Using the stress function in the flow of generalized Newtonian fluids through conduits with non-circular or multiply connected cross sections

TL;DR

This paper explores whether the spatial distribution of stress in generalized Newtonian fluids flowing through complex conduit geometries can be derived from the pressure field and conduit shape alone, enabling simplified analytical solutions.

Contribution

It extends the stress function approach to non-circular and multiply connected geometries, providing a pathway for analytical or semi-analytical solutions for complex flows.

Findings

Stress distribution depends on pressure and geometry, not fluid rheology.

Analytical solutions for stress, strain rate, and flow rate are feasible for complex geometries.

Method simplifies analysis of non-Newtonian flows in intricate conduit shapes.

Abstract

We investigate the possibility that the spatial dependency of stress in generalized Newtonian flow systems is a function of the applied pressure field and the conduit geometry but not of the fluid rheology. This possibility is well established for the case of a one-dimensional flow through simply connected regions, specifically tubes of circular uniform cross sections and plane thin slits. If it can also be established for the more general case of generalized Newtonian flow through non-circular or multiply connected geometries, such as the two-dimensional flow through conduits of rectangular or elliptical cross sections or the flow through annular circular pipes, then analytical or semi-analytical or highly accurate numerical solutions; regarding stress, rate of strain, velocity profile and volumetric flow rate; for these geometries can be obtained from the stress function, which can be easily obtained from the Newtonian case, in combination with the constitutive rheological relation for the particular non-Newtonian fluid, as done previously for the case of the one-dimensional flow through simply connected regions.