On the Representation of Stokes Flows
Werner Kratz · SIAM Journal on Mathematical Analysis · 1991
In this paper representations of Stokes flows in dimensions 2 and 3, which reduce the Stokes equations to the Laplace equation for an auxiliary function, are given. While it is known that two- and three-dimensional Stokes flows may be reduced to biharmonic problems, the representations here are new. The main result of this paper reads as follows: Given a domain $G \subseteq \mathbb{R}^3 $, which is star-shaped with respect to the origin, and functions $\vec v:G \to \mathbb{R}^3 $, $p:G \to \mathbb{R}$, then $\vec v$ and p represent a Stokes flow with velocity field $\vec v$ and pressure p in G (i.e., $\Delta \vec v = {\operatorname{grad}}p$ and ${\operatorname{div}}\vec v = 0$ in G) if and only if $\vec v$ and p are of the form \[ \vec v(\vec x) = \tilde \vec v(\vec x) - \frac{1}{3}\left\{ {{\operatorname{div}}\tilde \vec v(\vec x) \cdot \vec x + \vec x \times {\operatorname{curl}}\tilde \vec v(\vec x)} \right\},\quad p(\vec x) = - \frac{4}{3}{\operatorname{div}}\tilde \vec v(\vec x),\] where $\tilde \vec v$ is a harmonic function (i.e., $\Delta \tilde \vec v = 0$) in G.