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arxiv: 2604.27388 · v1 · submitted 2026-04-30 · ⚛️ physics.flu-dyn

Recognition: unknown

Asymmetric freezing of a sliding droplet on an inclined surface

Chander Shekhar Sharma, George Karapetsas, Kirti Chandra Sahu, Sivanandan Kavuri

Authors on Pith no claims yet

Pith reviewed 2026-05-07 09:48 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords asymmetric freezingsliding dropletinclined surfacefrozen morphologylubrication approximationice cuspcontact angle hysteresisStefan number
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The pith

Sliding prior to and during early freezing governs the asymmetry of frozen droplets on inclined cold surfaces.

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

This paper uses numerical simulations based on the lubrication approximation to study a liquid droplet sliding and freezing on an inclined cold surface. It establishes that the sliding motion before and in the initial stages of freezing is the primary driver of the final asymmetric frozen shape, featuring a tilted ice cusp. A sympathetic reader would care because the process explains irregular ice shapes that appear on sloped surfaces in nature and engineering, and it identifies how early motion sets features that later stages cannot easily change. The work varies substrate inclination, wettability, effective Bond number, and Stefan number to show how gravity and capillarity compete while liquid remains mobile. It concludes that early-time contact-angle hysteresis and the decomposition of flow into capillary versus gravity contributions control pinning, thinning, and the final morphology.

Core claim

The central claim is that sliding prior to and during the early stages of freezing plays a dominant role in governing the asymmetry of the frozen droplet. A tilted ice cusp forms at the droplet tip due to the competition between gravitational forces and capillary resistance, with its orientation and magnitude strongly dependent on substrate wettability and inclination. Greater inclination and increased wettability enhance asymmetry in droplet morphology. Further, highly wetting substrates favor capillary-driven retraction and induce transient liquid motion opposite to gravity during freezing. The evolution of contact-angle hysteresis at both the solid surface and the liquid-ice interface and

What carries the argument

The lubrication approximation model that couples gravity, capillarity, solidification kinetics, and contact-angle hysteresis to track sliding-induced deformation and interface evolution before and during freezing.

If this is right

  • Greater substrate inclination or wettability produces larger tilted cusps and stronger post-freezing contact-angle contrasts.
  • Higher Stefan numbers shorten the time liquid remains mobile, thereby reducing sliding-induced deformation and cusp size.
  • Highly wetting surfaces trigger capillary retraction that can drive transient uphill liquid motion while freezing proceeds.
  • Decomposition of flow shows gravity contributions control receding-edge thinning while capillary forces govern pinning at the advancing side.
  • The framework predicts that contact-line hysteresis at both solid-liquid and liquid-ice interfaces is fixed mostly by early-time dynamics.

Where Pith is reading between the lines

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

  • Surface designs that pin droplets within the first seconds of contact could reduce unwanted asymmetry in practical ice accretion.
  • Sudden changes in surface tilt after nucleation but before full freezing offer a direct experimental test of the early-motion hypothesis.
  • The same competition between gravity and capillarity during early freezing may govern asymmetric ice on other inclined or moving substrates, such as turbine blades.
  • Relaxing the lubrication assumption in targeted regions near the cusp could reveal whether three-dimensional effects alter the predicted cusp angle.

Load-bearing premise

The lubrication approximation remains valid throughout the sliding and freezing process and the chosen solidification kinetics and contact-angle hysteresis models sufficiently capture the real physics without requiring full three-dimensional flow resolution.

What would settle it

High-resolution experiments in which sliding is mechanically suppressed only during the first moments of freezing yet the final cusp angle and asymmetry still match the sliding case would falsify the claim that early sliding dominates morphology.

Figures

Figures reproduced from arXiv: 2604.27388 by Chander Shekhar Sharma, George Karapetsas, Kirti Chandra Sahu, Sivanandan Kavuri.

