Variational approach for resolving the flow of generalized Newtonian fluids in circular pipes and plane slits

TL;DR

This paper introduces a new variational method, independent of previous approaches, to solve the flow of generalized Newtonian fluids in pipes and slits, validated against traditional methods and supporting variational principles.

Contribution

A novel, general variational approach based on the Dirichlet principle for analyzing generalized Newtonian fluid flow, independent of Euler-Lagrange methods.

Findings

Results agree with traditional fluid mechanics methods

Method applicable to nine types of fluids

Supports the use of variational principles in fluid flow analysis

Abstract

In this paper, we use a generic and general variational method to obtain solutions to the flow of generalized Newtonian fluids through circular pipes and plane slits. The new method is not based on the use of the Euler-Lagrange variational principle and hence it is totally independent of our previous approach which is based on this principle. Instead, the method applies a very generic and general optimization approach which can be justified by the Dirichlet principle although this is not the only possible theoretical justification. The results that were obtained from the new method using nine types of fluid are in total agreement, within certain restrictions, with the results obtained from the traditional methods of fluid mechanics as well as the results obtained from the previous variational approach. In addition to being a useful method in its own for resolving the flow field in circular pipes and plane slits, the new variational method lends more support to the old variational method as well as for the use of variational principles in general to resolve the flow of generalized Newtonian fluids and obtain all the quantities of the flow field which include shear stress, local viscosity, rate of strain, speed profile and volumetric flow rate. The theoretical basis of the new variational method, which rests on the use of the Dirichlet principle, also provides theoretical support to the former variational method.