Analytical solution of the low Reynolds third-grade non-Newtonian fluids flow inside rough circular pipes

Abstract

The present paper focuses on finding an analytical solution for fully developed third-grade non-Newtonian fluids flows inside rough circular pipes at low Reynolds numbers (Stokes flows). The wall roughness is modeled by two different periodic morphologies based on sinusoidal and triangular geometries. In this study, the relative roughness (ratio of the roughness amplitude to the pipe hydraulic diameter) is selected to be a small value, which is appropriate for the perturbation analysis. The governing parameters including the axial and radial velocity profiles, stream function, wall shear stress, pressure gradient, and friction factor are expressed in analytical formulas and they are compared to the smooth pipe. The effect of the relative roughness, the wall wave number, and the non-Newtonian parameter on the governing parameters are investigated. The results show that modeling the roughness by triangular geometry has a better prediction of pressure drop regarding the basic solution of the smooth pipe.