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

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 uniform plane slits. The derivation is based on the application of Weissenberg-Rabinowitsch-Mooney-Schofield (WRMS) 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. We also compared the derived analytical solutions of Carreau and Cross fluids to the analytical solutions of power law fluids with comparable rheology and observed logical trends in the relation between the corresponding flow rates as a function of the applied pressure field.