Equilateral triangular pipe flow ($w=\frac{p_z}{4a\mu }(x^2-3y^2)(x-a)$)

For a straight pipe with cross-section $0<x<a$, $|y|<x/\sqrt{3}$, steady unidirectional incompressible flow reduces the Navier-Stokes equations to $\mu {\nabla }_{\perp }^{2}w=p_z$, with constant axial pressure gradient. The displayed cubic polynomial vanishes on every wall, and its transverse Laplacian is $p_z/\mu$. The axial drag per unit length on each of the three equal walls is $-p_z{a}^2/(3\sqrt{3})$. This follows by integrating the viscous traction on one wall; summing the three forces equals the pressure gradient times the triangular area.