solid to fluid-fluid strengths $s_{\mathrm{fs}}/s_{\mathrm{ff}}$ sets the equilibrium contact angle through a calibrated curve obtained by fitting whole droplet shapes to Young-Laplace solutions. If the claim is right, droplet shapes on leaves, pillared surfaces, and chemically patterned substrates can be simulated with the contact angle emerging from the physics rather than being imposed as a boundary condition, which matters for spray retention, microfluidics, and wetting studies.

What carries the argument

The load-bearing object is the pairwise force term $\mathbf{F}^{(\mathrm{pf})}_i = (H/m_i) \sum_j s_{ij} f_{ij}(\|\mathbf{x}_{ij}\|/H) \, \mathbf{x}_{ij}/\|\mathbf{x}_{ij}\|$, where $f_{ij}$ is a quartic polynomial force profile (equation 8) with a repulsive core, an attractive tail, and a zero crossing placed at $1.1\Delta x$ for fluid-fluid pairs and $2\Delta x$ for fluid-solid pairs. The quartic is the lowest-degree polynomial satisfying the endpoint and smoothness constraints, and the fluid-fluid zero crossing is motivated by the nearest-neighbour distance in an optimally packed particle arrangement. The explicit factor $H$ outside the sum makes $s_{ij}$ have the units of an interfacial tension, which is what renders the simulated surface tension independent of resolution. On top of that force, the calibration machinery consists of the virial-pressure measurement of Laplace pressure, the droplet-oscillation frequency, and whole-shape Young-Laplace fitting to the density isosurface $\langle \rho(\mathbf{x}) \rangle = \rho_0/2$, whose vertices give the contact angle.

What would settle it

Run the calibrated model at particle widths smaller than $2\times 10^{-5}$ m or larger than $8\times 10^{-5}$ m on a flat surface, measure the Laplace-pressure slope $\sigma = 30.96 s_{\mathrm{ff}}$ and the fitted Young-Laplace contact angle for a fixed $s_{\mathrm{fs}}/s_{\mathrm{ff}}$, and check whether both stay on the calibration curves; any drift with resolution would falsify the scale-independence claim. A second check is to compare the volume enclosed by the density isosurface with the known droplet volume near the contact line, since the paper already discards droplets whose isosurface volume is off by more than 10 percent.

Watch

Extended reading notes

Core claim

The central discovery is that the pairwise force profiles chosen in equation (8), together with the $H$-scaled force term in equation (7), remove the resolution dependence that has plagued earlier PF-SPH formulations and turn the simulated surface tension and contact angle into controllable, calibration-ready quantities. In the model, surface tension is produced by fluid-fluid cohesive forces and wetting by fluid-solid adhesive forces, both implemented as short-range repulsive-attractive polynomial forces between particles; the prefactor $H$ in the force term gives the interaction strength the units of N/m, so the resulting surface tension does not change when the particle spacing is refined. The paper calibrates $\sigma = 30.96 s_{\mathrm{ff}}$ by measuring Laplace pressure in static spherical droplets and by matching the Rayleigh oscillation frequency of perturbed droplets, and it calibrates the contact angle by fitting semi-analytical Young-Laplace profiles to the density isosurface of simulated sessile droplets. Across different surface tensions and volumes the measured contact angles collapse onto a single curve in the ratio $s_{\mathrm{fs}}/s_{\mathrm{ff}}$, supporting the model's scale independence. The same machinery then produces sessile droplet shapes on a square-pillared superhydrophobic surface, on a hydrophobic surface with a hydrophilic stripe, and on a reconstructed wheat leaf with roughness and hairs.

Load-bearing premise

Every contact-angle measurement and case-study conclusion assumes that the density isosurface at half the reference density, $\langle \rho(\mathbf{x}) \rangle = \rho_0/2$, reliably marks the liquid-gas interface and that the contact line lies exactly two particle widths ($2\Delta x$) from the substrate, so a systematic bias of that isosurface near the contact line would shift the calibrated $s_{\mathrm{fs}}/s_{\mathrm{ff}}$-to-angle curve and the computed droplet shapes.

