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Stability of a liquid drop trapped between two parallel planes. (English) Zbl 0627.53004

In the last years many problems involving liquid drops and capillarity phenomena have been considered due to their relationship to the methods of the calculus of variations [see R. Finn, Equilibrium capillary surfaces (Springer 1986; Zbl 0583.35002)]. In this paper the author considers the physical problem of a drop of liquid trapped between two parallel planes in the absence of gravity. As in all other problems involving drops, the forces involved are surface tension, which is proportional to the area of the free surface of the liquid, and the wetting energy, which is proportional to the area on the planes wetted by the drop. In this problem admissible configurations are those which preserve the volume of the liquid.
The energy functional associated with the problem of a drop trapped between two homogeneous parallel planes \(P_ 1\) and \(P_ 2\), in the absence of gravity is: \[ E(\Omega)=A(\Sigma)-a_ 1A(S_ 1)-a_ 2A(S_ 2), \] where \(\Omega\) is the region in space occupied by the drop, \(\Sigma\) is the liquid-air interface, \(S_ i\) is the region wetted on the plane \(P_ i\), \(a_ i\) is a constant determined by the materials involved, and A denotes area.
The admissible family of sets on which we have to look for local minima is the one with volume of \(\Omega\) constant. Using the variational formulation a criterion is derived for a solution of the Euler-Lagrange equation to be a local minimum for the energy functional, thus a stable solution. A simpler formulation of the stability criterion is then applied to the problem of the trapped drop, in order to solve the question in the case of perpendicular contact angles.
If h is the separation of the two planes, the cylindrical trapped drop is stable if its volume is larger than \(h^ 3/\pi\) and unstable if its volume is smaller. A similar problem was considered by Lord Rayleigh [On the capillary phenomena of jets, Scientific Paper, vol. 1, Cambridge University Press (1899), pp. 377-401] who obtained the condition \(h^ 3/4\pi\) under the assumption that admissible perturbations do not change the radius of the circles wetted by the drop of the two planes, that is the two circles are prescribed.
Reviewer: M.Emmer

MSC:

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
49K30 Optimality conditions for solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
35B32 Bifurcations in context of PDEs

Citations:

Zbl 0583.35002
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