The structure and kinematics of a non-Newtonian fluid flow in a pipe with a sudden expansion

This paper deals with a problem of a laminar steady-state flow of a non-Newtonian incompressible fluid in a pipe with a sudden expansion. The flow is described by a system of dimensionless equations in terms of stream function and vorticity in a cylindrical coordinate system: an equation of vorticity transfer and Poisson's equation for stream function. Rheological properties of the medium are defined by the Ostwald-de Waele model. The problem is solved numerically. The false transient method is applied to obtain a steady-state solution to the problem. The equations are discretized in accordance with the finite-difference method based on the alternating direction scheme. The final system of equations is solved by the tridiagonal matrix algorithm.
Flow structures of Newtonian, pseudoplastic, and dilatant fluids are found to include twodimensional flow zones before and after expansion plane. A recirculation region occurs in the inner corner. To assess the effect of the Reynolds number, expansion ratio, and power-law index on the lengths of the two-dimensional flow zones and recirculation region, the graphs are plotted over a wide range of variation in the parameters. Local pressure losses are presented as functions of the governing parameters of the problem.