Editorial extensions

If this is right

  • Surface tension can be prescribed directly as $s_{\mathrm{ff}} = \sigma/30.96$ at any tested resolution, replacing per-resolution tuning with a single linear calibration.
  • Equilibrium contact angle can be set by interpolating the ratio $s_{\mathrm{fs}}/s_{\mathrm{ff}}$ from the calibrated curve, with the angle emerging naturally at the contact line rather than being imposed.
  • The model reproduces the suspended-to-collapsed wetting-state transition on square-pillared surfaces when the droplet has a small impact velocity, matching the qualitative behaviour of the experimental study it follows.
  • Chemically patterned substrates are handled by simply varying the adhesive strength $s_{\mathrm{fs}}$ across the surface, as demonstrated by the smooth contact-angle transition around a hydrophilic stripe.
  • Because no interface tracking is required, the same code runs on physically rough and chemically heterogeneous surfaces such as the reconstructed wheat leaf, giving a contact-line shape that would otherwise require a separate free-boundary calculation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the scale independence extends below the tested particle widths, the same calibration curve could predict surface tension and contact angle for microfluidic droplets without re-fitting; the paper only demonstrates the linear law between $2\times 10^{-5}$ m and $8\times 10^{-5}$ m.
  • The whole-shape Young-Laplace fitting used for calibration could serve as a standard benchmark for comparing other pairwise force profiles, since the paper notes the literature lacks consensus without performing that comparison itself.
  • The paper's own 10 percent volume filter suggests the density isosurface can mislocate the interface; a more accurate interface reconstruction near the contact line would likely sharpen the contact-angle calibration and the wheat-leaf contact-line visualisation.
  • The wheat-leaf contact line could be fed directly into an evaporation or retention model as an initial condition; the paper lists this as future work rather than a demonstrated capability.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The manuscript proposes a pairwise-force smoothed particle hydrodynamics (PF-SPH) model with new polynomial force profiles (Eq. 8) and an H-scaled interaction (Eq. 7) intended to make simulated surface tension and contact angle scale-independent. The model is calibrated by measuring surface tension from Laplace pressure in static spherical droplets and from ellipsoidal droplet oscillation frequencies, both giving a linear relation sigma = 30.96 s_ff. The contact angle is calibrated by fitting semi-analytical Young-Laplace profiles to density isosurfaces of sessile droplets and plotting the fitted angle against s_fs/s_ff. The method is then demonstrated on a pillared superhydrophobic-type surface, a chemically patterned surface with a hydrophilic stripe, and a reconstructed wheat leaf, with the contact line obtained without explicit tracking.

Significance. If the calibration is reliable, the paper makes a useful contribution: the linear sigma-s_ff relation is corroborated by two independent tests, the H-scaling addresses a known resolution dependence in PF-SPH, and the case studies show the model can handle complex substrates without explicit contact-line tracking. The open-source implementation is a further strength. However, the contact-angle calibration is partly self-referential and rests on an isosurface definition whose accuracy near the substrate is not established. These issues affect the central predictive claim, so the contribution is not yet fully supported, although the underlying modelling idea is promising and the Laplace-pressure and oscillation results provide a solid core.

major comments (4)
  1. [Section 3.3, Eq. (13), Figures 10-11] The contact-angle calibration is partly circular: theta_CA is defined as the best-fit parameter of a Young-Laplace profile fitted to the simulated isosurface, and the same fitted values are then presented as agreement with Young-Laplace. This establishes that the simulated shapes are Young-Laplace-like, but it does not independently validate the model's wetting behaviour. Please provide an independent measurement of the contact angle (for example, a tangent construction at the contact line, or comparison with an independent numerical method) or explicitly reframe the claim as calibration rather than validation.
  2. [Section 2.2.4 versus Eq. (12)] The interface in Eq. (12) is defined from a fluid-only density sum, whereas the density evolved in Eq. (3) and used in the pressure and pairwise forces includes fixed solid particles. Near the substrate the fluid-only sum is truncated, so the half-density isosurface may be displaced by an O(Delta x) offset relative to the density field that actually controls the forces. Since the contact line is defined as the intersection of this isosurface with the 2 Delta x offset surface (Section 4.3), any systematic isosurface bias propagates into the calibration curve and the wheat-leaf contact line. Please test the sensitivity of the calibration to an alternative interface definition based on the full density field, or demonstrate convergence with resolution.
  3. [Figures 10 and 11] Droplets are excluded when the isosurface-enclosed volume differs from the true droplet volume by more than 10%, which is direct evidence that the isosurface can be inaccurate. The filter may bias the s_fs/s_ff-theta_CA curve if exclusions correlate with the force ratio. Please report the number and parameter values of excluded cases and show that the calibration is unchanged when the threshold is varied or when a corrected interface is used.
  4. [Section 4.2, Figure 14] The contact angle along the contact line is measured using the surface normal of the isosurface at a fixed height H/2 above the substrate, rather than at the contact line itself. This offset is not physically calibrated and may depend on resolution; it should be justified or replaced by a definition tied to the actual contact line, especially because the transition plot is used to support the chemically patterned surface claim.
minor comments (5)
  1. [Section 3.1] The claim that sigma = 30.96 s_ff is resolution-independent over Delta x in [2e-5, 8e-5] m is stated but not shown in any figure or table; please include the supporting data or a convergence plot.
  2. [Figures 10 and 11] The main text should state how many droplets were excluded by the 10% volume filter and whether exclusions are concentrated at particular s_fs/s_ff values.
  3. [Section 4.1, Figure 13] The comparison with Dupuis and Yeomans (2005) is qualitative; please state this explicitly and describe any quantitative metrics used.
  4. [References] The reference list contains a rendering issue ('M¨ uller') that should be corrected to 'Müller'.
  5. [Section 3.3] Equation (12) is introduced as an isosurface of the SPH-interpolated density, but the text does not specify the smoothing length used for the interpolation grid in the marching cubes step; a sentence clarifying this would help reproducibility.

Circularity Check

1 steps flagged · score 4.0 of 10

Contact-angle validation is self-referential (theta_CA is the Young-Laplace fit parameter), while surface-tension calibration is independently checked and the complex-surface case studies are not circular.

