The shape of a pendant liquid drop

Abstract
A characterization, is given for the most general equilibrium configuration of a symmetric pendent liquid drop. It is shown that for any vertex height u 0 the vertical section can be continued globally as an analytic curve, without limit sets or double points. For small |u 0 | it is proved the section projects simply on the axis u = 0; for large |u 0 | the section is shown to have near the vertex the general form of a succession of circular arcs joined near the axis by small arcs of large curvature. The section contracts at first toward a certain hyperbola, the 'circular arcs’ gradually changing shape but remaining, until a certain fixed height (asymptotically as u 0 — ∞), within a narrow band surrounding the hyperbola. The continuation of the section eventually projects simply on u = 0, separates from the hyperbola, and continues in an oscillatory manner to infinity. The properties described above are studied quantitatively. It is conjectured that as |u 0 | -> ∞ the section converges uniformly (as a point set) to a solution U(r) with simple projection (for all r > 0 ) on = 0 and an isolated singularity at r = 0. A preliminary (weak) form of the conjecture is proved. The (liquid drop) solutions are also studied from the point of view of their global embedding in the manifold of all formal solutions of the equations. From this point of view, the vertex of the drop appears as a transition point marking a change in qualitative appearance. It is conjectured that the only global solution without double points, in this extended sense, is the singular solution referred to above.

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