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2008 - 2009
on
“Self-Propelled Motions of Solids in Viscous Fluids: Mathematical
Analysis and Control Problems”
on
August 22, 2008
by
Prof. Marius Tucsnak
Nancy
Universites-IECN INRIA-CORIDA
at
11.00 am , L H – 1, Department of Mathematics
Indian Institute of Science, Bangalore
Abstract
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’s 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’s 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].
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