  1. fitted input called prediction [Section 3.3 (Eqs. 12-13, Figures 9-11)]
    "Then, to determine the contact angle of the droplet, we fit the shape of an axisymmetric sessile droplet determined by the Young-Laplace (Y-L) equation (Danov et al., 2016) to the vertices of the SPH droplet isosurface ... With this validation of the droplet shape and calibration of the contact angle, we now have a simple procedure for specifying the surface tension σ and contact angle θ_CA in the PF-SPH scheme."

    The contact angle is not measured by an independent protocol; θ_CA is the free parameter that best fits a Y-L solution to the SPH isosurface. Therefore the paper's claim that flat-surface SPH interfaces agree with Y-L solutions is a restatement of the fitting procedure, and the s_fs/s_ff versus θ_CA calibration curve in Figures 10-11 is an interpolation of those fitted values. This makes the flat-surface contact-angle 'validation' self-referential. The later rough-surface and wheat-leaf computations are not circular because they use the calibrated s_fs/s_ff as an input and compute the droplet shape without fitting θ_CA, though they inherit any bias in the interface definition (Eq. 12) used for calibration.

full rationale

The surface-tension branch of the derivation is not circular: σ is calibrated from the Laplace-pressure slope (Fig. 4) and independently corroborated by Rayleigh oscillation frequencies (Fig. 5), i.e. two different physical laws. The new force profile and H-scaling are a modelling ansatz, not a hidden reuse of the target result. The only substantive circularity is the contact-angle validation in Section 3.3, where the Young-Laplace profile itself supplies the definition of θ_CA via least-squares fitting, so 'good agreement' with Y-L is partly built into the measurement. This does not destroy the central claim, because the calibration curve is then used as an input for complex-surface predictions that are not fitted. Limitations worth noting, but not circular: the fluid-only density isosurface (Eq. 12) is used to measure interfaces although solid particles enter the simulated density sums (Section 2.2.4), and the 10% volume-exclusion filter in Figures 10-11 shows the isosurface can be inaccurate; this is a correctness risk for the contact-angle curve and case-study contact lines, not a definitional equivalence. The self-citation to Whebell et al. (2025) supplies only the wheat-leaf geometry from a microCT-based reconstruction and is not load-bearing for any theoretical claim. Overall score 4: one fitted-input-as-validation step, but the model's independent content (dynamic surface tension, scale-independence collapse, complex-surface demonstrations) remains.

Assumptions & free parameters 4 free parameters · 7 assumptions · 0 invented entities

The model introduces no new physical entities; the pairwise force is a numerical device. The central free parameters are the calibration constants and the zero-crossing positions of the force profiles. The key assumptions are the empirical PF-SPH ansatz and the isosurface representation of the interface.

free parameters (4)
  • surface tension calibration constant 30.96 = 30.96 (dimensionless)
    Measured slope from virial pressure versus 2/R for various s_ff; converts model interaction strength to physical surface tension. It is an empirical constant specific to this force profile and kernel.
  • contact angle calibration curve s_fs/s_ff vs theta_CA = interpolated from Figure 10
    Simulated droplets with different s_fs/s_ff are fit to Young-Laplace profiles to obtain contact angle; this lookup is used to set s_fs for target contact angles.
  • fluid-fluid zero crossing kappa*_ff = 1.1 = 1.1 (multiples of Delta-x)
    Chosen from FCC packing geometry (rhombic dodecahedral Voronoi volume gives rest distance approximately 1.1 Delta-x); determines polynomial coefficient alpha_ff = -23.5.
  • fluid-solid zero crossing kappa*_fs = 2 = 2.0 (multiples of Delta-x)
    Chosen to prioritize cohesion over adhesion at the contact line; determines alpha_fs = -11.
assumptions (7)
  • domain assumption Weakly compressible barotropic Navier-Stokes equations govern droplet flow
    Section 2.1; the continuum model is standard for SPH, with artificial speed of sound c0 = 10 v_max.
  • ad hoc to paper Pairwise forces can emulate surface tension and wetting at SPH resolution
    Section 2.4; the basis is empirical, explicitly acknowledged ('The basis for these forces in SPH is empirical'). No derivation from molecular interactions.
  • domain assumption The density isosurface at rho0/2 represents the liquid-gas interface
    Equation (12); used for all shape measurements and case-study rendering; validated only indirectly via the 10% volume filter.
  • domain assumption The Wendland kernel with H/Delta-x = 4 is a suitable interpolation kernel
    Section 2.2; standard in SPH, but the choice affects the pairwise force calibration constants.
  • domain assumption Dummy boundary particles with zero velocity approximate the no-slip condition
    Section 2.2.4; known to be consistent only as H approaches 0; for rough surfaces this is approximate.
  • standard math Young-Laplace semi-analytical solutions (Danov et al., 2016) are the correct benchmark for sessile droplet shapes
    Section 3.3; used to define the contact angle and to validate the shape.
  • domain assumption The equilibrium contact angle depends only on the ratio s_fs/s_ff (not on volume, surface tension, or resolution)
    Section 3.3; supported by Figures 10 and 11 showing collapse by ratio, but the patterned-surface result (target 45 degrees, measured 55 degrees) shows the transfer is imperfect.

