Poiseuille flow through a circular-segment duct

Steady, fully developed Poiseuille flow of an incompressible Newtonian fluid in a circular-segment duct is analysed. The problem is formulated in bipolar coordinates and the axial-velocity field is reduced to a Laplace problem for a harmonic correction. A Fourier cosine transform with respect to the unbounded coordinate yields an elementary boundary-value problem in the remaining variable, which is solved in transform space and inverted by residue calculus. This procedure leads to an explicit series representation for the axial velocity. The volumetric flow rate and the associated Poiseuille number are then obtained in a form suitable for computation, and closed-form expressions are derived for several opening angles. In addition, a $$6/6$$ 6 / 6 Padé approximant for the Poiseuille number as a function of the central angle is constructed, which reproduces the analytical results with high accuracy over the full range of circular-segment geometries.