Mathematical modeling of Stokes flow in petal shaped pipes

A theoretical model is developed to characterize fully developed laminar flow (Stokes flow) in idealized petal shaped pipes by regarding the pipe wall as a circular boundary having sinusoidal perturbation (or, equivalently, surface roughness). Built upon the method of perturbation, the model quantifies the effects of the relative roughness and wave number of the pipe boundary on the Stokes flow. Approximate solutions of the velocity field, pressure gradient, and static flow resistivity are obtained. The same approach together with the method of Fourier transform is used to deal with low Reynolds flow in pipes having other cross-sectional morphologies such as triangle and square. Results obtained from computational fluid dynamics simulations are used to validate the theoretical model predictions, with good agreement achieved. The presence of surface roughness causes periodic fluctuation and global offset of velocity distribution and enlarges both the pressure gradient and static flow resistivity in petal shaped pipes.