Axial Laminar Flow of Non-Newtonian Fluids in Narrow Eccentric Annuli

The analysis of laminar flow of power-law non- Newtonian fluids in narrow, eccentric annuli is employed in this paper to discuss the problems of lubricant flow in journal bearings and of errors introduced by eccentricity in experimental studies with concentric annuli on extruders and wellbore annuli. The velocity profile and pressure loss-flow rate equations are developed for the laminar flow region. In addition, the expected error in flow rate and pressure-loss measurements for concentric annuli as a result of eccentricity is determined. For example, a 10 per cent displacement of the core of an almost concentric annulus would cause a 1.8 per cent decrease in the observed pressure loss for a fluid with a power-law exponent n of 0.25. The corresponding increase in the observed volumetric flow rate would be 7.5 per cent. Introduction Non-Newtonianism and eccentricity occur simultaneously in two engineering problems:flow of lubricants in journal-bearings and pressure-reducing bushings, andflow of non-Newtonian fluids in plastic extruders and wellbore annuli. The lubricants used for moving parts are often non-Newtonian in character - often they are plastic in behavior. A solution to the problem of flow of non-Newtonian fluids in narrow eccentric annuli is particularly pertinent to this problem. In all experimental studies of laminar flow of fluids in concentric annuli, such as in extruders and well casings, the error due to eccentricity must be estimated or studied. A number of publications have dealt with this problem for Newtonian fluids; however, I am not aware of work for non-Newtonian fluids. This work is directed to the non-Newtonian problem. Before the solution to the problem is given, the pertinent conclusions from the work on Newtonian fluids will be reviewed. Heyda and Redberger and Charles have published general solutions to the problem of the laminar flow of Newtonian fluids in eccentric annuli, apparently without knowing of the earlier work of Caldwell and Bairstow and Berry, which is reported by Dryden, et al. Although several mathematical routes are encompassed by the work of these authors, the results appear to be equivalent. Redberger and Charles show that the error caused by eccentricity in concentric annuli is negligible for small diameter ratios (K less than 0.5); however, for large diameter ratios (K - 1), the error in the predicted flow rate can be as great as 100 per cent or more. Partial solutions to the problem are available from the work of Dryden, Tao and Donovan and Piercy, et al. Tao and Donovan examined the case of flow in narrow, eccentric annuli (K - 1) with and without rotation of the annular core. These authors also reviewed previous work on this subject and verified their approach with experimental data. Dryden gives the solution for the limiting case of complete eccentricity or tangency. Piercy, et al. published an early solution to the problem of narrow eccentric annular flow. The conclusions of Redberger and Charles and the experimental proof of Tao and Donovans both suggest that the region of large diameter ratios (K - 1) is of main interest and that the parallel planes approximation to the solution in this region is satisfactory. This method will now be extended to the laminar flow of non-Newtonian fluids in narrow eccentric annuli. THEORETICAL SOLUTION The geometrical aspects of the problem are illustrated in Fig. 1. To represent the non-Newtonian fluid the power-law model was selected. (1) This model has many disadvantages which have been pointed out; nevertheless, As simplicity, its frequent and wide applicability justify its use in this work. Fredrickson and Birds and Savins have used it as a basis for a theoretical study of laminar flow of non-Newtonian fluids in concentric annuli. SPEJ P. 277ˆ