Annular Capillary Surfaces: Properties and Approximation Techniques
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The capillary surface formed within a symmetric annular tube is analyzed. Assuming identical contact angles along each boundary, we consider surfaces u(x,y) that satisfy the capillary problem on an annular region. Several qualitative properties of u are determined and in particular, the behaviour of u is examined in the limiting cases of the annular domain approaching a disk as well as a thin ring. The iterative method of Siegel is also applied to the boundary value problem and convergence is demonstrated under conditions which include a contact angle of zero. Moreover, some geometries still yield interleaving iterates, allowing for upper and lower bounds to be placed on the boundary values of u. However, the interleaving properties no longer hold universally and for other geometries, another more complex behaviour is described. Finally, a numerical method is designed to approximate the iterative scheme.