다양한 영역에서의 유체의 표면장력파

Abstract

We study the capillary oscillations at the surface of a perfect incompressible fluid constrained in a solid container. In addition to two contact line boundary conditions, three different geometries are considered for the domain whose boundary consists of the free surface and solid walls of the container. For each case, the frequencies and solutions for the oscillations of the free boundary are computed numerically and some results can be extended to other geometries. Moreover, this thesis includes the proof of well-posedness of solutions for the corresponding initial value problems. More precisely, this thesis is organized as follows:

In Chap.2, we compute the natural frequencies for the oscillation of the free boundary of a perfect incompressible fluid in presence of capillary forces and partially in contact with a solid container. First, we study the case when the fluid occupies the domain {(x,y):y≤h(x,t)} with h(x,t)=0 for

x

>a and h(x,t) the free boundary for

≤a. We deduce an integro-differential evolutionary equation for the linearized free boundary and impose two different boundary conditions: the condition that the contact line between free boundary and the solid is pinned and the condition that the contact line can move vertically with a contact angle π/2. For both cases, we compute the natural oscillation frequencies for the free surface and compare the results with the frequencies of oscillation in the absence of solid walls. Secondly, we study the effect of having two parallel solid walls at

=a+b on the natural frequencies of oscillation of the free boundary.

In Chap.3, we compute the normal frequencies and normal modes for the oscillation of the free surface of a perfect incompressible fluid inside a semi-infinite container with a circular orifice. In doing that, a dual integral equation system involving the Bessel functions must be solved. We discuss two contact line conditions: pinned-edge condition and free-edge condition.

In Chap.4, We study the capillary oscillations of the surface of a 2D drop attached to a fan-shaped pillar. The fluid flow is modeled by means of a velocity potential and we assume a no-flux condition at the liquid-solid interface. The natural oscillation frequencies and oscillation modes are computed for two different physical situations depending on the contact line behavior: free-end and pinned-end. We also study the linearized initial value problem and prove well-posedness results in both free-end and pinned-end cases. Hence, for capillary oscillations when the fluid is in partial contact with a solid, not only initial conditions must be prescribed but also the behavior of the contact line.