= Solution
The only <velocity> component is axial, so <incompressibility> gives $\partial_zw=0$. Thus $w=w(x,y)$, and the steady convective <acceleration> is $w\partial_z(w\mathbf k)=0$. The transverse <momentum> equations give $p_x=p_y=0$, so $p=p(z)$. The axial <momentum> equation is therefore
$$
\boxed{\mu(w_{xx}+w_{yy})=p'(z).}
$$
Its left side depends only on $x,y$, while its right side depends only on $z$. Equality throughout the pipe <forces> both to be one constant, say $G$. Consequently \b[the axial <pressure gradient> is constant], and $p(z)=Gz+p_*$. This is absence of dependence on a coordinate, rather than probabilistic independence.
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