A Comparison of Differential and Integral Descriptions of the Annular Flow of a Power-Law Fluid
Authors: Wallick G.C. , Savins J.G. 
Published: December 2007

Abstract
Some physical processes may be described mathematically in both differentialand integral equation form. Formulation choice for numerical solution often isbased upon personal preference rather than upon problem characteristics. Wecompare differential and integral methods for the numerical description of thesteady-state flow of a non-Newtonian, power-law fluid through an annulus. Forthis application, our data indicate that the integral formulation is superiorboth in solution accuracy and computational efficiency. Our integral solutionmethod is a generalization of an earlier analytic solution that was restrictedto integer values of the power-law model parameter N. The new method ispower-law model parameter N. The new method is more directly applicable inpractical applications and is valid for all N, integer and non-integer.

Introduction
In many instances differential and integral equations may be used with equalvalidity for the mathematical description of a physical precess. The choice ofmethods often is dictated more by the past experience and predilection of theanalyst than past experience and predilection of the analyst than by the natureof the problem. Yet the efficiency and efficacy of the solution process may bestrongly dependent upon the problem formulation selected. As an example of thisprocedural dichotomy we will consider the numerical description of thesteady-state isothermal axial flow of an incompressible time independentnon-Newtonian fluid through the annular spacing between two fixed concentriccylinders of radii Ri and R, R greater than Ri.* We assume that the cylindersare infinite in length (no end effects) and that the flow is produced by theapplication of a constant pressure gradient in the axial z-direction. This flowproblem has been treated by a number of investigators, and has practicalapplication, e.g., flow of drilling fluids, extrusion of molten plastics, etc. Fredrickson and Bird have shown that, subject to the above assumptions, theflow equation may be written in the form
...........(1)
where J = -dp/dz= constant p represents the pressure, the radial coordinate, and = z pressure, the radial coordinate, and = z represents the shearingstress. We seek a solution of Eq. 1 subject to the adherence boundaryconditions
...........(2)
where v = vz is the axial flow velocity. For this flow problem it can beshown that
.............(3)
where is the shear-dependent viscosity function, and that the shear rate maybe expressed in the forms
..............(4)
The minus sign is used in Eq. 4 to insure that and always have the samesign, greater than 0. In principle, the flow problem outlined here may besolved for any non-Newtonian fluid for which the shear-dependent viscosityfunction can be established as a known analytic function of the rate of shearfrom an investigation of any of the viscometric flows. However, it isconvenient for our purpose to use the particular viscometric function
.............(5)
which is referred to as the power-law model. The parameters n and Kcharacterize the relationship between shear rate and shear stress for a powerlaw liquid. The parameter n is a measure of the departure from Newtonianbehavior. If n less than 1, the flow behavior is of the "shearthinning" type; if n greater than 1, it is of the "shearthickening" category.