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

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.