Abstract

We develop a leading-order continuum 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 obtain a lubrication-scale 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 coordinated leading-order coupling among the Bond (Bo), Péclet (Pe), and Damköhler (Da) numbers. This coupling defines a characteristic inclination-angle-dependent scaling trajectory Γ(θ) in the (Bo, Pe, Da) space: material parameters set the system’s position along this curve, while the geometric constraint organizes the ordering of hydrodynamic, transport, and confinement regimes. We further derive leading-order crossover criteria associated with transport transitions (Pe ≃ 1) and reactive-confinement loss (Da ≃ 1), providing explicit regime boundaries that can be evaluated for representative parameter ranges. A representative parameterization of an ultrathin atmospheric electrolyte film is then used to make these crossovers explicit, yielding illustrative inclination thresholds that depend on the chosen parameter set. Coupling the deterministic structure to a minimal stochastic closure captures intermittent wet–dry dynamics under environmental forcing. In this closure, inclination selectively accelerates the drying pathway through the drainage time (and thus drying rate λdry), while rewetting remains primarily humidity-controlled, to leading order, providing a scaling-based description of wet-state persistence and time-of-wetness versus θ. The resulting framework provides a continuum-scale physical description of confined films under geometric asymmetry, relevant to wetting, interfacial drainage, confined transport, and thin-film systems in which symmetry breaking and coupled interfacial–transport processes coexist.

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