Take the 2-minute tour ×  The Green's functions of Stokes flow
represent solutions of the continuity equation $\nabla\cdot {\bf
u}=0$ and the singularly forced Stokes equation $$-\nabla P+\mu
\nabla^2{\bf u}+{\bf g}\delta({\bf x-x_0})=0 \tag{*}$$  where
${\bf g}$ is an arbitrary constant, ${\bf x_0}$ is an arbitrary
point, and $\delta$ is the three-dimensional delta function.
Introducing the Green's function ${\bf G}$, we write the
solution of (*) in the form $$u_i({\bf
x})=\frac{1}{8\pi\mu}G_{ij}({\bf x,x_0})g_j$$  The following is
how to find $G_{ij}$ in the free-space infinite unbounded flow
from C. POZRIKIDIS's Boundary Integral and Singularity Methods
for Linearized Viscous Flow  Replace the delta function with $$
\delta(\hat{x})=-\frac{1}{4\pi}\nabla^2\left(\frac{1}{r}\right)
\tag{1} $$ where $r=|\hat{x}|$, $\hat{x}=x-x_0$. Recalling that
the pressure is a harmonic function, and balancing the
dimensions of the pressure term with those of the delta function
in (1), we set $$ P=-\frac{1}{4\pi}g\cdot\nabla