Analytical solutions for the flow of Carreau and Cross fluids in circular pipes and thin slits

TL;DR

This paper derives analytical formulas for the flow of Carreau and Cross fluids in pipes and slits, validating them against numerical and variational solutions, thus advancing understanding of non-Newtonian fluid dynamics in confined geometries.

Contribution

It presents new analytical expressions for non-Newtonian fluid flow in pipes and slits, validated against numerical and variational methods, enhancing modeling accuracy.

Findings

Analytical solutions match numerical and variational results well.

Validation confirms the effectiveness of the derived expressions.

Supports the use of the variational method for non-Newtonian flow analysis.

Abstract

In this paper, analytical expressions correlating the volumetric flow rate to the pressure drop are derived for the flow of Carreau and Cross fluids through straight rigid circular uniform pipes and long thin slits. The derivation is based on the application of Weissenberg-Rabinowitsch-Mooney-Schofield method to obtain flow solutions for generalized Newtonian fluids through pipes and our adaptation of this method to the flow through slits. The derived expressions are validated by comparing their solutions to the solutions obtained from direct numerical integration. They are also validated by comparison to the solutions obtained from the variational method which we proposed previously. In all the investigated cases, the three methods agree very well. The agreement with the variational method also lends more support to this method and to the variational principle which the method is based upon.