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Cite this review

Pith. "Pith review of Computing sessile droplet shapes on arbitrary surfaces with a new pairwise force smoothed particle hydrodynamics model." pith.science (2026). https://pith.science/paper/HNB6J2RY

@misc{pith2026241203810,
  author       = {Pith},
  title        = {Pith review of: Computing sessile droplet shapes on arbitrary surfaces with a new pairwise force smoothed particle hydrodynamics model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HNB6J2RY}},
  note         = {Machine review of arXiv:2412.03810}
}
read the original abstract

The study of the shape of droplets on surfaces is an important problem in the physics of fluids and has applications in multiple industries, from agrichemical spraying to microfluidic devices. Motivated by these real-world applications, computational predictions for droplet shapes on complex substrates -- rough and chemically heterogeneous surfaces -- are desired. Grid-based discretisations in axisymmetric coordinates form the basis of well-established numerical solution methods in this area, but when the problem is not axisymmetric, the shape of the contact line and the distribution of the contact angle around it are unknown. Recently, particle methods, such as pairwise force smoothed particle hydrodynamics (PF-SPH), have been used to conveniently forego explicit enforcement of the contact angle. The pairwise force model, however, is far from mature, and there is no consensus in the literature on the choice of pairwise force profile. We propose a new pair of polynomial force profiles with a simple motivation and validate the PF-SPH model in both static and dynamic tests. We demonstrate its capabilities by computing droplet shapes on a physically structured surface, a surface with a hydrophilic stripe, and a virtual wheat leaf with both micro-scale roughness and variable wettability. We anticipate that this model can be extended to dynamic scenarios, such as droplet spreading or impaction, in the future.

Figures

Figures reproduced from arXiv: 2412.03810 by the authors.

Figure 1
Figure 1. A side view of a 3D particle simulation in which a droplet of fluid particles (blue) is [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Distance-dependent pairwise force profiles [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Virial pressure calculated using equation ( [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (13 more)
Figure 4
Figure 4. Figure 4: Laplace pressure validation using calculated virial pressures of spherical droplets. [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: Calibrating the cohesive force strength 𝑠ff to measured surface tension in two different tests. Circles show surface tensions calculated from the Laplace pressure 𝑃 = 2𝜎/𝑅 (see [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]
Figure 6
Figure 6. Figure 6: The oscillating diameters of an inviscid, axis-aligned ellipsoidal droplet over 30ms, [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: The power spectrum of the oscillating droplet’s diameters (see Figure [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]
Figure 8
Figure 8. Figure 8: Schematic of the transformation from the cylindrical polar world coordinates to the [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]
Figure 9
Figure 9. Figure 9: An example of a Young-Laplace profile (a solution to the system ( [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]
Figure 10
Figure 10. Figure 10: Contact angles, measured by fitting Young-Laplace profiles to sessile droplet density [PITH_FULL_IMAGE:figures/full_fig_p020_10.png]
Figure 11
Figure 11. Figure 11: Contact angles, measured by fitting Young-Laplace profiles to sessile droplet density [PITH_FULL_IMAGE:figures/full_fig_p021_11.png]
Figure 12
Figure 12. Figure 12: A top-down view of the pillared substrate used for the simulation in Figure [PITH_FULL_IMAGE:figures/full_fig_p023_12.png]
Figure 13
Figure 13. Figure 13: A comparison of nearly identical simulations of a droplet settling on the physically [PITH_FULL_IMAGE:figures/full_fig_p023_13.png]
Figure 14
Figure 14. Figure 14: A droplet settles on a flat hydrophobic surface ( [PITH_FULL_IMAGE:figures/full_fig_p025_14.png]
Figure 15
Figure 15. Figure 15: A pairwise force SPH simulation of a 0.27𝜇L water droplet settled on the surface of a wheat leaf after 12 ms. The droplet’s liquid-gas interface is rendered by polygonising an isosurface of the density field, ⟨𝜌(x)⟩ = 𝜌0/2. The leaf surface is rendered similarly, usin…
Figure 16
Figure 16. Figure 16: The contact line of the droplet on a wheat leaf depicted in Figure [PITH_FULL_IMAGE:figures/full_fig_p028_16.png]

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Works this paper leans on

58 extracted references · 41 canonical work pages

  1. [1]

    Computer Simulation of Liquids

    Allen, MP and Tildesley, DJ (1989). Computer Simulation of Liquids. Clarendon Press, Oxford, corrected edition. ISBN 0-19-855645-4

    work page 1989

  2. [2]

    Coalescing and break-up of viscous drops with surface tension through the Smoothed Particle Hydrodynamics

    Antuono, M; Marrone, S; and Colagrossi, A (2025). Coalescing and break-up of viscous drops with surface tension through the Smoothed Particle Hydrodynamics . Computer Methods in Applied Mechanics and Engineering, 442:118014. doi:10.1016/j.cma.2025.118014

    work page arXiv 2025

  3. [3]

