Interface Equations for Capillary Rise in Random Environment

TL;DR

This paper derives equations describing how quenched disorder affects capillary rise in 2D and 3D environments, introducing a systematic formalism that accounts for local geometric fluctuations and mobility variations.

Contribution

It develops a projection formalism to include disorder in interface dynamics equations for capillary rise, providing new linearized equations with distinct noise characteristics.

Findings

Derived local Fourier space equations for meniscus and contact line dynamics.

Identified effective noise terms proportional to velocity, differing from previous models.

Validated that deterministic parts align with existing studies in certain limits.

Abstract

We consider the influence of quenched noise upon interface dynamics in 2D and 3D capillary rise with rough walls by using phase-field approach, where the local conservation of mass in the bulk is explicitly included. In the 2D case the disorder is assumed to be in the effective mobility coefficient, while in the 3D case we explicitly consider the influence of locally fluctuating geometry along a solid wall using a generalized curvilinear coordinate transformation. To obtain the equations of motion for meniscus and contact lines, we develop a systematic projection formalism which allows inclusion of disorder. Using this formalism, we derive linearized equations of motion for the meniscus and contact line variables, which become local in the Fourier space representation. These dispersion relations contain effective noise that is linearly proportional to the velocity. The deterministic parts of our dispersion relations agree with results obtained from other similar studies in the proper limits. However, the forms of the noise terms derived here are quantitatively different from the other studies.