Active Jurin's law

TL;DR

This paper generalizes Jurin's law for capillary rise to active fluids, showing how internal stresses from activity can enhance, suppress, or stabilize capillary heights, leading to new phase behaviors.

Contribution

It introduces a theoretical framework for active Jurin's law using active nematics, revealing how activity modifies capillary rise and predicts complex stability phenomena.

Findings

Active stresses alter the classical capillary height.

Phase diagram shows regimes of activity-enhanced and suppressed rise.

Multiple steady states and activity-induced bistability are possible.

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.