We propose a novel formulation of capillarity, which geometrically extends the Bosanquet equation to irregular geometries, taking the effect of inertia and the dynamic contact angle into account. The governing equation is an integrodifferential equation that is solved numerically and compared with computer simulations, experimental data, and other cases available in the literature. The numerical examples investigated in this work show that contrary to flat channels and tubes, inertial effects decay much slower in corrugated channels and tubes due to the walls' geometrical fluctuations. We also draw the paramount conclusion that the true solution for Jurin's height in irregular capillaries is path-dependent and highly sensitive to the initial conditions, and no single static-equilibrium solution can necessarily be attributed to the eventual position of the meniscus. Resulting from the non-linear dynamics, the multiple equilibria in the presence of gravity for irregular capillaries can only be analyzed if the effect of inertia is considered, which has largely been neglected in the literature thus far.
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