Figure 1
Figure 1. Figure 1: Schematic of a sessile droplet freezing on a solid substrate inclined at an angle view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of the procedure used to determine contact angles during (a) view at source ↗
Figure 3
Figure 3. Figure 3: Illustration of the procedure used to determine the liquid–ice contact angle ( view at source ↗
Figure 4
Figure 4. Figure 4: Evolution of the droplet shape, h (solid lines), and the freezing front, s (dot￾dashed lines), on a substrate with (a) α = 0◦ and (b) α = 75◦ . The normalized time, t/tf , corresponding to each row is indicated in panel (a). Here, the equilibrium contact angle is fixed at θeq = 17.5 ◦ , with all other parameters listed in view at source ↗
Figure 5
Figure 5. Figure 5: Temporal evolution of (a, c) the droplet–substrate contact angle ( view at source ↗
Figure 6
Figure 6. Figure 6: Effect of the effective Bond number (Bom = Bo sinα) on (a) the cusp angle θCusp and (b) the droplet–substrate contact angle hysteresis (θa − θr) at t/tf = 0 (solid lines) and t/tf = 1 (dot-dashed lines) for different equilibrium contact angles. The remaining dimensionless parameters are listed in view at source ↗
Figure 7
Figure 7. Figure 7: Droplet shapes at different normalized times for various effective Bond numbers view at source ↗
Figure 8
Figure 8. Figure 8: Temporal evolution of (a) the advancing ( view at source ↗
Figure 9
Figure 9. Figure 9: Temporal evolution of (a) the advancing ( view at source ↗
Figure 10
Figure 10. Figure 10: Variation of (a) the cusp angle (θCusp) at the droplet tip, measured relative to the z-axis, as a function of substrate inclination (α) for different equilibrium contact angles. Here, the experimental results from Kumar et al. (2025) are included for comparison. (b) Variation of the difference between advancing and receding contact angles (θa − θr) with inclination angle (α) for different equilibrium cont… view at source ↗
Figure 11
Figure 11. Figure 11: Temporal evolution of droplet shape (h, solid) and freezing front (s, dot–dashed) on an inclined substrate (α = 75◦ ) for (a) θeq = 17.5 ◦ and (b) θeq = 26.5 ◦ . The remaining dimensionless parameters are listed in view at source ↗
Figure 12
Figure 12. Figure 12: Evolution of the droplet shape, h (solid lines), and the freezing front position, s (dot-dashed lines) for (a) θeq = 15.3 ◦ and (b) θeq = 21.2 ◦ . The normalized time, t/tf , corresponding to each row is indicated in panel (a). Here, α = 60◦ , and the remaining dimensionless parameters and the corresponding total freezing times (tf ) are provided in Tables 2 and 5, respectively. θeq = 15.3 ◦ and θeq = 21.… view at source ↗
Figure 13
Figure 13. Figure 13: Temporal evolution of (a) the advancing ( view at source ↗
Figure 14
Figure 14. Figure 14: Temporal evolution of (a) the bulk velocity (¯u view at source ↗
Figure 15
Figure 15. Figure 15: Effect of the Stefan number (Ste) on (a) the cusp angle (θCusp) and (b) the difference between the advancing and receding contact angles (θa − θr) at t/tf = 1, for a droplet on substrates with different inclination angles (α). Here, the equilibrium contact angle is θeq = 26.5 ◦ . The remaining dimensionless parameters and the corresponding total freezing times (tf ) are listed in tables 2 and 6, respectively view at source ↗
Figure 16
Figure 16. Figure 16: Comparison of the equilibrium shape of a droplet sliding on a substrate inclined view at source ↗
Figure 17
Figure 17. Figure 17: Evolution of the freezing front, s (dot-dashed lines), and the shape, h (solid line) of the droplet undergoing freezing on a horizontal (α = 0◦ ) cold substrate. The dimensionless parameters considered in the simulations are ϵ = 0.2, Ste = 0.04, Tv = 0.5, An = 8, Ds = ΛS = Bi = Bo = 1, and Dw = 0. This set of parameters corresponds to those used by Zadravzil et al. (2006). Here, the dimensionless total fr… view at source ↗
Figure 18
Figure 18. Figure 18: Evolution of the freezing front, s (dot-dashed lines), and the droplet shape, h (solid lines), on a cold substrate inclined at α = 75◦ . The remaining dimensionless parameters are Bo = 0.4, Ste = 1.49 × 10−3 , An = 10, Ds = 0.9, ΛS = 3.89, ΛW = 675, Bi = 0.16, Dw = 2.94, and ϵ = 0.2. The total dimensionless freezing time is tf = 201. The results obtained using 12001, 25601, and 30001 grid points are indis… view at source ↗
Figure 19
Figure 19. Figure 19: Evolution of the droplet shape, h (solid lines), and the freezing front, s (dot–dashed lines), on a substrate inclined at α = 75◦ for (a) θeq = 12◦ and (b) θeq = 17.5 ◦ . The normalized time, t/tf , corresponding to each row is indicated in panel (a). All other parameters are provided in view at source ↗
read the original abstract

