Pressure Poisson equation for incompressible flow (${\nabla }^2(p/\rho )=-{\partial }_i{\partial }_j(u_i{u}_j)$)

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.

The quadrupolar kernel of the pressure Poisson equation for incompressible flow allows mixed pressure-velocity correlations with an $r^{-3}$ tail. Its derivative in the evolution of a longitudinal triple correlation allows an $r^{-4}$ tail, potentially changing the Loitsyansky integral. The existence of a nonlocal kernel does not fix the tail coefficient or prevent cancellations. An isotropic mean pressure multipole can vanish while a mixed higher-order correlation remains nonzero.