Dimensionless non-Newtonian fluid mechanics

I present an alternative reasoning for the choice of characteristic quantities to be employed in the non-dimensionalization of the governing equations of non-Newtonian fluid flow problems. The usual non-dimensionalization procedure generates well known dimensionless groups such as the Reynolds number, Deborah or Weissenberg number, Carreau number, Bingham number, and capillary number. The groups that represent dimensionless rheological properties (e.g. Deborah number, Carreau number and Bingham number) involve flow quantities such as characteristic velocities or deformation rates. Consequently, for a fixed flowing material, the values of these groups change with the flow rate. In the alternative procedure, the resulting dimensionless rheological groups are actually dimensionless rheological properties, and thus remain fixed for a given flowing material. Moreover, each set of values of these dimensionless rheological properties defines a class of rheologically equivalent materials. The proposed non-dimensionalization procedure is physically more sound, and renders simpler both the application of dimensionless results to engineering situations and the comparisons between numerical and experimental results in scientific investigations.