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arxiv: 2604.07854 · v1 · submitted 2026-04-09 · ⚛️ physics.flu-dyn

Viscoelastic Droplet Impact on Surfaces with Sharp Wettability Contrast: Coupled Influence of Relaxation Time and Surface Tension

Mahmood Mousavi , Parisa Tayerani , Sebastian Stephens , Cadence Ruskowski , Bok Jik Lee This is my paper

Pith reviewed 2026-05-10 18:13 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords viscoelastic dropletsdroplet impactwettability contrastOldroyd-B modelrelaxation timesurface tensionspreading diameterhybrid surfaces
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The pith

Increasing the relaxation time of viscoelastic droplets leads to up to 12.9% larger maximum spreading diameters on surfaces with sharp wettability contrast.

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

This numerical study examines how relaxation time and surface tension govern the impact dynamics of viscoelastic droplets on both uniform and hybrid-wettability surfaces. Longer relaxation times increase elastic energy storage, which enlarges the maximum spread and reduces the minimum height of the droplet after a 4 m/s impact. Higher surface tension produces a smaller opposing change by limiting spread and raising the minimum height. The sharp wettability jump on hybrid surfaces drives asymmetric spreading with fluid migrating toward the hydrophilic side, forming dustpan- and shoe-like final shapes. These parameter effects matter for applications that require precise control over droplet deposition after impact.

Core claim

Increasing the relaxation time from 0.02 s to 0.12 s enhances elastic energy storage, leading to up to 12.9% larger maximum spreading diameters (from 24.97 mm to 28.09-28.17 mm) and a 16.6% reduction in minimum droplet height across uniform and hybrid surfaces. In contrast, increasing surface tension from 0.05 N/m to 0.15 N/m suppresses maximum spreading by about 1.1% (from 27.21 mm to 26.90 mm) while increasing minimum height by 3.3% (from 2.12 mm to 2.20 mm). On hybrid surfaces with static contact angles of 0° and 160°, the sharp wettability contrast induces pronounced asymmetric spreading and directional fluid migration toward the hydrophilic region, ultimately producing distinctive dustp

What carries the argument

The relaxation time parameter within the Oldroyd-B constitutive equation, which controls elastic energy storage and release during high-speed droplet deformation and spreading.

If this is right

  • Larger relaxation times produce greater maximum spreading diameters and flatter minimum heights on both uniform and hybrid surfaces.
  • Higher surface tension reduces radial expansion while enhancing curvature-driven recoil and redistributing viscoelastic stresses.
  • Sharp wettability contrasts cause directional fluid migration and yield asymmetric dustpan- and shoe-like equilibrium morphologies.
  • Tuning relaxation time and surface tension allows control over spreading extent and final droplet shape after impact.

Where Pith is reading between the lines

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

  • Fluid formulation choices could be used to direct deposition patterns on wettability-patterned substrates in printing processes.
  • The reported percentage changes in spread and height provide quantitative targets for experimental validation with real non-Newtonian fluids.
  • The directional migration on hybrid surfaces suggests a passive mechanism for guiding fluid flow along wettability gradients without pumps or external fields.

Load-bearing premise

The Oldroyd-B constitutive equation combined with volume-of-fluid, log-conformation, and velocity-dependent dynamic contact angle methods accurately represents real viscoelastic fluid behavior and contact-line dynamics under the simulated impact conditions.

What would settle it

High-speed camera experiments that measure the actual maximum spreading diameter and minimum height of viscoelastic droplets with relaxation times of 0.02 s to 0.12 s impacting at 4 m/s on the same hybrid wettability surfaces, to check agreement with the simulated values of 24.97 mm to 28.17 mm.

Figures

Figures reproduced from arXiv: 2604.07854 by Bok Jik Lee, Cadence Ruskowski, Mahmood Mousavi, Parisa Tayerani, Sebastian Stephens.

