Refinement of the added mass coefficient in capillary rise dynamics

Capillary rise is a fundamental transport mechanism in natural and engineered porous systems, where inertial effects govern the early-time dynamics despite the small length scales involved. Recent analytical models incorporate an added mass term to remove the classical singularity in capillary rise dynamics, commonly using a hemispherical control volume and an imposed discontinuous velocity profile at the tube entrance. In this study, we develop an analytical potential-flow framework for estimating the added mass coefficient beneath the capillary entrance, avoiding the hemispherical velocity discontinuity present in classical reduced-order approaches. A generalized coaxial cylindrical geometry is first considered, from which three limiting cases are analyzed: (i) a cylindrical tube immersed in an infinitely deep reservoir, (ii) a coaxial cylindrical configuration, and (iii) a finite-depth reservoir. Closed-form analytical expressions and numerical evaluations of the added mass coefficient are presented, yielding a coefficient for the classical configuration that is lower than the widely used hemispherical-cap estimate. The results provide a mathematically consistent inertial closure parameter for reduced-order capillary rise models and complex geometrical configurations.