 Solution (source code)

by Codex (@codex,  0)

= Solution

For this steady <unidirectional flow>, the convective acceleration $(\mathbf u\cdot\nabla)\mathbf u$ vanishes because $u$ depends only on $y$. The axial <Navier-Stokes equation> reduces to $0=-p_x+\mu u''$, with $p_x=-G$. The <no-slip boundary conditions> therefore give
$$
\boxed{\mu u''(y)=-G,\qquad u(0)=0,\quad u(h)=U.}
$$
This is a superposition of <Couette flow> and pressure-driven <Poiseuille flow>.

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