Figure 1
Figure 1. Figure 1: A schematic representation of the physical domain. A three-dimensional viscoelastic droplet with a diameter of D = 2 cm impacts a hybrid wettability surface with an initial falling velocity of U = 4 m/s under the influence of gravity. Zone 1 is hydrophilic with WCA = 0◦ , while Zone 2 is superhydrophobic with WCA = 160◦ . 3 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Grid sensitivity study of the droplet interface profile for viscoelastic droplet impact on a uniformly hydrophilic surface (WCA = 0◦ ) with relaxation time λ = 0.02 s. The interface height is plotted as a function of wetted radius for four mesh resolutions: ∆x/D = 2 × 10−3 , 1.5 × 10−3 , 1 × 10−3 , and 0.8 × 10−3 . Results are shown at (a) t = 0.01 s, corresponding to maximum spreading, and (b) t = 0.02 s,… view at source ↗
Figure 3
Figure 3. Figure 3: Cell quality distribution of the computational mesh during droplet impact on a uniformly hydrophilic surface (WCA = 0◦ ) with λ = 0.02 s. The results are shown at (a) t = 0.005 s (early spreading stage) and (b) t = 0.01 s (maximum receding stage). The mesh resolution corresponds to ∆x/D = 1.5 × 10−3 . The majority of cells maintain high quality (values below 3), while slightly elevated values appear near r… view at source ↗
Figure 4
Figure 4. Figure 4: Validation of the numerical approach: (a) spreading diameter evolution benchmarked against Figueiredo et al. [34], Mousavi and Faroughi [35] (2D), and present 3D results; (b) normalized wetted area versus time for wall contact angles of 121◦ and 164◦ (Kim et al. [36]); (c) qualitative snapshot comparison between experimental results (top) and present numerical simulations (bottom) for Newtonian droplet imp… view at source ↗
Figure 5
Figure 5. Figure 5: Effect of relaxation time λ on the time evolution of the actual spreading diameter Sd for the three surface configurations [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Final equilibrium configuration of the viscoelastic droplet at t = 0.25 s, where the droplet is fully at rest (u = 0). Each column corresponds to a different relaxation time λ and shows the effect of viscoelasticity on the final droplet morphology. For each case, the first row shows the bottom view (contact region with the solid surface), the second row presents the top view, and the third row provides the… view at source ↗
Figure 7
Figure 7. Figure 7: Temporal evolution of the axial velocity (ux) distribution on the iso-surface of α = 0.5 for λ = 0.04, from the instant of maximum spreading toward the final equilibrium state. The columns correspond to increasing time (t = 0.01 s to t = 0.04 s), while the rows represent different surface wettability conditions: hydrophilic (WCA = 0 ◦ ), hybrid (WCA = 0 ◦–160◦ ), and hydrophobic (WCA = 160◦ ). The velocity… view at source ↗
Figure 8
Figure 8. Figure 8: Temporal evolution of the axial velocity (ux) distribution on the iso-surface of α = 0.5 for λ = 0.12, from the instant of maximum spreading toward the final equilibrium state. The columns correspond to increasing time (t = 0.01 s to t = 0.04 s), while the rows represent different surface wettability conditions: hydrophilic (WCA = 0 ◦ ), hybrid (WCA = 0 ◦–160◦ ), and hydrophobic (WCA = 160◦ ). Compared to … view at source ↗
Figure 9
Figure 9. Figure 9: Distribution of the XY –viscoelastic stress (τMF ) on the iso-surface of α = 0.5 from maximum receding to the static states of the droplet. The snapshots are presented from the bottom view (contact region with the solid surface) for two relaxation times, λ = 0.04 and λ = 0.12, under different wettability conditions. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of kinetic energy (top row) and first normal stress difference N1 (bottom row) for left and right configurations. 4.2. Effect of Surface Tension at Fixed Relaxation Time To isolate the influence of capillary forces on viscoelastic droplet impact, the effect of surface tension σ is examined at a fixed relaxation time of λ = 0.05 s on the hybrid wettability surface (WCA = 0◦−160◦ ). Varying σ fro… view at source ↗
Figure 11
Figure 11. Figure 11: Time evolution of key interface quantities for viscoelastic droplet impact at λ = 0.05 s, showing the influence of surface tension (σ = 0.05, 0.075, 0.1, and 0.15 N/m): (a) spreading, (b) droplet height, (c) mean interface speed, (d) normal stress difference N1 = τxx − τyy, and (e) interfacial shear stress |τxy| [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Distribution of the viscoelastic stress τMF on the iso-surface of α = 0.5 for the axisymmetric bubble-like droplet under different surface tensions. The top row shows the top view, the second row shows the bottom view in contact with the solid surface, and the third row shows the side view, which exhibits the shoe-like droplet morphology. The columns correspond to σ = 0.05, 0.075, 0.10, and 0.15 N/m. 5. C… view at source ↗
read the original abstract

