Asymptotic solutions of the flow of a Johnson-Segalman fluid through a slowly varying pipe using double perturbation strategy

A double perturbation strategy is presented to solve the asymptotic solutions of a Johnson-Segalman （J-S） fluid through a slowly varying pipe. First, a small parameter of the slowly varying angle is taken as the small perturbation parameter, and then the second-order asymptotic solution of the flow of a Newtonian fluid through a slowly varying pipe is obtained in the first perturbation strategy. Second, the viscoelastic parameter is selected as the small perturbation parameter in the second perturbation strategy to solve the asymptotic solution of the flow of a J-S fluid through a slowly varying pipe. Finally, the parameter effects, including the axial distance, the slowly varying angle, and the Reynolds number, on the velocity distributions are analyzed. The results show that the increases in both the axial distance and the slowly varying angle make the axial velocity slow down. However, the radial velocity increases with the slowly varying angle, and decreases with the axial distance. There are two special positions in the distribution curves of the axial velocity and the radial velocity with different Reynolds numbers, and there are different trends on both sides of the special positions. The double perturbation strategy is applicable to such problems with the flow of a non-Newtonian fluid through a slowly varying pipe.