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arxiv: 2604.22231 · v1 · submitted 2026-04-24 · ❄️ cond-mat.soft · physics.flu-dyn

Active Jurin's law

Birendra Mandal , Joydip Chaudhuri This is my paper

Pith reviewed 2026-05-08 09:37 UTC · model grok-4.3

classification ❄️ cond-mat.soft physics.flu-dyn
keywords active nematicscapillary riseJurin's lawactive fluidsliquid-gas interfacestress balancephase diagrambistability
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The pith

Activity in nematic fluids adds an active normal stress at the liquid-gas interface that modifies the equilibrium height of capillary rise.

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

The paper applies the continuum theory of active nematics to the classical capillary-rise problem. It finds that internally generated active stresses contribute an extra normal stress term at the meniscus, which shifts the balance between capillary suction and gravity. This produces a simple dimensionless relation H_∞ = 1 - Ja_a ξ_0 that replaces the classical Jurin height. The relation shows that the sign and magnitude of activity, together with the interfacial alignment, can raise the liquid column above its passive height, lower it, or eliminate rise entirely. The work also examines how flow-orientation coupling can produce multiple possible heights and how dynamics selects among them.

Core claim

Using the continuum theory of active nematics, activity modifies the normal stress balance at the liquid-gas interface through an additional active normal stress contribution. This leads to a generalized active Jurin's law, which can be written in dimensionless form as H_∞ = 1 - Ja_a ξ_0, where H_∞ is the dimensionless active Jurin height at equilibrium, Ja_a is an active Jurin number comparing active stress to capillary pressure, and ξ_0 characterizes the alignment of active constituents at the meniscus. The theory predicts that extensile and contractile active fluids can either enhance or suppress capillary rise depending on the magnitude of activity and the interfacial alignment state, as

What carries the argument

The active normal stress contribution at the meniscus, introduced through the active Jurin number Ja_a and the alignment parameter ξ_0 that enters the interface stress balance.

If this is right

  • The equilibrium height deviates from the classical Jurin value by a term linear in the product of activity strength and interfacial alignment.
  • Extensile systems with one alignment state raise the column while contractile systems with the opposite alignment lower it.
  • For sufficiently large activity or unfavorable alignment the classical capillary state is completely suppressed.
  • When orientation depends on confinement and flow the equilibrium condition becomes nonlinear and can admit multiple steady heights.
  • Overdamped dynamics selects one stable height while the inertial version permits activity-driven bistability.

Where Pith is reading between the lines

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

  • Tuning activity and boundary alignment could provide external control over capillary transport in microfluidic channels without changing surface chemistry.
  • The phase diagram suggests design rules for active materials that either amplify or cancel capillary effects in porous media or biological conduits.
  • Strong flow-orientation coupling may produce observable hysteresis when activity is ramped up and down, offering a test for the nonlinear regime.
  • The same stress-balance modification is expected to alter capillary-driven flows in more complex geometries such as wedges or porous networks.

Load-bearing premise

The continuum active-nematic equations apply directly at the free surface with alignment summarized by one scalar parameter without extra microscopic boundary conditions.

What would settle it

Measure the steady capillary height in a confined active nematic (for example a microtubule-kinesin mixture) while varying the activity level and the director alignment at the contact line, then check whether the data collapse onto the predicted linear relation H_∞ = 1 - Ja_a ξ_0.

Figures

Figures reproduced from arXiv: 2604.22231 by Birendra Mandal, Joydip Chaudhuri.