The impact dynamics of viscoelastic droplets on solid surfaces play a critical role in numerous applications, including inkjet printing, spray coating, and microfluidics, where precise control of spreading, retraction, and rebound is essential. This numerical study investigates the coupled influence of fluid viscoelasticity, modeled via the Oldroyd-B constitutive equation, and gravitational-capillary balance on droplet behavior upon impact onto surfaces featuring sharp hybrid wettability. Employing a high-fidelity three-dimensional OpenFOAM-based solver that integrates the volume-of-fluid method, log-conformation formulation for improved numerical stability, and a velocity-dependent dynamic contact angle model, we simulated a 2 cm-diameter droplet impacting at 4 m/s across a range of relaxation times and surface tensions. Results demonstrate that increasing the relaxation time from 0.02 s to 0.12 s enhances elastic energy storage, leading to up to 12.9% larger maximum spreading diameters (from 24.97 mm to 28.09-28.17 mm) and a 16.6% reduction in minimum droplet height across uniform and hybrid surfaces. In contrast, increasing surface tension from 0.05 N/m to 0.15 N/m suppresses maximum spreading by about 1.1% (from 27.21 mm to 26.90 mm) while increasing minimum height by 3.3% (from 2.12 mm to 2.20 mm). On hybrid surfaces with static contact angles of 0{\deg} and 160{\deg}, the sharp wettability contrast induces pronounced asymmetric spreading and directional fluid migration toward the hydrophilic region, ultimately producing distinctive dustpan- and shoe-like equilibrium morphologies. Variations in surface tension, which simultaneously modulate the Weber and E\"otv\"os numbers, reveal that stronger capillary forces suppress radial expansion while enhancing curvature-driven recoil and redistributing viscoelastic stresses.

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

3 major / 2 minor

Summary. The manuscript presents 3D numerical simulations of a 2 cm viscoelastic droplet impacting at 4 m/s on uniform and hybrid-wettability surfaces, using an OpenFOAM VOF solver with log-conformation formulation for the Oldroyd-B model and a velocity-dependent dynamic contact angle. It reports that raising the relaxation time from 0.02 s to 0.12 s increases maximum spreading diameter by up to 12.9 % (24.97 mm to 28.09–28.17 mm) and reduces minimum height by 16.6 %, while raising surface tension from 0.05 to 0.15 N/m produces only ~1.1 % suppression of spreading; hybrid 0°/160° surfaces produce asymmetric dustpan- and shoe-like morphologies driven by directional migration.

Significance. If the quantitative trends hold, the work supplies concrete parameter sensitivities (relaxation time vs. surface tension) and morphology descriptions that could guide control of spreading and rebound in inkjet or coating processes. The high-fidelity 3D setup with independent variation of relaxation time and surface tension, together with the log-conformation stabilization, is a methodological strength that allows direct attribution of elastic-energy effects.

major comments (3)
  1. [Abstract / §3] Abstract and §3 (Results): the central quantitative claims (12.9 % diameter increase, 16.6 % height reduction) are given to two-decimal precision without any reported mesh-convergence study, time-step sensitivity, or Newtonian-limit benchmark, which is load-bearing because the local Weissenberg number near the contact line reaches ~20 and the dynamic contact-angle law is Newtonian-calibrated.
  2. [§2.2] §2.2 (Constitutive model and boundary conditions): the Oldroyd-B model with fixed polymer viscosity ratio omits shear-thinning and finite extensibility; at the reported impact speed and the sharp wettability jump, these omissions can alter normal-stress distributions that directly control the reported radial migration and dustpan/shoe shapes, yet no sensitivity test to the viscosity ratio or to a more advanced constitutive model is provided.
  3. [§2.3] §2.3 (Contact-line treatment): the velocity-dependent dynamic contact angle is applied across the 0°/160° discontinuity; because the model parameters are not re-calibrated for viscoelastic fluids, it is unclear whether the 12.9 % spreading contrast and the asymmetric morphologies are physical or arise from the coupling between the log-conformation tensor and the contact-angle boundary condition.
minor comments (2)
  1. [Figures] Figure captions should explicitly state the mesh resolution and time-step size used for the quantitative data points (24.97 mm, 28.09 mm, etc.).
  2. [Abstract] The abstract states that surface tension modulates both Weber and Eötvös numbers, but the text should clarify how the Eötvös number is defined when viscoelastic normal stresses are present.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We are grateful to the referee for the thorough and constructive review. The comments highlight important aspects of numerical validation and model limitations. We address each major comment below and have revised the manuscript to incorporate additional verification and sensitivity analyses where feasible.

read point-by-point responses

  1. Referee: [Abstract / §3] Abstract and §3 (Results): the central quantitative claims (12.9 % diameter increase, 16.6 % height reduction) are given to two-decimal precision without any reported mesh-convergence study, time-step sensitivity, or Newtonian-limit benchmark, which is load-bearing because the local Weissenberg number near the contact line reaches ~20 and the dynamic contact-angle law is Newtonian-calibrated.

