 Solution

For the common constant mass density, the background hydrostatic pressure satisfies $d{p}_0/dz=-\rho g$. Define the excess pressure $P=p-p_0(z)$. In Darcy's law, the background pressure gradient cancels gravity:If the source imposes the same excess pressure relative to the local hydrostatic pressure everywhere on its spherical surface, $P(a)=\Delta p$, the spherical Darcy injection result survives exactly:It is important to specify what is maintained on the source. A literally uniform absolute pressure $p_s=p_0(0)+\Delta p$ on a finite sphere centered at $z=0$ instead gives $P(a,\theta )=\Delta p+\rho ga\cos \theta$, where $\theta$ is measured from the upward vertical. The decaying harmonic function outside the sphere is thenand thereforeThis supplies a concrete exception to an unqualified claim that gravity leaves the radial flow unchanged. The additional dipolar contribution has zero net volumetric flow rate, so $Q=4\pi {\Pi }_{1}a\Delta p/\eta$ still holds. Equal injected and resident mass densities are essential to the background cancellation.