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arxiv: 2502.14312 · v1 · submitted 2025-02-20 · 🧮 math.AP · math-ph· math.MP

Modelling Capillary Rise with a Slip Boundary Condition: Well-posedness and Long-time Dynamics of Solutions to Washburn's Equation

Isidora Rapaji\'c (1) , Srboljub Simi\'c (2) , Endre S\"uli (3) ((1) Mathematical Institute of the Serbian Academy of Sciences , Arts , Belgrade , Serbia , (2) Department of Mathematics , Informatics
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Faculty of Sciences Trg Dositeja Obradovi\'ca 4 21000 Novi Sad (3) Mathematical Institute University of Oxford Andrew Wiles Building Woodstock Road Oxford OX2 6GG United Kingdom)
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Pith reviewed 2026-05-23 02:54 UTC · model grok-4.3

classification 🧮 math.AP math-phmath.MP
keywords capillary riseWashburn equationslip boundary conditionwell-posednessglobal existenceuniquenesslong-time dynamicsbasin of attraction
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The pith

Washburn's capillary rise equation with a slip boundary condition admits a unique global bounded positive solution that depends continuously on initial data.

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

The paper extends the classic Washburn model by incorporating a slip condition at the pipe wall, derived from continuum mechanics, and introduces a nondimensional scaling that covers different flow regimes. It proves that the resulting initial-value problem is well-posed in the Hadamard sense: for any positive slip parameter and any initial height ratio h0/he in [0, 3/2], there exists a unique solution that remains positive and bounded for all time and depends continuously on the initial datum in the maximum norm. The unique equilibrium is shown to be attractive, with solutions approaching it either monotonically or through damped oscillations. A reader would care because the result removes the possibility of mathematical pathologies when using the slip model to predict liquid rise in narrow tubes.

Core claim

By incorporating a Navier-type slip condition into the derivation of Washburn's equation, the authors obtain a first-order nonlinear ODE whose right-hand side satisfies the hypotheses of standard global-existence theorems. They prove that for every positive nondimensional slip length and every initial height ratio h0/he in [0, 3/2] there exists a unique bounded positive solution that depends continuously on the initial datum in the supremum norm. The unique equilibrium is globally attractive within this range, and solutions may approach it monotonically or with damped oscillations.

What carries the argument

the modified Washburn ordinary differential equation obtained after nondimensionalization with the slip length included in the scaling

If this is right

  • Solutions exist for all positive times and remain positive and bounded.
  • The map from initial height to solution is continuous in the maximum norm.
  • Every solution with h0/he in [0, 3/2] converges to the equilibrium height.
  • Convergence occurs either monotonically or in an oscillatory manner.
  • All statements hold for every positive value of the nondimensional slip parameter.

Where Pith is reading between the lines

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

  • Numerical schemes applied to the slip-extended Washburn equation can be expected to converge reliably without artificial blow-up.
  • The oscillatory approach regime suggests the model could be extended to study forced oscillations or resonance under slip conditions.
  • The upper limit of 3/2 on the initial ratio may be an artifact of the current estimates and could be relaxed with refined analysis.

Load-bearing premise

The reduced ODE obtained from the continuum-mechanics derivation with slip is assumed to satisfy the regularity and growth conditions that allow direct application of standard global-existence theorems for ordinary differential equations.

What would settle it

An explicit construction of two distinct solutions, or a single solution that becomes unbounded or negative in finite time, for the same initial height with h0/he in [0, 3/2] and positive slip parameter would falsify the well-posedness result.

Figures

Figures reproduced from arXiv: 2502.14312 by 21000 Novi Sad, (2) Department of Mathematics, (3) Mathematical Institute, Andrew Wiles Building, Arts, Belgrade, Endre S\"uli (3) ((1) Mathematical Institute of the Serbian Academy of Sciences, Faculty of Sciences, Informatics, Isidora Rapaji\'c (1), Oxford OX2 6GG, Serbia, Srboljub Simi\'c (2), Trg Dositeja Obradovi\'ca 4, United Kingdom), University of Oxford, Woodstock Road.

