Inclination-Driven Thin-Film Hydrodynamics: Universal Trajectory in the {Da, Pe, Bo} Space

TL;DR

This paper presents a comprehensive theoretical model for thin-film hydrodynamics on inclined surfaces, unifying various physical effects and revealing a universal trajectory in key dimensionless parameter space.

Contribution

It introduces a unified thin-film equation incorporating inclination effects, capillarity, intermolecular forces, and stochastic wetting, linking the parameters through a universal trajectory in (Da, Pe, Bo) space.

Findings

Coupling of Da, Pe, Bo numbers defines a universal trajectory.

Inclination induces dominant advection in thin-film dynamics.

Stochastic modeling captures intermittent wet-dry cycles.

Abstract

We develop a unified theoretical framework for thin-film hydrodynamics on inclined solid substrates, integrating capillarity, intermolecular forces, gravitational symmetry breaking, confined transport and stochastic wetting into a single formulation. Starting from lubrication theory with capillary curvature and disjoining-pressure interactions, we derive a general thin-film equation that incorporates inclination-driven advection, nanoscale stabilization and humidity-controlled source-sink fluxes. A dimensionless analysis shows that, within the long-wave lubrication approximation, inclination induces a leading-order coupling of the Bond, Peclet and Damkohler numbers. This coupling defines a characteristic trajectory in the parameter space (Da, Pe, Bo), determined by the structure of the lubrication flux. Coupling this deterministic framework to a minimal stochastic formulation captures the intermittent wet-dry dynamics characteristic of ultrathin films under environmental forcing. The framework provides a general surface-physics description of confined films under geometric asymmetry, applicable across wetting, interfacial drainage, reactive confinement and soft-matter systems.