On the Oseen fundamental solution and asymptotic profile of exterior flows

J. Math. Fluid Mech., 22 (2020), 1-19
https://doi.org/10.1007/s00021-019-0474-x

The Oseen equations are associated with flow past an object which moves with small translational velocity $\zeta \in {\mathbb R}^d$ within a viscous liquid. In this paper, we are interested in the fundamental solution of the 2- and 3-d Oseen equations and its role in explaining the behavior at infinity of such fluid flows. After a review of the fundamental solutions of several differential operators, we obtain a general expression for the Oseen fundamental solution without making any restrictions on the magnitude and orientation of $\zeta$. The existence of a wake region and the anisotropic behavior of the fluid flow are then explained by simplified expressions and precise estimates of the velocity component. Using a trivial extension of the velocity and pressure to the whole space and the framework of tempered distributions, we also deduce the integral representation and the asymptotic expansion of weak solutions of the Oseen exterior problem.