Active Jurin's law
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 \(H_{\infty} = 1 - \mathrm{Ja}_a ξ_0\), 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 \(ξ_0\) 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. From this relation we construct a phase diagram in the \((\mathrm{Ja}_a,ξ_0)\) 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.