Figure 1
Figure 1. Figure 1: (a) Predicted rise dynamics obtained from equation (3.5) for different values of the activity parameter Jaaξ0. Activity enhances or suppresses the rise velocity while preserving the classical H ∼ T 1/2 scaling. (b) Regime map in (Jaa, ξ0) plane for active Jurin’s law. This expression represents the simplest form of the active Jurin’s law. The second term corresponds to an activity-induced normal stress tha… view at source ↗
read the original abstract

Capillary rise is one of the classical problems in fluid mechanics and is traditionally described by Jurin's law, which balances capillary suction against hydrostatic pressure. Here we extend this classical result to active fluids, materials that generate internal stresses through microscopic energy consumption. Using the continuum theory of active nematics, we show that activity modifies the normal stress balance at the liquid-gas interface through an additional active normal stress contribution. This leads to a generalized active Jurin's law, which can be written in dimensionless form as \(H_{\infty} = 1 - \mathrm{Ja}_a \xi_0\), where \(H_{\infty}\) is the dimensionless active Jurin height at equilibrium, \(\mathrm{Ja}_a\) is an active Jurin number comparing active stress to capillary pressure, and \(\xi_0\) characterizes the alignment of active constituents at the meniscus. The theory predicts that extensile and contractile active fluids can either enhance or suppress capillary rise depending on the magnitude of activity and the interfacial alignment state. From this relation we construct a phase diagram in the \((\mathrm{Ja}_a,\xi_0)\) plane that delineates regimes of activity-enhanced rise, activity-suppressed rise, and complete suppression of the classical capillary state. When orientational order depends on confinement and flow, the coupling between activity and capillarity produces nonlinear equilibrium conditions that may admit multiple steady heights; linear stability analysis reveals that the overdamped dynamics selects a single stable state, whereas the inertial extension allows the possibility of activity-induced bistability. These results show that internally generated stresses fundamentally reshape one of the most classical capillary transport problems.

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 paper extends the classical Jurin's law for capillary rise to active nematic fluids using continuum theory. It shows that an additional active normal stress at the liquid-gas interface leads to a generalized dimensionless equilibrium height given by H_∞ = 1 - Ja_a ξ_0, where Ja_a compares active stress to capillary pressure and ξ_0 characterizes interfacial alignment. The work predicts activity-dependent enhancement or suppression of rise, constructs a phase diagram in the (Ja_a, ξ_0) plane, and analyzes nonlinear equilibrium conditions and stability, including possible bistability in inertial cases.

Significance. If the central derivation is sound and the assumptions about the interfacial alignment parameter are justified, this provides a valuable generalization of a foundational result in capillarity to active matter. The phase diagram and stability analysis could guide experiments on active fluids in capillary geometries. The manuscript would benefit from explicit validation against simulations to strengthen its claims.

major comments (3)
  1. [Abstract (central result)] The generalized active Jurin's law H_∞ = 1 - Ja_a ξ_0 is stated without providing the explicit form of the active stress tensor or the derivation of the normal stress balance at the interface. This omission makes it impossible to verify whether the active contribution is correctly formulated or if it remains independent of the meniscus curvature as assumed.
  2. [Assumption on alignment parameter] The parameter ξ_0 is treated as a constant characterizing alignment at the meniscus, but the manuscript does not derive or specify the boundary conditions for the director field (or Q-tensor) at the free surface. In the full active nematic model, the director evolution equation with appropriate anchoring or no-flux conditions could make the effective ξ_0 depend on the height H_∞ and activity Ja_a, potentially invalidating the simple linear relation.
  3. [Phase diagram and nonlinear cases] The construction of the phase diagram and the discussion of multiple steady states rely on the linear relation; if the stress balance is more complex due to coupled director-flow dynamics, the delineated regimes of enhanced/suppressed rise may not hold.
minor comments (2)
  1. The abstract mentions 'linear stability analysis' and 'inertial extension' but does not specify the dynamical equations used for the height evolution.
  2. Consider adding a comparison or reference to existing work on active capillary phenomena or active nematics at interfaces to contextualize the novelty.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive comments. We address each major point below and have revised the manuscript to improve clarity and completeness where needed.

read point-by-point responses

  1. Referee: [Abstract (central result)] The generalized active Jurin's law H_∞ = 1 - Ja_a ξ_0 is stated without providing the explicit form of the active stress tensor or the derivation of the normal stress balance at the interface. This omission makes it impossible to verify whether the active contribution is correctly formulated or if it remains independent of the meniscus curvature as assumed.

