Poiseuille flow through parabolic segment and lens-shaped ducts

This work investigates the steady, laminar, two-dimensional Poiseuille flow of a Newtonian fluid through parabolic segment and lens-shaped ducts. The governing equations are solved using the finite element method. Using parabolic coordinates and the method of separation of variables, special analytical solutions are derived. For ducts with small aspect ratios, a systematic perturbation method is applied to obtain approximate solutions, while for ducts with large aspect ratios, the Maclaine–Cross formula is employed to determine the limiting value of the friction factor–Reynolds number product. The flow rate and the friction factor–Reynolds number product are computed for various aspect ratios. The numerical and analytical solutions show excellent agreement.