A variational formulation of a floating body
- Eine variationelle Formulierung eines schwimmenden Körpers
Roth, Magdalena Maria; Wagner, Alfred (Thesis advisor)
Aachen : Publikationsserver der RWTH Aachen University (2010)
Dissertation / PhD Thesis
Aachen, Techn. Hochsch., Diss., 2010
In this thesis we consider a two-dimensional motion with a floating body from a variational point of view. We consider an irrotational flow of an inviscid, homogeneous and incompressible liquid of finite depth acted on by gravity and surface tension. We look for 2R-periodic waves which propagate steadily with phase velocity c from the left to the right without alteration of form. The rigid body which will be a sphere throughout this thesis is assumed to move in the same direction having the same speed. Since the wave profile of a steady wave and the velocity potential are stationary with respect to a reference frame in uniform horizontal motion we can restrict ourselves to the time-independent case. Moreover, a volume constraint is considered. We consider the energy functional depending on the velocity potential, the surface of the fluid and the position of the rigid body. Since we consider a floating body forces due to gravity acting on the body and adhesion forces between fluid and body play a role. Therefore, the energy functional consists of the kinetic energy of the fluid, the potential energy of the fluid and the body and the energy due to adhesion and cohesion forces. We assume the interface between fluid and air, and the interface between fluid and rigid body respectively to be given non-parametrically and formulate the problem of a steady motion with a floating body as an obstacle problem. We begin with a brief introduction to the physical background of our model and derive the energy functional which we will examine during the rest of this thesis and give a precise statement of our problem. We proceed with the examination of the static case when no kinetic energy is involved. We define the appropriate function space in which we seek a minimizer and show existence in the static case under certain assumptions. Furthermore, making use of the Isoperimetric inequality we show that the contact set of a minimizer between fluid and body consists of finitely many components. Here, the number of components only depends on given constants. We compute the Euler-Lagrange equations of the energy functional in the static case. In this context we can determine the contact angles between fluid and solid. Moreover, we are able to verify the classical principle of Archimedes when no surface tension force is present and derive necessary conditions otherwise. Then we return to the full problem and prove the existence of a minimizer by showing the lower-semicontinuity of the energy functional. We show an integrability result for the velocity potential, namely that its gradient is integrable up to a power strictly greater than 4. We apply a very general result of Mitrea and Mayboroda about the regularity of the solution of the Neumann boundary problem in Lipschitz domains using Besov scales. This integrability is used to show that the contact set still consists of finitely many components. Moreover, a minimizer of our functional indeed presents a weak solution of our problem meaning that it satisfies the boundary conditions in a weak sense.