    Comparison of surface tension generation methods in smoothed particle hydrodynamics for dynamic systems

    Arai, E; Tartakovsky, A; Holt, RG; Grace, S; and Ryan, E (2020). Comparison of surface tension generation methods in smoothed particle hydrodynamics for dynamic systems. Computers & Fluids, 203:104540. doi:10.1016/j.compfluid.2020.104540

    arXiv 2020

  4. [4]

    Smoothed particle hydrodynamics for coastal engineering problems

    Barreiro, A; Crespo, A; Dom \'i nguez, J; and G \'o mez-Gesteira , M (2013). Smoothed particle hydrodynamics for coastal engineering problems. Computers & Structures, 120:96--106. doi:10.1016/j.compstruc.2013.02.010

    work page doi:10.1016/j.compstruc.2013.02.010 2013

  5. [5]

    Surface tension and wetting at free surfaces in smoothed particle hydrodynamics

    Blank, M; Nair, P; and P \"o schel, T (2024). Surface tension and wetting at free surfaces in smoothed particle hydrodynamics. Journal of Fluid Mechanics, 987:A23. doi:10.1017/jfm.2024.410

    work page doi:10.1017/jfm.2024.410 2024

  6. [6]

    Variational and momentum preservation aspects of smooth particle hydrodynamic formulations

    Bonet, J and Lok, TS (1999). Variational and momentum preservation aspects of smooth particle hydrodynamic formulations. Computer Methods in Applied Mechanics and Engineering, 180:97--115. doi:10.1016/S0045-7825(99)00051-1

    doi:10.1016/s0045-7825(99)00051-1 1999

  7. [7]

    Physics of Wetting: Phenomena and Applications of Fluids on Surfaces

    Bormashenko, EY (2017). Physics of Wetting: Phenomena and Applications of Fluids on Surfaces . De Gruyter, Berlin & Boston. ISBN 978-3-11-044481-0. doi:doi:10.1515/9783110444810

    work page doi:10.1515/9783110444810 2017

  8. [8]

    A continuum method for modeling surface tension

    Brackbill, J; Kothe, D; and Zemach, C (1992). A continuum method for modeling surface tension. Journal of Computational Physics, 100:335--354. doi:10.1016/0021-9991(92)90240-Y

    doi:10.1016/0021-9991(92)90240-y 1992

Show all 58 references
  1. [9]

    Static drop on an inclined plate: Analysis by the finite element method

    Brown, R; Orr, F; and Scriven, L (1980). Static drop on an inclined plate: Analysis by the finite element method. Journal of Colloid and Interface Science, 73:76--87. doi:10.1016/0021-9797(80)90124-1

    1980 doi

  2. [10]

    Progress in modeling wetting phenomena on structured substrates

    Chamakos, NT; Sema, DG; and Papathanasiou, AG (2021). Progress in modeling wetting phenomena on structured substrates. Archives of Computational Methods in Engineering, 28:1647--1666. doi:10.1007/s11831-020-09431-3

    2021 doi

  3. [11]

    Fast and accurate SPH modelling of 3D complex wall boundaries in viscous and non viscous flows

    Chiron, L; De Leffe, M; Oger, G; and Le Touz \'e , D (2019). Fast and accurate SPH modelling of 3D complex wall boundaries in viscous and non viscous flows. Computer Physics Communications, 234:93--111. doi:10.1016/j.cpc.2018.08.001

    2019 doi

  4. [12]

    Theoretical considerations on the free-surface role in the smoothed-particle-hydrodynamics model

    Colagrossi, A; Antuono, M; and Le Touz \'e , D (2009). Theoretical considerations on the free-surface role in the smoothed-particle-hydrodynamics model. Physical Review E, 79:056701. doi:10.1103/PhysRevE.79.056701

    2009 doi

  5. [13]

    Underwater Explosions

    Cole, RH (1948). Underwater Explosions. Princeton University Press, Princeton, N.J

    1948

  6. [14]

    DualSPHysics : Open-source parallel CFD solver based on smoothed particle hydrodynamics ( SPH )

    Crespo, A; Dom \'i nguez, J; Rogers, B; G \'o mez-Gesteira , M; Longshaw, S; Canelas, R; Vacondio, R; Barreiro, A; and Garc \'i a-Feal , O (2015). DualSPHysics : Open-source parallel CFD solver based on smoothed particle hydrodynamics ( SPH ). Computer Physics Communications, ...

    2015 doi

  7. [15]

    GPUs , a new tool of acceleration in CFD : Efficiency and reliability on smoothed particle hydrodynamics methods

    Crespo, AC; Dominguez, JM; Barreiro, A; G \'o mez-Gesteira , M; and Rogers, BD (2011). GPUs , a new tool of acceleration in CFD : Efficiency and reliability on smoothed particle hydrodynamics methods. PLoS ONE, 6:e20685. doi:10.1371/journal.pone.0020685

    2011 doi

  8. [16]

    Shape analysis of a rotating axisymmetric drop in gravitational field: Comparison of numerical schemes for real-time data processing

    Danov, KD; Dimova, SN; Ivanov, TB; and Novev, JK (2016). Shape analysis of a rotating axisymmetric drop in gravitational field: Comparison of numerical schemes for real-time data processing. Colloids and Surfaces A: Physicochemical and Engineering Aspects, 489:75--85. doi:10.1...

