Universality in low Reynolds number flows:theory and applications

Peter Wittwer Département de Physique ThéoriqueUniversité de Genève

reading:

R. P. Feynman, Vol. II

G. K. Batchelor, An Introduction to Fluid Mechanics

L. Landau, E. Lifchitz, Mécanique des fluides

M. Van Dyke, An Album of Fluid Motion

collaborations:

Guillaume Van Baalen

Frédéric Haldi

Sebastian Bönisch

Vincent Heuveline

─ Introduction to the problem ─ Asymptotic analysis

─ Applications

Exterior Flows

Navier-Stokes

Re=0.16

Re=1.54

Re=56.5

Re=118

Re=7000

Case of finite volume

Case of infinite volume, I

Case of infinite volume, II

Asymptotic analysis

Results (d=2)

Interpretation:

Results (d=3)

Two steps:

─ construct downstream asymptotics

dynamical system invariant manifold theory renormalization group universality

─ determines asymptotics everywhere

Vorticity:

Vorticity equation

Fourier transform

Diagonalize

Stable and unstable modes

use contraction mapping principle

Large time asymptotics:

Two steps:

─ construct downstream asymptotics

dynamical system invariant manifold theory renormalization group universality

─ determines asymptotics everywhere

Determines asymptotics everywhere:

Applications

in collaboration with:

Sebastian BönischRolf Rannacher

Vincent Heuveline

Heidelberg & Karlsruhe

Adaptive boundary conditions

To second order:

Comparison with Experiment:

Cloud Microphysics and Climate

M. B. Baker, SCIENCE, Vol. 276, 1997

Work in progress:• d=2 case with lift (numerical)

• d=2 second order asymptotics (theory)

• d=3 (numerical)

• d=2, 3: free fall problem (numerical)

• d=3 case with rotation at infinity (theory; see P. Galdi

(2005) for recent results)

Other research groups:

• d=2 time periodic (theory)