Some problems in the mathematical theory of fluid mechanics
Dashti, Masoumeh (2008) Some problems in the mathematical theory of fluid mechanics. PhD thesis, University of Warwick.Full text not available from this repository.
Official URL: http://webcat.warwick.ac.uk/record=b2338533~S15
This thesis addresses three problems related to the mathematical theory of fluid mechanics.
Firstly, we consider the three-dimensional incompressible Navier-Stokes equations
with an initial condition that has H1-Sobolev regularity. We show that there is
an a posteriori condition that, if satisfied by the numerical solutions of the equations,
guarantees the existence of a strong solution and therefore the validity of the numerical
computations. This is an extension of a similar result proved by Chernyshenko,
Constantin, Robinson & Titi (2007) to less regular solutions not considered by them.
In the second part, we give a simple proof of uniqueness of fluid particle trajectories
corresponding to the solution of the d-dimensional Navier Stokes equations, d = 2, 3,
with an initial condition that has H(d/2)−1-Sobolev regularity. This result has been
proved by Chemin & Lerner (1995) using the Littlewood-Payley theory for the flow in
the whole space Rd. We provide a significantly simpler proof, based on the decay of
Sobolev norms ( of order more than (d/2)−1 ) of the velocity field after the initial time,
that is also valid for the more physically relevant case of bounded domains.
The last problem we study is the motion of a fluid-rigid disk system in the whole
plane at the zero limit of the rigid body radius. We consider one rigid disk moving
with the fluid flow and show that when the radius of the disk goes to zero, the solution
of this system converges, in an appropriate sense, to the solution of the Navier-Stokes
equations describing the motion of only fluid in the whole plane. We then prove that
the trajectory of the centre of the disk, at the zero limit of its radius, coincides with a
fluid particle trajectory. We also show an equivalent result for the limiting motion of
a spherical tracer in R3, over a small enough time interval.
|Item Type:||Thesis or Dissertation (PhD)|
|Subjects:||Q Science > QA Mathematics|
|Library of Congress Subject Headings (LCSH):||Fluid mechanics, Navier-Stokes equations|
|Official Date:||September 2008|
|Institution:||University of Warwick|
|Theses Department:||Mathematics Institute|
|Supervisor(s)/Advisor:||Robinson, James C. (James Cooper), 1969-|
|Sponsors:||University of Warwick|
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