We investigate the asymmetric freezing of a liquid droplet sliding on an inclined cold surface using numerical simulations based on the lubrication approximation. The combined effects of gravity, capillarity, and solidification kinetics on droplet motion, interfacial deformation, and the resulting frozen morphology are examined through systematic variations in substrate inclination, wettability, effective Bond number, and Stefan number. Our results show that sliding prior to and during the early stages of freezing plays a dominant role in governing the asymmetry of the frozen droplet. A tilted ice cusp forms at the droplet tip due to the competition between gravitational forces and capillary resistance, with its orientation and magnitude strongly dependent on substrate wettability and inclination. Greater inclination and increased wettability enhance asymmetry in droplet morphology. Further, highly wetting substrates favor capillary-driven retraction and induce transient liquid motion opposite to gravity during freezing. The evolution of contact-angle hysteresis at both the solid surface and the liquid-ice interface underscores the importance of early-time dynamics, when the unfrozen liquid remains mobile and gravitational effects are most pronounced. Decomposition of the liquid motion into capillary and gravity-driven contributions provides physical insight into contact-line pinning, receding-edge thinning, and the development of asymmetric liquid-ice contact angles. Increasing the Stefan number accelerates freezing, limits sliding-induced deformation, and reduces both the cusp angle and the post-freezing contact-angle contrast. Overall, this study establishes a physical framework for understanding the morphology of frozen droplets on inclined substrates.

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 / 3 minor

Summary. The paper uses lubrication-approximation simulations to examine the asymmetric freezing of a droplet sliding on an inclined cold surface. Systematic parameter sweeps in inclination angle, wettability, effective Bond number, and Stefan number are performed, with the central claim that sliding prior to and during early freezing dominates the resulting frozen asymmetry. Key reported features include formation of a tilted ice cusp at the advancing tip, wettability-dependent capillary retraction opposite gravity, contact-angle hysteresis evolution at both substrate and liquid-ice interfaces, and force decomposition separating capillary and gravitational contributions to contact-line motion.

Significance. If the lubrication reduction remains accurate, the work supplies a physically interpretable framework for how gravity-capillary competition and early-time mobility shape frozen morphologies on inclined substrates. The explicit decomposition of liquid velocity into capillary and gravity-driven parts, together with the parametric dependence on Stefan number, offers testable predictions for cusp angle and post-freezing contact-angle contrast that could inform anti-icing design. The absence of any grid-convergence data, 3D benchmark, or experimental comparison, however, keeps the quantitative significance provisional.

major comments (3)
  1. [§2] §2 (Governing equations and lubrication reduction): The lubrication approximation is invoked for the entire sliding-plus-freezing process, yet no a-priori estimate or post-hoc check is given for the slope or Reynolds-number limits at the receding contact line and during cusp formation, where the reported thinning and transient retraction occur. This assumption is load-bearing for the claim that sliding dominates asymmetry.
  2. [Results] Results (parametric sweeps and asymmetry metrics): No grid-convergence study, mesh-refinement test, or comparison against full Navier-Stokes or phase-field simulations is reported for the cusp angle, contact-angle contrast, or asymmetry index. Without these, it is impossible to separate the physical effect of early sliding from possible numerical artifacts in the lubrication model.
  3. [§4] §4 (Force decomposition): The decomposition of liquid motion into capillary and gravity-driven components is used to explain pinning and retraction, but the manuscript does not quantify the relative magnitudes or show that the neglected inertial terms remain small throughout the early-time window emphasized in the abstract.
minor comments (3)
  1. [Methods] The definition of the effective Bond number and its relation to the classical Bond number should be stated explicitly in the methods section rather than only in the abstract.
  2. [Figures] Figure captions for the morphology snapshots should include the specific values of inclination, wettability, Bo, and Ste used in each panel to allow direct comparison with the text.
  3. [§2] A brief statement on the numerical treatment of the moving liquid-ice interface (e.g., whether a sharp or diffuse interface is employed) would clarify the solidification kinetics implementation.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive review. The comments highlight important aspects of numerical validation and model assumptions that we will address in the revised manuscript. We respond point by point to the major comments below.

