Analytical solution for the viscous flow of small Reynolds numbers in rough pipes

In view of the viscous flow fields of small Reynolds numbers in rough circular tubes and petal circular tubes, the rough surface in the tube was considered as a smooth surface subjected to small disturbance. The perturbation method was used to expand the perturbation of fluid parameters under small disturbance. The boundary conditions with complex morphologies were expanded into the Taylor series, and the smooth boundary conditions were approximately obtained. Then the fluid mechanics equations were solved simultaneously to give the approximate solution of the pressure gradient under the premise of the 1 st-order perturbation expansion, and the static flow resistance and tortuosity of the pipeline were obtained. The results show that the fluid parameters determined with the modified perturbation method agree very well with those through numerical simulation, and the theoretical approximate solution of the flow field in the rough pipe is validated.