Dynamic Capillarity Fundamentals

• Dynamic capillarity is the study of time-dependent capillary forces influencing fluid interfaces in non-equilibrium conditions across scales.
• It incorporates rate-dependent corrections to classical capillary laws, enhancing models of porous media, thin films, and deformable structures.
• Applications include improving multiphase flow simulations and microfluidic designs through experimental validation and advanced computational frameworks.

Dynamic capillarity refers to the time-dependent, non-equilibrium behavior of capillary forces in fluid systems, particularly when interfaces, wettability, and flow geometry induce or respond to rapid changes. Unlike quasi-static capillarity, where surface tension and pressure equilibrate instantaneously, dynamic capillarity incorporates rate effects, history dependence, contact-line relaxation, and transient micro/macrostructure evolution. This concept arises in multiphase flow, porous media, thin films, deformable microstructures, and micro/nanofluidic contexts, and has broad implications for both fundamental theory and modeling in continuum mechanics, hydrodynamics, and soft matter physics.

1. Governing Principles and Rate-Dependent Capillary Laws

Dynamic capillarity manifests when the timescale for interface relaxation (by molecular diffusion, film rearrangement, or meniscus motion) is not negligible compared to the imposed flow or morphological evolution. Rate effects often emerge in models as a deviation from equilibrium capillary pressure, frequently via constitutive relations such as

$$p_c^{\mathrm{dyn}}(S) = p_c^{\mathrm{stat}}(S) - \tau(S)\,\partial_t S,$$

where $p_c^{\mathrm{stat}}(S)$ is the static relation (e.g., Young–Laplace), $S$ is saturation or interface position, and $\tau(S)$ is a dynamic capillarity coefficient with pressure$\times$time dimensions (Tantardini et al., 9 Nov 2025, Mitra et al., 2019).

In conservation laws and porous media equations, inclusion of a pseudo-parabolic term,

$- \nabla \cdot [M(S)\tau(S)\nabla(\partial_t S)],$

regularizes shocks and accounts for delayed interface response, leading to well-posedness even in compositionally complex and three-phase flows (Tantardini et al., 9 Nov 2025, Graf et al., 2018).

Microscale and nanoscale experiments confirm that even at high curvatures and extreme stresses, dynamic capillary effects persist, with continuum relations holding up to correction for boundary effects such as negative slip lengths and immobile molecular layers (Vincent et al., 2015, Gruener et al., 2010).

2. Dynamic Capillarity in Porous and Deformable Media

In porous systems, dynamic capillarity is crucial for:

• Smoothing of saturation fronts and suppression of unphysical shocks in multiphase transport (Mitra et al., 2019).
• Coupling of transient meniscus motion and film flow with nonlocal, history-dependent interface configurations.
• Rate-dependence in capillary entry pressure and permeability, influenced by film morphology and trap sizes (Tantardini et al., 9 Nov 2025).
• Representative elementary volume (REV) criteria set by the amplitude and spectrum of capillary pressure fluctuations, demanding time-space averaging theories incorporating transient interface statistics (McClure et al., 2020).

Elastocapillary deformations introduce further complexity: imbibition or drainage can create transient strains in soft or porous matrices, governed by coupled partial differential equations linking capillary pressure, viscous flow, and elastic response (Sanchez et al., 2023, Nasouri et al., 2018). Notably, the dynamic capillary tension in the fluid is only partially transmitted to the matrix during imbibition, with the Laplace contribution only half that in steady-state condensation due to the linear pressure drop profile (Sanchez et al., 2023). In poroelastocapillary rise, wettability-induced softening can drastically accelerate sheet coalescence and truncate fluid uptake (Nasouri et al., 2018).

3. Dynamic Capillarity in Thin Films, Contact Lines, and Free Boundaries

Viscous thin films subject to curvature gradients experience dynamic capillarity through the fourth-order thin-film equation,

$$\partial_t h + \frac{\gamma}{3\eta} \partial_x [h^3 \partial_x^3 h] = 0,$$

where $\gamma$ is surface tension and $\eta$ the viscosity (Salez et al., 2012). Perturbations (e.g., a step in thickness) relax self-similarly, with the interface profiles and capillary energy exhibiting explicit time-dependent scaling laws.

Contact line dynamics fundamentally exhibit dynamic capillarity: the meniscus cannot adjust its shape instantaneously at finite velocity. This is modeled by rate-dependent dynamic contact angles (e.g., molecular kinetics: $\cos\theta_d-\cos\theta_s+\xi \mathrm{Ca}=0$ or hydrodynamic: $\theta_d^3-\theta_s^3-\kappa\mathrm{Ca}=0$), which modify the instantaneous capillary pressure (Yelkhovsky et al., 2018). For rapid flows, or under conditions deviating from the lubrication limit, dynamic capillarity produces overshoot, oscillations, or delayed pinning in meniscus motion, and is necessary for accurate modeling of droplet ejection, wetting/dewetting transitions, and spontaneous instability (Wollman et al., 2012, Fargette et al., 2013).