    2016 doi

  9. [17]

    Improving convergence in smoothed particle hydrodynamics simulations without pairing instability: SPH without pairing instability

    Dehnen, W and Aly, H (2012). Improving convergence in smoothed particle hydrodynamics simulations without pairing instability: SPH without pairing instability. Monthly Notices of the Royal Astronomical Society, 425:1068--1082. doi:10.1111/j.1365-2966.2012.21439.x

    2012

  10. [18]

    Neighbour lists in smoothed particle hydrodynamics

    Dom \'i nguez, JM; Crespo, AJC; G \'o mez-Gesteira , M; and Marongiu, JC (2011). Neighbour lists in smoothed particle hydrodynamics. International Journal for Numerical Methods in Fluids, 67:2026--2042. doi:10.1002/fld.2481

    2011 doi

  11. [19]

    DualSPHysics : From fluid dynamics to multiphysics problems

    Dom \'i nguez, JM; Fourtakas, G; Altomare, C; Canelas, RB; Tafuni, A; Garc \'i a-Feal , O; Mart \'i nez-Est \'e vez , I; Mokos, A; Vacondio, R; Crespo, AJC; Rogers, BD; Stansby, PK; and G \'o mez-Gesteira , M (2022). DualSPHysics : From fluid dynamics to multiphysics problems....

    2022 doi

  12. [20]

    Spray retention on whole plants: Modelling, simulations and experiments

    Dorr, GJ; Forster, WA; Mayo, LC; McCue, SW; Kempthorne, DM; Hanan, J; Turner, IW; Belward, JA; Young, J; and Zabkiewicz, JA (2016). Spray retention on whole plants: Modelling, simulations and experiments. Crop Protection, 88:118--130. doi:10.1016/j.cropro.2016.06.003

    2016 doi

  13. [21]

    Impaction of spray droplets on leaves: Influence of formulation and leaf character on shatter, bounce and adhesion

    Dorr, GJ; Wang, S; Mayo, LC; McCue, SW; Forster, WA; Hanan, J; and He, X (2015). Impaction of spray droplets on leaves: Influence of formulation and leaf character on shatter, bounce and adhesion. Experiments in Fluids, 56:143. doi:10.1007/s00348-015-2012-9

    2015 doi

  14. [22]

    Modeling droplets on superhydrophobic surfaces: Equilibrium states and transitions

    Dupuis, A and Yeomans, JM (2005). Modeling droplets on superhydrophobic surfaces: Equilibrium states and transitions. Langmuir, 21:2624--2629. doi:10.1021/la047348i

    2005 doi

  15. [23]

    Smoothed particle hydrodynamics: Theory and application to non-spherical stars

    Gingold, RA and Monaghan, JJ (1977). Smoothed particle hydrodynamics: Theory and application to non-spherical stars. Monthly Notices of the Royal Astronomical Society, 181:375--389. doi:10.1093/mnras/181.3.375

    1977 doi

  16. [24]

    Axisymmetric Fluid-Liquid Interfaces: Tables Giving the Shape of Sessile and Pendant Drops and External Menisci, with Examples of Their Use

    Hartland, S and Hartley, RW (1976). Axisymmetric Fluid-Liquid Interfaces: Tables Giving the Shape of Sessile and Pendant Drops and External Menisci, with Examples of Their Use . Elsevier Scientific Pub. Co, Amsterdam & New York. ISBN 978-0-444-41396-3

    1976

  17. [25]

    Rival contact-angle models and the spreading of drops

    Hocking, LM (1992). Rival contact-angle models and the spreading of drops. Journal of Fluid Mechanics, 239:671--681. doi:10.1017/S0022112092004579

    1992 doi

  18. [26]

    Isomorphism linking smooth particles and embedded atoms

    Hoover, WG (1998). Isomorphism linking smooth particles and embedded atoms. Physica A: Statistical Mechanics and its Applications, 260:244--254. doi:10.1016/S0378-4371(98)00357-4

    1998 doi

  19. [27]

    A parallel SPH implementation on multi-core CPUs

    Ihmsen, M; Akinci, N; Becker, M; and Teschner, M (2011). A parallel SPH implementation on multi-core CPUs . Computer Graphics Forum, 30:99--112. doi:10.1111/j.1467-8659.2010.01832.x

    2011

  20. [28]

    Numerical simulation of two-phase droplets on a curved surface using Surface Evolver

    Jamali, M and Tafreshi, HV (2021). Numerical simulation of two-phase droplets on a curved surface using Surface Evolver . Colloids and Surfaces A: Physicochemical and Engineering Aspects, 629:127418. doi:10.1016/j.colsurfa.2021.127418

    2021

  21. [29]

