 Pressure Poisson equation for incompressible flow

Taking the divergence of the constant-density Navier-Stokes equation and using incompressibility gives the displayed elliptic Poisson equation for pressure. Its Green function is nonlocal. For an isolated eddy, the far pressure has a quadrupolar contribution proportional to $r^{-3}$, obtained by differentiating the kernel $1/r$ twice. Boundary conditions can contribute additional pressure fields.