An Analysis of Capillary Flow In Finite Length Interior Corners

Abstract

We analyze the mathematical robustness of slow massively parallel interior corner flows in low gravity environments. An interior corner provides a preferential orientation in low gravity environments. This is a luxury usually only found on earth. It also provides a passive pumping mechanism due to geometry of a conduit. The driving force for this flow is a pressure difference due to local surface curvature gradients. An alternative reasoning is that due to the geometrical constraints the interior corner surface energy is unbounded below. This results in the liquid wicking into corners indefinitely. Interior corner flow's main quantity of interest is the meniscus height h(z,t). With this variable one can calculate an average velocity w, flow rate Q, and volume of liquid in the corner V. Our study is different from most as it is highly in-depth look at finite domains, while the majority of previous solutions focus on similarity solutions of infinite, or semi-infinite domains. Boundary conditions, more specifically the functions that are assigned to the governing equation, play an integral role to meniscus height. We study a simplified problem of corners initially filled with quiescent liquid at t = 0, and boundary conditions are instantaneously applied when t > 0. Approximate asymptotic expressions are found for this process, but more importantly a method of approximating nonlinear heat equations as a sequence of linear heat equations is proved as a viable method for engineering purposes. Time varying boundary conditions are analyzed using a method of model-approximation. This is where we simply remove the nonlinearity of the governing equation and insert a fitting term n. The method works surprising well for a range of constant and time varying boundary conditions. In all cases the relative error between solutions is less than 10%. This is a major theme of the thesis, that is, force initial value boundary value problems to be linear via a substitution and achieve results sufficient for engineering analysis. For parallel corners, volume can transfer between corners in a multiple corner system. This motivates formulating an ODE governing the average height H(t) instead of meniscus height h(z,t). We formulate an N corner start-up problem similar to the analysis of a single corner. This solution is only true for a quasi-steady process for creeping flows. In order to feed a corner fluid, manifold tubing is required. Tubing presents a drastic geometrical difference where manifold resistance is much greater then the