Inclination-Driven Thin-Film Dynamics: Geometry-Induced 2 Regime Ordering in the (<em>Bo</em>, <em>Pe</em>, <em>Da</em>) Space

We develop a unified theoretical framework for thin-film hydrodynamics on inclined 14 solid substrates, integrating capillarity, intermolecular forces, gravitational symmetry 15 breaking, confined transport, and stochastic wetting into a single formulation. Starting 16 from lubrication theory with capillary curvature and disjoining-pressure interactions, we 17 obtain a general thin-film equation that incorporates inclination-driven advection, na- 18 noscale stabilization, and humidity-controlled source–sink fluxes. A dimensionless anal- 19 ysis shows that, within the long-wave lubrication approximation, inclination induces a 20 leading-order coupling among the Bond, Péclet, and Damköhler numbers. This coupling 21 defines a characteristic inclination-parameterized trajectory Γ(θ) in the (Bo, Pe, Da) space: 22 material parameters set the system’s position along this curve, while the geometric con- 23 straint governs the ordering of hydrodynamic, transport, and confinement regimes. We 24 further derive quantitative crossover criteria associated with transport transitions (Pe ≃ 25 1) and reactive-confinement loss (Da ≃ 1), providing explicit regime boundaries that can 26 be evaluated for representative parameter ranges. A representative parameterization of 27 an ultrathin atmospheric electrolyte film is then used to make these crossovers explicit, 28 yielding illustrative inclination thresholds for the onset of transport reorganization and 29 reactive-confinement loss. 30 Coupling the deterministic structure to a minimal stochastic closure captures intermittent 31 wet–dry dynamics under environmental forcing. In this closure, inclination selectively ac- 32 celerates the drying pathway through the drainage time (and thus λdry), while re-wetting 33 remains primarily humidity-controlled, providing a leading-order basis for wet-state per- 34 sistence and time-of-wetness versus θ. The resulting framework provides a general phys- 35 ical description of confined films under geometric asymmetry, relevant to wetting, inter- 36 facial drainage, confined transport, and thin-film systems in which symmetry breaking 37 and coupled interfacial–transport processes coexist across scales.