Poiseuille flow through a semi-parabolic duct

We study steady, fully developed Poiseuille flow of a Newtonian fluid in semi-parabolic ducts. An exact solution is obtained in parabolic coordinates for a special case. For small aspect ratios, a matched asymptotic expansion captures the right-angle corner layer. High-order finite-element computations validate both analyses and reproduce the benchmark. Exact expressions are provided for the volumetric flow rate and the Poiseuille number. Additional results are presented for the mean wall shear stress, compactness, and geometrical correction factor, providing a broader hydraulic characterisation of this geometry of ducts, while also illustrating the close analogy between the present fluid-mechanics problem and corresponding formulations in solid mechanics.