Stokeslet

Definition

Let $\mu > 0$ be the dynamic viscosity and let $F \in \R^3$ be a vector, the point force. For $x \in \R^3 \setminus \set{0}$ write $r = \abs{x}$. The Stokeslet with force $F$ at the origin is the pair $(u, p)$ given by $$u(x) = \frac{1}{8 \pi \mu} \Big( \frac{F}{r} + \frac{(F \cdot x) \, x}{r^3} \Big), \qquad p(x) = \frac{F \cdot x}{4 \pi r^3}.$$ Let $\delta_0$ be the Dirac delta at the origin. The Stokeslet is the solution, decaying at infinity, of the Stokes equations with a point force in the sense of distributions: $$-\mu \Delta u + \nabla p = F \, \delta_0, \qquad \nabla \cdot u = 0 \quad \text{in } \R^3.$$ The matrix $G_{ij}(x) = \frac{1}{8 \pi \mu} \big( \frac{\delta_{ij}}{r} + \frac{x_i x_j}{r^3} \big)$ with $u = G F$ is the Oseen tensor.