Two Problems in Liquid-Solid Interaction

University Of Pittsburgh, Pittsburgh PA

Investigators

Abstract

Over the past decades, the study of the motion of a viscous liquid interacting with solid bodies has become one of the main focuses of applied research. The objective of this project is to address two fundamental aspects in this area. The first regards the vibration-induced motion of a rigid body in a viscous liquid, when the vibration is produced by a time-periodically displaced mass inside the body. The wide area of applications of this phenomenon includes biomedical engineering and design of micro- and nano-technological equipment. The second aspect concerns the effect of flow-induced oscillations on a spring-mounted structure. These type of questions are of the utmost relevance in the study of the equilibrium and stability of suspension bridges and, in some cases, of their collapse, like in the notorious Tacoma Narrows Bridge disaster. This award will also provide opportunities for the involvement of graduate students in the project's research. Both projects fall into the realm of non-linear analysis and their study requires the use of fresh mathematical ideas that, possibly, will be employed also in other areas of applied nonlinear PDE's. First and foremost is the investigation of (local and global) Hopf bifurcation in a liquid-solid interaction problem in presence of a continuum spectrum of the relevant linearized operator. To date, this question represents an entirely uncharted territory. Another and not less important mathematical feature of the proposed research consists in the detailed study of the range of nonlinear elliptic operators (related to Navier-Stokes-like boundary-value problems) that are Fredholm of negative index. Interestingly enough, this kind of analysis may lead to the rigorous quantitative understanding of how and why a vibrating sphere can move in a viscous liquid. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.