Unsteady flow of an inelastic power-law fluid in a circular tube

377

Fluid Mechanics, 10 (1982) 377-379 Elsevier Scientific Publishing Company, Amsterdam-Printed in The Netherlands

Journal of Non-Newtonian

Short communication UNSTEADY ELOW OF AN INELASTIC A CIRCULAR TUBE

AUTHORS’ DUPUIS

REPLY

TO

COMMENTS

POWER-LAW

BY L.

FLUID

CHOPLIN

IN

AND

M.

R.T. BALMER and M.A. FIORINA Rheology L&oratory College of Engineering und Applied Science, The University of WisconsinMilwuukee, Milwaukee, Wisconsin 53201 (U.S.A.)

(Received October 25, 1981)

We acknowledge the fact that our 1980 article in Journal of NonNewtonian Fluid Mechanics (Vol. 7, pp. 189- 194) contained calculational errors and we are grateful to Choplin and Dupuis [l] for bringing this to our attention. Upon reviewing our records of this work we found that the error occurred when we made the final computer run to generate the graphics for Figs. l-7. At this point the dimensionless print out time step and the calculational time step were inadvertently set equal to each other. This meant that the calculational time step was 0.1 which was much too large for accurate calculations at the small T values in Figs. l-5 and Fig. 7. Figure6 is accurate as printed because it represents calculational results at very large r values wherein a calculational time step of 0.1 yields sufficient accuracy. Figures 8 and 9 on pulsatile flow were worked up separately from the other figures and all the results contained therein had time steps of 0.008 or less. Therefore these two figures should be without significant error. During the early development of this work we had studied the effect of time step on calculational accuracy using time steps of 0.1, 0.01 and 0.001 (Mr. Fiorina’s thesis contains the details of this study). We had reached the same conclusions as drawn by Choplin and Dupuis, i.e. that a time step of 0.01 generally results in errors less than about 1% for the Newtonian case when compared to the analytical solution of Szymanski. The apparent

378

DIMENSIONLESS VELOCITY,@

DIMENSIONLESS VELOCITY ‘4’

Fig. 1. Start-up

velocity profile development

for a fluid with a power-law

index n=O.25.

Fig. 2. Start-up

velocity profile development

for a fluid with a power-law

index n=O.S.

agreement between numerical and analytical solutions for n = 1.0 shown in our original Fig. 3 was due to a presumed equivalence based on our earlier test runs with smaller time steps. It is incorrect as shown in our original article. We include here a re-run of Figs. l-5 with a computational time step of 0.01. These figures do not include curves for 7 = 0.1 because errors up to

Fig. 3. Start-up

velocity profile development

for a fluid with a power-law

index PI= 1.0.

Fig. 4. Start-up

velocity profile development

for a fluid with a power-law

index n = 1S.

379 about 6% were seen there for the case of n = 1.0. Errors were about 1% or less for 730.2. We wish to emphasize that to our knowledge our article contained no

0

1

2

3

I

5

I

7

8

9

10

DIMENSIONLESS VELOCITY, @

Fig. 5. Start-up velocity profile development

fdr a fluid with a power-law index n=2.5.

error in theory or technique. The error pointed out by Choplin and Dupuis was one of numerical execution only. We apologize for any inconvenience this may have caused. Reference 1 L. Choplin and M. Dupuis, J. Non-Newtonian

Fluid Mech., 10 (1981) 373-375.