read point-by-point responses

  1. Referee: §2 (Governing equations and lubrication reduction): The lubrication approximation is invoked for the entire sliding-plus-freezing process, yet no a-priori estimate or post-hoc check is given for the slope or Reynolds-number limits at the receding contact line and during cusp formation, where the reported thinning and transient retraction occur. This assumption is load-bearing for the claim that sliding dominates asymmetry.

    Authors: We agree that explicit checks are needed to support the lubrication approximation, especially near the receding contact line. In the revised manuscript we will add a-priori estimates of the maximum interface slope and local Reynolds numbers evaluated at the receding contact line and during early cusp formation. Post-hoc verification of these quantities throughout the early-time window will also be included to confirm that the assumptions remain valid and that sliding indeed dominates the observed asymmetry. revision: yes

  2. Referee: Results (parametric sweeps and asymmetry metrics): No grid-convergence study, mesh-refinement test, or comparison against full Navier-Stokes or phase-field simulations is reported for the cusp angle, contact-angle contrast, or asymmetry index. Without these, it is impossible to separate the physical effect of early sliding from possible numerical artifacts in the lubrication model.

    Authors: We acknowledge that a grid-convergence study is essential. We will include mesh-refinement tests in the revised manuscript and demonstrate convergence of the cusp angle, contact-angle contrast, and asymmetry index with respect to spatial resolution. These tests will help confirm that the reported features arise from the physics rather than numerical artifacts. Direct comparisons with full Navier-Stokes or phase-field simulations lie outside the scope of the present lubrication study but can be considered in future work. revision: yes

  3. Referee: §4 (Force decomposition): The decomposition of liquid motion into capillary and gravity-driven components is used to explain pinning and retraction, but the manuscript does not quantify the relative magnitudes or show that the neglected inertial terms remain small throughout the early-time window emphasized in the abstract.

    Authors: We will revise §4 to quantify the relative magnitudes of the capillary and gravitational contributions to the liquid velocity during the early-time regime. We will also provide estimates of the inertial terms (via local Reynolds numbers) to demonstrate that they remain small compared with the retained terms, thereby justifying the quasi-steady force decomposition and supporting the physical interpretation of contact-line pinning and retraction. revision: yes

Circularity Check

0 steps flagged

No circularity: results emerge from independent numerical solution of lubrication model

full rationale

The paper reports outcomes of direct numerical simulations of the lubrication equations with explicit parametric sweeps over inclination angle, wettability, Bond number, and Stefan number. The central claim—that pre- and early-freezing sliding governs frozen asymmetry—is obtained by solving the time-dependent evolution equations rather than by any redefinition of inputs, by fitting a parameter and relabeling its output as a prediction, or by a self-citation chain that closes on itself. No equation is shown to be equivalent to its own inputs by construction, and the model assumptions (lubrication reduction, contact-angle hysteresis, solidification kinetics) are stated independently of the reported morphologies. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on the lubrication approximation for thin-film flow together with standard dimensionless groups (Bond and Stefan numbers) and phenomenological models for contact-angle hysteresis and solidification kinetics.

free parameters (2)
  • effective Bond number
    Dimensionless group varied systematically to control the relative strength of gravity and capillarity; not fitted to a specific data set but chosen as an input parameter.
  • Stefan number
    Dimensionless measure of latent-heat release versus sensible heat; varied to control freezing speed.
axioms (2)
  • domain assumption Lubrication approximation is valid for the droplet thickness and velocity scales considered
    Invoked as the foundation for the entire numerical model of droplet motion, deformation, and freezing.
  • domain assumption Contact-angle hysteresis can be modeled with fixed advancing and receding angles at both solid-liquid and liquid-ice interfaces
    Used to capture pinning and receding-edge thinning during sliding and freezing.

pith-pipeline@v0.9.0 · 5563 in / 1364 out tokens · 48750 ms · 2026-05-07T09:48:20.372166+00:00 · methodology

discussion (0)

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