    Simulating anisotropic droplet shapes on chemically striped patterned surfaces

    Jansen, HP; Bliznyuk, O; Kooij, ES; Poelsema, B; and Zandvliet, HJW (2012). Simulating anisotropic droplet shapes on chemically striped patterned surfaces. Langmuir, 28:499--505. doi:10.1021/la2039625

    2012 doi

  22. [30]

    On the determination of molecular fields.--- I

    Jones, JE (1924). On the determination of molecular fields.--- I . From the variation of the viscosity of a gas with temperature. Proceedings of the Royal Society of London. Series A, 106:441--462. doi:10.1098/rspa.1924.0081

    1924

  23. [31]

    A smoothed particle hydrodynamics model for droplet and film flow on smooth and rough fracture surfaces

    Kordilla, J; Tartakovsky, AM; and Geyer, T (2013). A smoothed particle hydrodynamics model for droplet and film flow on smooth and rough fracture surfaces. Advances in Water Resources, 59:1--14. doi:10.1016/j.advwatres.2013.04.009

    2013 doi

  24. [32]

    Fluid mechanics , volume 6 of Course of Theoretical Physics

    Landau, LD and Lifshitz, EM (1987). Fluid mechanics , volume 6 of Course of Theoretical Physics . Pergamon Press, Oxford, England; New York, 2nd edition. ISBN 978-0-08-033933-7

    1987

  25. [33]

    Molecular Dynamics: With Deterministic and Stochastic Numerical Methods , volume 39 of Interdisciplinary Applied Mathematics

    Leimkuhler, B and Matthews, C (2015). Molecular Dynamics: With Deterministic and Stochastic Numerical Methods , volume 39 of Interdisciplinary Applied Mathematics . Springer International Publishing, Cham. ISBN 978-3-319-16374-1. doi:10.1007/978-3-319-16375-8

    2015 doi

  26. [34]

    Dissipative particle dynamics with attractive and repulsive particle-particle interactions

    Liu, M; Meakin, P; and Huang, H (2006). Dissipative particle dynamics with attractive and repulsive particle-particle interactions. Physics of Fluids, 18:017101. doi:10.1063/1.2163366

    2006 doi

  27. [35]

    A numerical approach to the testing of the fission hypothesis

    Lucy, LB (1977). A numerical approach to the testing of the fission hypothesis. The Astronomical Journal, 82:1013. doi:10.1086/112164

    1977 doi

  28. [36]

    Theoretical analysis of the no-slip boundary condition enforcement in SPH methods

    Macia, F; Antuono, M; Gonzalez, LM; and Colagrossi, A (2011). Theoretical analysis of the no-slip boundary condition enforcement in SPH methods. Progress of Theoretical Physics, 125:1091--1121. doi:10.1143/PTP.125.1091

    2011 doi

  29. [37]

    Simulating droplet motion on virtual leaf surfaces

    Mayo, LC; McCue, SW; Moroney, TJ; Forster, WA; Kempthorne, DM; Belward, JA; and Turner, IW (2015). Simulating droplet motion on virtual leaf surfaces. Royal Society Open Science, 2:140528. doi:10.1098/rsos.140528

    2015 doi

  30. [38]

    Simulating free surface flows with SPH

    Monaghan, JJ (1994). Simulating free surface flows with SPH . Journal of Computational Physics, 110:399--406. doi:10.1006/jcph.1994.1034

    1994

  31. [39]

    Smoothed particle hydrodynamics

    Monaghan, JJ (2005). Smoothed particle hydrodynamics. Reports on Progress in Physics, 68:1703--1759. doi:10.1088/0034-4885/68/8/R01

    2005 doi

  32. [40]

    Shock simulation by the particle method SPH

    Monaghan, JJ and Gingold, R (1983). Shock simulation by the particle method SPH . Journal of Computational Physics, 52:374--389. doi:10.1016/0021-9991(83)90036-0

    1983 doi

  33. [41]

    Simulating surface tension with smoothed particle hydrodynamics

    Morris, JP (2000). Simulating surface tension with smoothed particle hydrodynamics. International Journal for Numerical Methods in Fluids, 33:333--353. doi:10.1002/1097-0363(20000615)33:3<333::AID-FLD11>3.0.CO;2-7

    2000 doi

  34. [42]

    Interactive blood simulation for virtual surgery based on smoothed particle hydrodynamics

    M \"u ller, M; Schirm, S; and Teschner, M (2004). Interactive blood simulation for virtual surgery based on smoothed particle hydrodynamics. Technology & Health Care, 12:25--31. doi:10.3233/thc-2004-12103

    2004 doi

  35. [43]

    Dynamic capillary phenomena using incompressible SPH

    Nair, P and P \"o schel, T (2018). Dynamic capillary phenomena using incompressible SPH . Chemical Engineering Science, 176:192--204. doi:10.1016/j.ces.2017.10.042

    2018 doi

  36. [44]

    Droplet retention on an incline

    Ravi Annapragada, S; Murthy, JY; and Garimella, SV (2012). Droplet retention on an incline. International Journal of Heat and Mass Transfer, 55:1457--1465. doi:10.1016/j.ijheatmasstransfer.2011.09.057

    2012 doi

  37. [45]

    On the capillary phenomena of jets

    Rayleigh, L (1879). On the capillary phenomena of jets. Proceedings of the Royal Society of London, 29:71--97. doi:10.1098/rspl.1879.0015

  38. [46]

    A pair potential force model for surface tension calculations with smoothed particle hydrodynamics

    Santacruz-Yunga , E; Guerrero-Rodr \'i guez , B; Silva-Rojas , P; P \'e rez-Roa , R; Sigalotti, LDG; Trejo, C; and Plaza, E (2025). A pair potential force model for surface tension calculations with smoothed particle hydrodynamics. Computational Particle Mechanics. doi:10.1007...

