= Rectangular duct flow
{title2=$\nabla_\perp^2w=-G/\mu$}
<Rectangular duct flow> is <steady flow> of a <Newtonian fluid> down a straight rectangular pipe under a constant axial <pressure gradient> $-G$. For stationary walls and <unidirectional flow>, <incompressibility> eliminates axial dependence and the <Navier-Stokes equation> becomes a two-dimensional <Poisson equation> with zero <Dirichlet boundary conditions>. Subtracting a <plane Poiseuille flow> profile leaves a <harmonic function> that repairs the side-wall values using <separation of variables>. Uniqueness follows from <Green's first identity>: the difference of two solutions has zero boundary values and zero <integral> of the squared <gradient>, and hence vanishes.
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