Speaker

:

Professor Marius Tucsnak

Subject Area

:

Mathematics

Venue

:

Lecture Hall - I, Dept of Mathematics

Time

:

11.00 am

Date  

:

August 22,2008 (Friday)

Title

:

Self-Propelled Motions of Solids in Viscous Fluids: Mathematical Analysis and Control Problems

The aim of this presentation is to highlight some recent advances on the mathematical analysis and the control of self-propelled motions of solids in a fluid. We study a model consisting in a solid undergoing an undulatory deformation, which is immersed in a viscous incompressible fluid. The motion of the fluid is governed by the incompressible Navier-Stokes equations and the standard conservation~Rs laws of linear and angular momentum rule the dynamics of the structure. The time variation of the fluid domain (due to the motion of the structure) is not known a priori, so we deal with a free boundary value problem. The displacement of
the solid is decomposed into a rigid part and a deformation (undulatory) part. The rigid part of the displacement results from the interaction of the fluid and the solid, whereas the deformation part is given. Since our aim is to possibly consider several immersed solids, the domain filled by the fluid is one of the unknowns. Therefore we have to tackle a free boundary value problem. The solutions are controlled by an input which is the shape of the solid. We first show that the initial and boundary value problem obtained by coupling the Navier-Stokes equations for the fluid to Newton~Rs law for the creature is well-posed in Sobolev type spaces. We next give an approximation scheme for the governing equations which is tested on some undulatory motions observed by the zoologists in order to get straight-line-swimming or turning. We finally tackle, from a control theoretic perspective the swimming of aquatic microorganisms. Since, the Reynolds number is this time very low, we consider a model based on the Stokes equations for the fluid. This presentation is essentially based on results from [1] and [2].