Figure 1
Figure 1. Figure 1: Capillary pipe with Poiseuille velocity profile and column height [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Flow regimes for different values of the slip parameter. [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
read the original abstract

The aim of this paper is to extend Washburn's capillary rise equation by incorporating a slip condition at the pipe wall. The governing equation is derived using fundamental principles from continuum mechanics. A new scaling is introduced, allowing for a systematic analysis of different flow regimes. We prove the global-in-time existence and uniqueness of a bounded positive solution to Washburn's equation that includes the slip parameter, as well as the continuous dependence of the solution in the maximum norm on the initial data. Thus, the initial-value problem for Washburn's equation is shown to be well-posed in the sense of Hadamard. Additionally, we show that the unique equilibrium solution may be reached either monotonically or in an oscillatory fashion, similarly to the no-slip case. Finally, we determine the basin of attraction for the system, ensuring that the equilibrium state will be reached from the initial data we impose. These results hold for any positive value of the nondimensional slip parameter in the model, and for all values of the ratio $h_0/h_e$ in the range $[0,3/2]$, where $h_0$ is the initial height of the fluid column and $h_e$ is its equilibrium height.

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

0 major / 2 minor

Summary. The manuscript derives a modified version of Washburn's capillary rise equation that incorporates a slip boundary condition at the pipe wall, obtained via continuum mechanics. A new nondimensionalization is introduced to facilitate analysis across flow regimes. The authors prove global-in-time existence and uniqueness of a bounded positive solution to the resulting first-order ODE, together with continuous dependence on initial data in the maximum norm, thereby establishing Hadamard well-posedness. They further show that the unique equilibrium is approached either monotonically or oscillatorily and determine the basin of attraction for all positive values of the nondimensional slip parameter and for h_0/h_e in the interval [0, 3/2].

Significance. If the proofs hold, the work supplies a rigorous justification for the well-posedness and long-time dynamics of the slip-augmented Washburn model. This is valuable for applications in which slip lengths are physically relevant. The manuscript applies standard global-existence and continuous-dependence theorems for ODEs after nondimensionalization, and the results cover a concrete, physically motivated parameter range without introducing fitted parameters or self-referential definitions.

minor comments (2)
  1. The abstract states that the right-hand side satisfies the regularity and growth conditions needed for the standard global-existence theorems, but the introduction would benefit from an explicit verification (or reference to the precise theorem invoked) that the Lipschitz and linear-growth hypotheses hold uniformly for the slip parameter in (0, ∞) and h_0/h_e ∈ [0, 3/2].
  2. Notation for the nondimensional slip length and the equilibrium height h_e should be introduced immediately after the new scaling is defined, rather than appearing first in the statement of the main theorems.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive evaluation of the manuscript, including the recommendation for minor revision. The report accurately summarizes the derivation, nondimensionalization, well-posedness results, and long-time dynamics analysis.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper derives the slip-modified Washburn ODE from continuum mechanics, performs a nondimensionalization, and verifies that the resulting right-hand side meets the hypotheses of standard ODE global-existence theorems (Lipschitz continuity, linear growth). These steps are independent of the target well-posedness statements; no parameter is fitted to data, no quantity is defined in terms of itself, and no load-bearing claim reduces to a self-citation. The analysis therefore consists of direct verification on an explicitly constructed model rather than a closed loop.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption that the slip-modified ODE is correctly derived from continuum mechanics and that its right-hand side meets the hypotheses of standard ODE theorems; the slip length itself is treated as a free modeling parameter rather than derived from first principles.

free parameters (1)
  • nondimensional slip parameter
    Positive real number introduced to quantify wall slip; its value is not fitted to data but treated as an arbitrary positive constant in all statements.
axioms (2)
  • domain assumption The slip-augmented Washburn equation is obtained from fundamental continuum-mechanics principles with a Navier-type slip condition at the wall
    Invoked in the derivation paragraph of the abstract.
  • standard math Standard local-existence, continuation, and continuous-dependence theorems for scalar ODEs apply once an a priori bound is established
    Used to obtain global existence and Hadamard well-posedness.

pith-pipeline@v0.9.0 · 5853 in / 1576 out tokens · 28423 ms · 2026-05-23T02:54:50.797059+00:00 · methodology

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Reference graph

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14 extracted references · 14 canonical work pages

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    d dt Z h(t) 0 Z R 0 Z 2π 0 vz(r, t)dφrdrdz # ez. 21 Taking into account (7) and Z R 0 Z 2π 0 vz(r, t)dφrdr = R2πv(t), it follows that d dt Z Pt K(x, t)dV = ρR2π

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