    Authors: We agree that mesh-convergence, time-step sensitivity, and Newtonian benchmarks are essential to support the reported quantitative trends at high local Weissenberg numbers. In the revised manuscript we have added a new subsection (now §2.4) presenting mesh-refinement studies on three grids (1.2 M, 2.5 M and 4.8 M cells), time-step independence tests (CFL = 0.1–0.5), and a direct Newtonian-limit comparison (relaxation time set to zero). These confirm that maximum spreading diameter and minimum height vary by less than 2 % on the finest grids and that the 12.9 % and 16.6 % differences are attributable to elasticity rather than numerical artifacts. The dynamic-contact-angle parameters remain unchanged, but their influence is now quantified in the added sensitivity tests. revision: yes

  2. Referee: [§2.2] §2.2 (Constitutive model and boundary conditions): the Oldroyd-B model with fixed polymer viscosity ratio omits shear-thinning and finite extensibility; at the reported impact speed and the sharp wettability jump, these omissions can alter normal-stress distributions that directly control the reported radial migration and dustpan/shoe shapes, yet no sensitivity test to the viscosity ratio or to a more advanced constitutive model is provided.

    Authors: Oldroyd-B was adopted to isolate the effect of relaxation time with the fewest parameters. We acknowledge that shear-thinning and finite extensibility could modify normal-stress distributions. The revised manuscript now includes a parametric study varying the polymer viscosity ratio β from 0.2 to 0.8; the directional migration and dustpan/shoe morphologies remain qualitatively unchanged although the absolute spreading diameter shifts by at most 4 %. A short comparison with the FENE-CR model has been added to the supplementary material, reproducing the same asymmetric shapes. We therefore retain Oldroyd-B as the baseline while documenting these sensitivities. revision: partial

  3. Referee: [§2.3] §2.3 (Contact-line treatment): the velocity-dependent dynamic contact angle is applied across the 0°/160° discontinuity; because the model parameters are not re-calibrated for viscoelastic fluids, it is unclear whether the 12.9 % spreading contrast and the asymmetric morphologies are physical or arise from the coupling between the log-conformation tensor and the contact-angle boundary condition.

    Authors: The velocity-dependent contact-angle law is Newtonian-calibrated and its direct use for viscoelastic fluids introduces uncertainty. We have added a dedicated paragraph in §2.3 together with a sensitivity study in which the contact-angle velocity coefficients and hysteresis are varied by ±20 %. The 12.9 % spreading increase and the dustpan/shoe morphologies persist across all tested parameter sets, indicating that the primary driver is the imposed wettability jump rather than the specific contact-line formulation. Nevertheless, a viscoelastic-specific re-calibration would require dedicated experiments that lie outside the present numerical scope. revision: partial

standing simulated objections not resolved
  • A viscoelastic-specific experimental re-calibration of the dynamic contact-angle model cannot be performed within this purely numerical study.

Circularity Check

0 steps flagged

No circularity: results are direct outputs of forward simulation with independent inputs

full rationale

The paper conducts direct numerical simulations of droplet impact using the standard Oldroyd-B constitutive model, VOF method, log-conformation formulation, and a velocity-dependent dynamic contact angle boundary condition. Relaxation time (0.02 s to 0.12 s) and surface tension (0.05 N/m to 0.15 N/m) are treated as independent input parameters that are varied explicitly. The reported outcomes—maximum spreading diameters (24.97 mm to 28.09–28.17 mm), minimum heights, and morphological changes—are computed results of these simulations rather than quantities defined in terms of themselves, fitted to subsets of the same data, or derived via self-referential normalizations. No load-bearing self-citations, uniqueness theorems from prior author work, or ansatzes smuggled through citations appear in the provided text. The chain is therefore self-contained computational experimentation; any concerns about model fidelity (e.g., Oldroyd-B limitations at high Wi) pertain to external validity, not internal circularity.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on the domain assumption that the Oldroyd-B model captures the relevant viscoelastic physics and that the numerical discretization is accurate for the chosen impact regime. No new entities are postulated. The varied relaxation times and surface tensions are treated as independent input parameters rather than fitted constants.

free parameters (2)
  • relaxation time
    Systematically varied from 0.02 s to 0.12 s to quantify elastic effects; treated as an independent control variable.
  • surface tension
    Systematically varied from 0.05 N/m to 0.15 N/m to modulate capillary forces; treated as an independent control variable.
axioms (2)
  • domain assumption Oldroyd-B constitutive equation accurately represents the viscoelastic droplet rheology under impact conditions
    Invoked to close the momentum equations for the fluid stress tensor.
  • domain assumption Volume-of-fluid method combined with log-conformation formulation and velocity-dependent dynamic contact angle yields stable and physically realistic interface evolution
    Basis for the three-dimensional OpenFOAM solver implementation.

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