182 research outputs found
NOTE ON THE PROBLEM OF MOTION OF VISCOUS FLUID AROUND A ROTATING AND TRANSLATING RIGID BODY
We consider the linearized and nonlinear systems describing the motion of incompressible flow around a rotating and translating rigid body Ɗ in the exterior domain Ω = ℝ3 \ D , where Ɗ ⊂ ℝ3 is open and bounded, with Lipschitz boundary. We derive the L∞-estimates for the pressure and investigate the leading term for the velocity and its gradient. Moreover, we show that the velocity essentially behaves near the infinity as a constant times the first column of the fundamental solution of the Oseen system. Finally, we consider the Oseen problem in a bounded domain ΩR := BR ∩ Ω under certain artificial boundary conditions on the truncating boundary ∂BR, and then we compare this solution with the solution in the exterior domain Ω to get the truncation error estimate
On the interplay between vortices and harmonic flows: Hodge decomposition of Euler's equations in 2d
Let be a compact manifold without boundary whose first homology is
nontrivial. Hodge decomposition of the incompressible Euler's equation in terms
of 1-forms yields a coupled PDE-ODE system. The -orthogonal components are
a `pure' vorticity flow and a potential flow (harmonic, with the dimension of
the homology). In this paper we focus on point vortices on a compact
Riemann surface without boundary of genus , with a metric chosen in the
conformal class.
The phase space has finite dimension . We compute a surface of
section for the motion of a single vortex () on a torus () with a
non-flat metric, that shows typical features of non-integrable 2-dof
Hamiltonians. In contradistinction, for flat tori the harmonic part is
constant. Next, we turn to hyperbolic surfaces (), having constant
curvature -1, with discrete symmetries. Fixed points of involutions yield
vortex crystals in the Poincar\'e disk. Finally we consider multiply connected
planar domains. The image method due to Green and Thomson is viewed in the
Schottky double. The Kirchhoff-Routh hamiltonian given in C.C. Lin's celebrated
theorem is recovered by Marsden-Weinstein reduction from to .
The relation between the electrostatic Green function and the hydrodynamical
Green function is clarified.
A number of questions are suggested
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