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Find the shape of a soap film (i.e., Minimal Surface) which will fill two inverted conical Funnels facing each other is known as Sinclair's soap film problem (Bliss 1925, p. 121). The soap film will assume the shape of a Catenoid.
See also Catenoid, Funnel, Minimal Surface
References
Bliss, G. A. Calculus of Variations. Chicago, IL: Open Court, pp. 121-122, 1925.
Isenberg, C. The Science of Soap Films and Soap Bubbles. New York: Dover, p. 81, 1992.
Sinclair, M. E. ``On the Minimum Surface of Revolution in the Case of One Variable End Point.'' Ann. Math. 8, 177-188, 1907.