Analysis of Capillary Flow in Interior Corners: Perturbed Power Law Similarity Solutions

The design of fluid management systems requires accurate models for fluid transport. In the low gravity environment of space, gravity no longer dominates fluid displacement; instead capillary forces often govern flow. This thesis considers the redistribution of fluid along an interior corner. Following a rapid reduction of gravity, fluid advances along the corner measured by the column length z = L(t), which is governed by a nonlinear partial differential equation with dynamical boundary conditions. Three flow types are examined: capillary rise, spreading drop, and tapered corner. The spreading drop regime is shown to exhibit column length growth L ~ t2/5, where a closed form analytic solution exists. No analytic solution is available for the capillary rise problem. However, a perturbed power law similarity solution is pursued to approximate an analytic solution in the near neighborhood of the exact solution for the spreading drop. It is recovered that L ~ t1/2 for the capillary rise problem. The tapered corner problem is not analytically understood and hence its corresponding L is undocumented. Based on the slender corner geometry, it is natural to hypothesize the tapered corner column length initially behaves like the capillary rise regime, but after sufficient time has elapsed, it transitions into the spreading drop regime. This leads to a conjecture that its column length growth L is restricted to t2/5 < L < t1/2. To verify this conjecture an explicit finite difference numerical solution is developed for all three regimes. As will be shown, the finite difference scheme converges towards the analytic solutions for the spreading drop and capillary rise regimes. From this we assume the finite difference scheme is accurate for corner flows of similar geometries, and thus apply this scheme the more onerous criteria of the tapered corner. Numerical results support the conjectured L behavior for the tapered corner. Understanding the dynamics of such flows and responses to various geometries offers design advantages for spacecraft waste-management systems, fuel control, hydration containment, cryogenic flows, and a myriad of other fluid applications.