4. Modeling Frameworks and Regularization Strategies

Dynamic capillarity is now universally incorporated into multiphase models by:

• Adding pseudo-parabolic $O(\nabla(\partial_t S))$ terms to transport equations (Tantardini et al., 9 Nov 2025, Graf et al., 2018, Karlsen et al., 2022).
• Augmenting capillary laws with dynamic corrections calibrated from pore-network simulations or experiment (Tantardini et al., 9 Nov 2025).
• Implementing regularized network and finite-volume models with explicit interface tracking and rate-dependent capillary relations (Yelkhovsky et al., 2018).
• Using Smoothed Particle Hydrodynamics (SPH) methods with capillary forces built in via pairwise interactions, enabling meshless simulation of fast rupture, coalescence, and instability (Nair et al., 2017).

For thin films and soft matter, coupling gradient elasticity and Cahn–Hilliard-type models with capillarity in the evolving (current) configuration yields a fully coupled system capable of capturing stress-driven phase separation, swelling, and large-strain effects (Roubíček, 2019).

In mathematical and computational analysis of dynamic capillarity, kinetic formulations, H-measures, stochastic velocity averaging, and quasi-Polish compactness tools are deployed to establish existence, uniqueness, and strong convergence in singular limits or fluctuating media (Karlsen et al., 2022, Graf et al., 2018).

5. Dimensionless Groups and Physical Regimes

Dynamic capillarity is quantified by several dimensionless numbers:

• Capillary number: $\mathrm{Ca} = \mu U/\sigma$ (viscous to capillary stresses), controlling dynamic contact angle deviation.
• Reynolds number: $\mathrm{Re} = \rho U R/\mu$, identifying inertia-dominated versus quasi-steady regimes.
• Dynamic capillarity numbers of form $\mathcal{D} = \tau \partial_t S$ or variations thereof, measuring the importance of capillary relaxation.
• Modified Weber ($\mathrm{We}^*$) and Suratman ($\mathrm{Su}^*$) numbers in droplet ejection map the inertia, viscous, and capillary interplay, setting thresholds for regime transitions (Wollman et al., 2012).
• The universal parameter $\zeta = \langle D \rangle \langle F \rangle$ combining averaged damping and forcing in generalized capillary rise, acting as a regime-classifying scalar (Saha et al., 23 Nov 2024).

These groups define not only qualitative behavior (e.g., monotonic vs. overshoot in imbibition, regime maps for droplet ejection) but also thresholds for non-equilibrium phenomena including shock formation, oscillatory fronts, or plateau states in saturation.

6. Applications, Optimization, and Extensions

Practical implications of dynamic capillarity span:

• Microfluidic design, with $\zeta$-based optimization for geometry, wettability, and flow enhancement/suppression (Saha et al., 23 Nov 2024).
• Carbon storage, contaminant remediation, and geological multiphase flow, where inclusion of dynamic capillarity regularizes and stabilizes the numerical simulation of compositional and phase behavior (Tantardini et al., 9 Nov 2025).
• Nano/microfluidic pumping, passive membrane transport, and the design of capillary valves, where continuum models hold to nanometric scales up to corrections for boundary immobilization (Vincent et al., 2015).
• Engineering of smart actuator and sensor systems that leverage elastocapillary instability on rapid (ms to sub-ms) timescales (Fargette et al., 2013).

Stochastic dynamic capillarity appears in equations for random forcing and on manifolds, where singular limits map SPDEs with regularized capillarity to stochastic conservation laws with uniquely characterized martingale solutions (Karlsen et al., 2022).

7. Experimental Observations and Validations

Experiments spanning scales from nanometers (Vycor glass, silicon nanochannels) to millimeters (capillary tubes, thin films, droplet ejection) robustly demonstrate:

• Persistence of classical capillarity (Kelvin, Young–Laplace, Washburn) down to 3–10 molecular diameters, with breakdown evident only in the emergence of nanometric sticking layers or modest negative slip length corrections (Vincent et al., 2015, Gruener et al., 2010).
• Deformation and kinetic strain fields that scale identically with meniscus advance, and exhibit abrupt jumps or regime changes associated with interface disappearance or flow-arrest (Sanchez et al., 2023).
• A broad spectrum of front profiles in two-phase flow through columns—ranging from monotone to non-monotone with overshoot and sustained plateaus—arising only with sufficiently large dynamic capillarity timescales (Mitra et al., 2019).
• Failures of classical models (Lucas–Washburn) to accurately predict interface evolution or velocity fields unless dynamic contact angle corrections and local dissipation zones are incorporated (Fiorini et al., 2023).

These results feed directly into model calibration, identification of rate-controlling mechanisms, and design choices in capillarity-driven transport and engineered systems.

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Dynamic capillarity thus represents a unifying framework for nonequilibrium interfacial dynamics across classical and emerging length scales, embedding rate dependence, memory, and microstructural evolution in both equations of state and continuum field theories. Its incorporation is essential for accurate modeling and control of multiphase, deformable, and stochastic fluid systems.