Source dipole ($\mathbf{u}=[3(\mathbf{M}\cdot \mathbf{r})\mathbf{r}/r^5-\mathbf{M}/r^3]/(4\pi )$)

A source dipole is the limit of an equal source and sink brought together with their source-strength times separation vector fixed at $\mathbf{M}$. Differentiating the three-dimensional point source field gives $\mathbf{u}=-(\mathbf{M}\cdot \nabla )[\mathbf{r}/(4\pi r^3)]$. It has velocity potential $-\mathbf{M}\cdot \mathbf{r}/(4\pi r^3)$, zero total volume flux, and $r^{-3}$ far-field decay. Unlike a stresslet, it is an irrotational potential singularity rather than a force-dipole flow.