Steady Oseen Flow in Exterior Domains

Abstract

As we observed in the Introduction to Chapter V, the Stokes approximation fails to describe the physical properties of a system constituted by an object B moving with a small, constant velocity in a viscous liquid, at least at large distances from B where the viscous effects become less important. In particular, for B a ball, the explicit solution one finds (see (V.0.3)) exhibits no wake behind the body and is, therefore, unacceptable from the physical viewpoint. Moreover, for B a circle (plane motion), the problem admits no solution except for the trivial one. In addition to this, as observed by Oseen (1927, p.165), for the solution (V.0.3) we obtain, after a simple calculation,

$$ \left| {\frac{{{\text{v}} \cdot \nabla {\text{v}}}}{{\Delta {\text{v}}}}} \right| \to \infty {\kern 1pt} {\text{as}}\left| x \right| \to \infty , $$

no matter how small v_∞ is, thus violating the assumption under which the Stokes equations are derived (see the Introduction to Chapter IV).

e vidi le fiammelle andare avante, lasciando retro a sè l’aer dipinto.

DANTE, Purgatorio XXIX, vv. 73–74