Kinematics of a power-law fluid flow in a pipe with a varying cross section

Fluid transportation through channels with varying cross-section is widely spread in a number of technical applications. This circumstance determines constant interest of researchers to study such flows.
This paper presents an investigation of the steady-state axisymmetric flow of an incompressible power-law fluid in a pipe of varying cross section with contraction followed by expansion. The mathematical formulation of the problem is developed using the equations in a cylindrical coordinate system in terms of vortex-stream function variables. Rheological behavior of the considered medium is described by the Ostwald-de Waele power-law model. To implement the numerical algorithm, a coordinate transformation is carried out. The problem is solved using the finite-difference method. An asymptotic time solution of the unsteady flow equation is applied to obtain steady-state fields of the vortex and stream function in the computational domain. To verify the developed numerical algorithm, an approximation convergence is examined on the sequence of square grids.
The calculations of the flow of three rheological media (Newtonian, pseudoplastic, and dilatant) are carried out. A parametric study is performed to reveal the effect of the Reynolds number and power-law index on the flow structure. The shear stress distributions on the wall are demonstrated at various geometric parameters.