Solution of the Kirchhoff–Plateau Problem

Abstract

The Kirchhoff–Plateau problem concerns the equilibrium shapes of a system in which a flexible filament in the form of a closed loop is spanned by a liquid film, with the filament being modeled as a Kirchhoff rod and the action of the spanning surface being solely due to surface tension. We establish the existence of an equilibrium shape that minimizes the total energy of the system under the physical constraint of noninterpenetration of matter, but allowing for points on the surface of the bounding loop to come into contact. In our treatment, the bounding loop retains a finite cross-sectional thickness and a nonvanishing volume, while the liquid film is represented by a set with finite two-dimensional Hausdorff measure. Moreover, the region where the liquid film touches the surface of the bounding loop is not prescribed a priori. Our mathematical results substantiate the physical relevance of the chosen model. Indeed, no matter how strong is the competition between surface tension and the elastic response of the filament, the system is always able to adjust to achieve a configuration that complies with the physical constraints encountered in experiments.

Soap film spanning an elastic link

“…Since we are dealing with approximating surfaces, we need to specify the notion of convergence of surfaces. We do this following Fried et al [13]. The problem we have to solve is connected to the fact that Λ[z k ], the closed subset in R 3 occupied by the whole link, changes along the minimizing sequence.…”

“…Proof. The proof can be made easily by following the proof presented in [13] taking into account the fact that we can study the two rods separately since it is sufficient to reduce the proof in an open neighbourhood well-cointained in each rod.…”

“…Then the mapping (s, ζ 1 , ζ 2 ) ↦ p[z](s, ζ 1 , ζ 2 ) is globally injective on int Ω.Proof. By(13) it suffices to show the global injectivity of p 1 and repeat the arguments for p 2 . Let us fix a configuration z 1 ∈ V 1 , by Theorem 1 there is a set I 0 of measure zero such that…”

Construction of exact minimal parking garages: nonlinear helical motifs in optimally packed lamellar structures

“…In many physical and biological cases, the helical motifs are mobile and can vary their relative orientation and position to further lower their energy. Additionally, this also opens the problem of considering an energy contribution from the boundary, as for example, the Euler-Plateau problem where one considers the boundary as an Euler elastica [19] or the Kirchhoff-Plateau problem where one considers the boundary as a Kirchhoff rod [20]. The systematic study of these relative forces and torques between motifs as well as the energy contribution of the boundaries are left to future work.…”

Importance and Effectiveness of Representing the Shapes of Cosserat Rods and Framed Curves as Paths in the Special Euclidean Algebra

“…. , N, that can be used to reconstruct the (discretized) placement of the cross sections of the rod in space, by giving an initial cross section as a vector R 0 in R 12 and successively applying equation (5). We then see that a piecewise constant finite-element approximation of the operator field L-which is a path in se(3)-provides a uniquely defined approximation of the placement of a rod through a discretization of its shape.…”