Dynamics and Kinetic Roughening of Interfaces in Two-Dimensional Forced Wetting

TL;DR

This paper investigates the dynamics and roughening behavior of forced wetting fronts in 2D disordered media using a phase field model, deriving a crossover length and analyzing interface roughness and growth exponents.

Contribution

It introduces a linearized interface equation for forced wetting, identifies a velocity-dependent crossover length, and compares full model solutions with linearized predictions.

Findings

Interfaces are superrough with a roughness exponent of ~1.35.

The crossover length scales as the inverse square root of velocity.

Numerical results agree with recent Hele-Shaw experiments.

Abstract

We consider the dynamics and kinetic roughening of wetting fronts in the case of forced wetting driven by a constant mass flux into a 2D disordered medium. We employ a coarse-grained phase field model with local conservation of density, which has been developed earlier for spontaneous imbibition driven by a capillary forces. The forced flow creates interfaces that propagate at a constant average velocity. We first derive a linearized equation of motion for the interface fluctuations using projection methods. From this we extract a time-independent crossover length $\xi_\times$, which separates two regimes of dissipative behavior and governs the kinetic roughening of the interfaces by giving an upper cutoff for the extent of the fluctuations. By numerically integrating the phase field model, we find that the interfaces are superrough with a roughness exponent of $\chi = 1.35 \pm 0.05$, a growth exponent of $\beta = 0.50 \pm 0.02$, and $\xi_\times \sim v^{-1/2}$ as a function of the velocity. These results are in good agreement with recent experiments on Hele-Shaw cells. We also make a direct numerical comparison between the solutions of the full phase field model and the corresponding linearized interface equation. Good agreement is found in spatial correlations, while the temporal correlations in the two models are somewhat different.