|
Under typical physical
conditions, the solution of the capillarity equation for a tube of circular section 𝒟 will
always exceed over 𝒟 the solution obtained for a concentric tube of the same material
and larger radius. We address here a question raised by M. Miranda, as to whether a
solution over a general domain 𝒟0 will exceed, over that section, the solution over
any domain 𝒟1 strictly containing 𝒟0. We show that whenever a domain 𝒟1 admits a
zero gravity solution surface in a variational sense for the given contact angle, and
has at some point a boundary curvature inward directed and exceeding the
ratio of perimeter to area of the section, there is then a subdomain 𝒟0 for
which a negative answer appears for all sufficiently small gravity g; that
occurs with height differences inversely proportional to g, uniformly over
𝒟0.
Under other conditions, positive answers appear. We provide an example in which
the limiting behavior as g → 0 reverses in a discontinuous way, with smooth
infinitesimal change of ∂𝒟0. Remarkably, the discontinuous change occurs
at a circular cylinder configuration, for which one normally expects stable
behavior.
The discussion includes some results that seem to have general geometric
interest; notably, we characterize in Theorem 5 all convex domains containing a
disk, and for which the ratio of perimeter to area is not less than for the
disk.
|