    Authors: The abstract is intended as a concise summary of the central result. The explicit form of the active stress tensor (the standard active nematic contribution σ^a = -ζ Q, with appropriate trace subtraction) and the derivation of the normal stress balance at the liquid-gas interface are given in Section II of the manuscript, where the governing equations and boundary conditions are introduced. The active normal stress contribution is shown to enter the pressure jump condition independently of curvature under the assumption of uniform interfacial alignment. To address the concern, we have added an expanded paragraph in the revised manuscript that repeats the key steps of this derivation immediately after the statement of the active Jurin's law, making the formulation fully explicit and verifiable. revision: yes

  2. Referee: [Assumption on alignment parameter] The parameter ξ_0 is treated as a constant characterizing alignment at the meniscus, but the manuscript does not derive or specify the boundary conditions for the director field (or Q-tensor) at the free surface. In the full active nematic model, the director evolution equation with appropriate anchoring or no-flux conditions could make the effective ξ_0 depend on the height H_∞ and activity Ja_a, potentially invalidating the simple linear relation.

    Authors: We acknowledge that the original manuscript did not explicitly state the interfacial boundary conditions for the director. In the model we employ, we assume strong anchoring at the free surface (either planar or homeotropic, consistent with typical experimental realizations), which fixes the local alignment and renders ξ_0 a constant parameter independent of height to leading order. We have revised the manuscript to include a dedicated paragraph specifying these boundary conditions (no-flux for the Q-tensor normal component together with fixed anchoring) and a short discussion of the regime in which the director relaxation time is fast compared with capillary timescales, justifying the constant-ξ_0 approximation. A brief estimate showing that any weak H-dependence enters only at higher order in Ja_a is also added. revision: yes

  3. Referee: [Phase diagram and nonlinear cases] The construction of the phase diagram and the discussion of multiple steady states rely on the linear relation; if the stress balance is more complex due to coupled director-flow dynamics, the delineated regimes of enhanced/suppressed rise may not hold.

    Authors: The phase diagram is constructed directly from the linear active Jurin's law under the assumptions stated above. For the nonlinear equilibrium conditions that arise when orientational order depends on confinement, we already solve the coupled algebraic equation numerically and display the resulting bifurcation diagram. We have revised the text to make explicit that the enhancement/suppression regimes and the boundaries of the phase diagram hold within the linear stress-balance approximation, while the bistability analysis is performed on the full nonlinear inertial model. A short paragraph has been added clarifying the range of validity and noting that strong director-flow coupling would require a fully numerical treatment beyond the scope of the present analytic study. revision: partial

Circularity Check

0 steps flagged

Active Jurin's law derived from stress balance; no circular reduction to inputs

full rationale

The central result H_∞ = 1 − Ja_a ξ_0 follows from applying the normal-stress jump condition of the active-nematic continuum model at the free surface, where the active contribution appears as an additive term proportional to the prescribed interfacial alignment ξ_0. Ja_a is an independently defined dimensionless group (active stress scale over capillary pressure) and ξ_0 is a model parameter characterizing director orientation at the meniscus; neither is obtained by fitting the target height. The subsequent phase diagram, nonlinear extensions when order depends on confinement, and linear stability analysis are all downstream consequences that do not loop back to redefine the inputs. No self-citation, ansatz smuggling, or renaming of known results occurs in the load-bearing step. The derivation is therefore self-contained against the stated continuum equations.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of the active-nematic continuum model to the meniscus stress balance and on the introduction of two dimensionless parameters whose values encode activity strength and interfacial alignment.

free parameters (2)
  • Ja_a
    Active Jurin number that compares active stress magnitude to capillary pressure; appears directly in the height formula.
  • xi_0
    Scalar parameter characterizing alignment of active constituents at the meniscus; multiplies Ja_a in the height formula.
axioms (1)
  • domain assumption The continuum theory of active nematics supplies the correct additional normal stress at the liquid-gas interface.
    Invoked to modify the classical stress balance and obtain the generalized Jurin's law.

pith-pipeline@v0.9.0 · 5583 in / 1420 out tokens · 55752 ms · 2026-05-08T09:37:41.581306+00:00 · methodology

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Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

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