 Solution

For a steady unidirectional field $\mathbf{u}=w\mathbf{k}$, incompressibility gives ${\partial }_{z}w=0$. Thus $w=w(x,y)$ and the convective acceleration vanishes. The transverse momentum equations give $p_x=p_y=0$, so $p=p(z)$. The axial equation isIts left side is independent of $z$ and its right side independent of $x,y$, so both equal a constant $G$. No slip against the translating wall gives $w=U$ on the boundary. Thus the asserted Poisson problem follows.

For the circular pipe, the axisymmetric solution isIt satisfies the boundary value and ${\Delta }_{\perp }{r}^2=4$. Integrating over the cross-section givesTherefore the laboratory-frame volume flux vanishes precisely whenFor a pressure decrease in the positive axial direction, this requires upstream wall translation.