    2025 doi

  39. [47]

    Smoothed particle hydrodynamics study of the roughness effect on contact angle and droplet flow

    Shigorina, E; Kordilla, J; and Tartakovsky, AM (2017). Smoothed particle hydrodynamics study of the roughness effect on contact angle and droplet flow. Physical Review E, 96:033115. doi:10.1103/physreve.96.033115

    2017 doi

  40. [48]

    Smoothed particle hydrodynamics in astrophysics

    Springel, V (2010). Smoothed particle hydrodynamics in astrophysics. Annual Review of Astronomy and Astrophysics, 48:391--430. doi:10.1146/annurev-astro-081309-130914

    2010 doi

  41. [49]

    Modeling of surface tension and contact angles with smoothed particle hydrodynamics

    Tartakovsky, A and Meakin, P (2005). Modeling of surface tension and contact angles with smoothed particle hydrodynamics. Physical Review E, 72:026301. doi:10.1103/PhysRevE.72.026301

    2005 doi

  42. [50]

    Pairwise force smoothed particle hydrodynamics model for multiphase flow: Surface tension and contact line dynamics

    Tartakovsky, A and Panchenko, A (2016). Pairwise force smoothed particle hydrodynamics model for multiphase flow: Surface tension and contact line dynamics. Journal of Computational Physics, 305:1119--1146. doi:10.1016/j.jcp.2015.08.037

    2016 doi

  43. [51]

    Evaporating droplets on inclined plant leaves and synthetic surfaces: Experiments and mathematical models

    Tredenick, EC; Forster, WA; Pethiyagoda, R; van Leeuwen , RM; and McCue, SW (2021). Evaporating droplets on inclined plant leaves and synthetic surfaces: Experiments and mathematical models. Journal of Colloid and Interface Science, 592:329--341. doi:10.1016/j.jcis.2021.01.070

    2021 doi

  44. [52]

    Tuning drop motion by chemical patterning of surfaces

    Varagnolo, S; Schiocchet, V; Ferraro, D; Pierno, M; Mistura, G; Sbragaglia, M; Gupta, A; and Amati, G (2014). Tuning drop motion by chemical patterning of surfaces. Langmuir, 30:2401--2409. doi:10.1021/la404502g

    2014 doi

  45. [53]

    C- CSF : Accurate , robust and efficient surface tension and contact angle models for single-phase flows using SPH

    Vergnaud, A; Oger, G; Le Touz \'e , D; DeLeffe, M; and Chiron, L (2022). C- CSF : Accurate , robust and efficient surface tension and contact angle models for single-phase flows using SPH . Computer Methods in Applied Mechanics and Engineering, 389:114292. doi:10.1016/j.cma.20...

    2022

  46. [54]

    An overview of smoothed particle hydrodynamics for simulating multiphase flow

    Wang, ZB; Chen, R; Wang, H; Liao, Q; Zhu, X; and Li, SZ (2016). An overview of smoothed particle hydrodynamics for simulating multiphase flow. Applied Mathematical Modelling, 40:9625--9655. doi:10.1016/j.apm.2016.06.030

    2016 doi

  47. [55]

    Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree

    Wendland, H (1995). Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Advances in Computational Mathematics, 4:389--396. doi:10.1007/BF02123482

    1995 doi

  48. [56]

    Data Fusion for a Multiscale Model of a Wheat Leaf Surface : A Unifying Approach Using a Radial Basis Function Partition of Unity Method

    Whebell, RM; Moroney, TJ; Turner, IW; Pethiyagoda, R; Wille, ML; Cooper-White , JJ; Kumar, A; Taylor, P; and McCue, SW (2025). Data Fusion for a Multiscale Model of a Wheat Leaf Surface : A Unifying Approach Using a Radial Basis Function Partition of Unity Method . SIAM Journa...

    2025 doi

  49. [57]

    Smoothed particle hydrodynamics ( SPH ) for complex fluid flows: Recent developments in methodology and applications

    Ye, T; Pan, D; Huang, C; and Liu, M (2019). Smoothed particle hydrodynamics ( SPH ) for complex fluid flows: Recent developments in methodology and applications. Physics of Fluids, 31:011301. doi:10.1063/1.5068697

    2019 doi

  50. [58]

    Smoothed particle hydrodynamics model for diffusion through porous media

    Zhu, Y and Fox, PJ (2001). Smoothed particle hydrodynamics model for diffusion through porous media. Transport in Porous Media, 43:441--471. doi:10.1023/a:1010769915901

    2001 doi

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.