Unified analytical framework and regime map for capillary rise in cylindrical assemblies adjacent to a flat surface

A generalized mechanics-based framework is developed to characterize capillary-driven equilibrium and meniscus morphology in geometrically constrained cylindrical assemblies, including double-cylinder, multi-cylinder, and cylinder–surface configurations. The model development and results focus on an array of cylinders adjacent to a flat surface as the most complex morphology. The proposed formulation introduces a consistent dimensionless representation in terms of normalized pitch, gap, and radius. This representation enables systematic exploration of the admissible equilibrium solution space across configurations, particularly for an array of cylinders adjacent to a flat surface. Regime boundaries are identified as transitions between mechanically feasible equilibrium states, arising from geometric constraints on meniscus continuity and curvature. To facilitate practical implementation, a compact approximation is derived, providing an accurate and computationally efficient surrogate for the full implicit solution (R² = 0.99, RMSE = 0.14). The model is validated against independent datasets from the literature and new experimental measurements obtained from a custom-built testbed, demonstrating agreement within 5 mm across a wide range of geometric and wetting conditions. Analysis of limiting case recovers classical solutions for two cylinders in contact with a surface, confirming the internal consistency of the framework. The results reveal that capillary rise is governed by a nontrivial coupling between geometric confinement and interfacial curvature, leading to distinct equilibrium regimes not captured by existing simplified models. The developed framework establishes a unified theoretical basis for capillary phenomena in constrained geometries and provides a basis for analyzing a wide range of interfacial systems, including fibrous media, porous structures, and microstructured surfaces.