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Comments on “unsteady flow of an inelastic power-law fluid in a circular tube”, by R.T. Balmer and M.A. Fiorina
373
Journal of Non-Newtonian Fluid Mechanics, 10 (1982) 373-375 Elsevier Scientific Publishing Company, Amsterdam-Printed in The Netherlands
Short communication COMMENTS ON “UNSTEADY FLOW OF AN INELASTIC POWER-LAW FLUID IN A CIRCULAR TUBE”, BY R.T. BALMER AND M.A. FIORINA
L. CHOPLIN
* and M. DUPUIS
Chemical Engineering Department, L.uval University, Quebec, Qc, Cl K 7P4 (Cunudu) (Received
June 17. 1981)
In a recent paper, Balmer and Fiorina [l] solved the momentum equations for the flow of an inelastic power-law fluid in a circular tube submitted to an oscillating and/or step change in pressure gradient. An implicit finite difference technique using a predictor-corrector scheme was used. No data are presently available in the literature on inelastic power-law fluids with which their results could be compared. However, it is possible to compare the limiting case, i.e. when the power-law index, n, is equal to unity (Newtonian case), with the analytical solution when the fluid is subjected to a step change in pressure gradient. This problem, known as the start-up flow, has been solved first by Szymanski [2]. Its solution is well known and it is part of many textbooks (see for example ref. [3]). The purpose of this communication is to criticize the results presented in the publication of Balmer and Fiorina [l]. They claimed that their results, when compared to the analytical solution in the case of the start-up flow problem, agree within 1%. We show here that their results are inaccurate by approximately 15%. It is possible to read (with a reasonable precision) some of the calculated values directly from Figs. l-5 of ref. [ 13. For example, the values corresponding to the evolution of the dimensionless centerline velocity can be obtained from the intercept of their numerical velocity profile with the axis. Let US consider the case of Fig. 3 of ref. [ 11, corresponding to the start-up flow of a Newtonian fluid. As shown, the numerical solution of Balmer and Fiorina fits very well the analytical solution (Szymanski [2]). However, we * To whom the correspondence 0377-0257/82/0000-0000/$02.75
should be addressed. 0 1982 Elsevier Scientific
Publishing
Company
Fig. 1. Ratio of centerline velocity to steady state centerline velocity versus dimensionless Analytical solution, ref. [2,3]; . -. -. time. Newtonian case. Start-up flow problem. - - - Numerical solution (50 time increments); Numerical solution (100 time increments); n Values taken from ref. [I].
will show that the development of the velocity profile given by the Szymanat all with that given by Balmer and ski’s solution do not correspond Fiorina. Figure 1 represents the ratio of centerline velocity to steady state centertime, T, for the line velocity ( +cL /( +cL Atea+ State) versus the dimensionless start-up flow of a Newtonian fluid *. Reported in this figure, are the analytical solution [2,3], our numerical solutions and values of the numerical solution of Balmer and Fiorina, as a function of the dimensionless time. We can see that considerable deviations exist between the analytical solution and the solution of Balmer and Fiorina, although they claimed an agreement within 1%. In the worst case, the deviation is approximately 15%, which is absolutely inacceptable. We determined the evolution of the centerline velocity with time, using a numerical solution based on an alternating direction implicit method. Because the problem is unidirectional, an appropriate choice of the number of time increments gave us a satisfactory result. In Fig. 1, we show two results: using 50 time increments (maximum error: 3%) and using 100 time increments (maximum error: C 1%). Not only the results reported by Balmer and Fiorina [l] for the Newtonian case are erroneous, but also their results for the non-Newtonian cases with the power-law index (n) values of 0.25, 0.5, 1.5 and 2.5. We estimated * We use the same nomenclature as in ref. [I]. ~=pr/R*p radius of circular tube, p: density and t: time.
with ~1: Newtonian
viscosity;
R:
375
that the deviations were of the same order of magnitude as for the Newtonian case. We do not report here our calculated results since the deviations between our results and those of Balmer and Fiorina are of the same magnitude as shown in Fig. 1. Finally we solved numerically the start-up flow problem with a superimposed oscillating pressure gradient and our results showed the same differences when compared to those of Balmer and Fiofina. Since these problems are relatively important, we thought that it was necessary to point out these inaccuracies. Further work is done on these problems using more adequate rheological models than the power-law model, whose principal disadvantage consists in predicting an infinite viscosity at zero-shear rate. Acknowledgments
This work has been carried out during a graduate course in Applied Mathematics in our department. We wish to thank Professor P.J. Carreau for this thoughtful comments. References I R.T. Balmer and M.A. Fiorina, J. Non-Newtonian Fluid Mech., 7 (1980) 189-198. 2 P. Szymanski, J. Math. Pure Appl., Serie 9, 11 (1932) 67- 107. 3 R.B. Bird, W.E. Stewart and E.N. Lightfoot, Transport Phenomena, John Wiley and Sons, New York, 1960, pp. 126- 130.
Journal of Non-Newtonian Fluid Mechanics, 10 (1982) 373-375 Elsevier Scientific Publishing Company, Amsterdam-Printed in The Netherlands
Short communication COMMENTS ON “UNSTEADY FLOW OF AN INELASTIC POWER-LAW FLUID IN A CIRCULAR TUBE”, BY R.T. BALMER AND M.A. FIORINA
L. CHOPLIN
* and M. DUPUIS
Chemical Engineering Department, L.uval University, Quebec, Qc, Cl K 7P4 (Cunudu) (Received
June 17. 1981)
In a recent paper, Balmer and Fiorina [l] solved the momentum equations for the flow of an inelastic power-law fluid in a circular tube submitted to an oscillating and/or step change in pressure gradient. An implicit finite difference technique using a predictor-corrector scheme was used. No data are presently available in the literature on inelastic power-law fluids with which their results could be compared. However, it is possible to compare the limiting case, i.e. when the power-law index, n, is equal to unity (Newtonian case), with the analytical solution when the fluid is subjected to a step change in pressure gradient. This problem, known as the start-up flow, has been solved first by Szymanski [2]. Its solution is well known and it is part of many textbooks (see for example ref. [3]). The purpose of this communication is to criticize the results presented in the publication of Balmer and Fiorina [l]. They claimed that their results, when compared to the analytical solution in the case of the start-up flow problem, agree within 1%. We show here that their results are inaccurate by approximately 15%. It is possible to read (with a reasonable precision) some of the calculated values directly from Figs. l-5 of ref. [ 13. For example, the values corresponding to the evolution of the dimensionless centerline velocity can be obtained from the intercept of their numerical velocity profile with the axis. Let US consider the case of Fig. 3 of ref. [ 11, corresponding to the start-up flow of a Newtonian fluid. As shown, the numerical solution of Balmer and Fiorina fits very well the analytical solution (Szymanski [2]). However, we * To whom the correspondence 0377-0257/82/0000-0000/$02.75
should be addressed. 0 1982 Elsevier Scientific
Publishing
Company
Fig. 1. Ratio of centerline velocity to steady state centerline velocity versus dimensionless Analytical solution, ref. [2,3]; . -. -. time. Newtonian case. Start-up flow problem. - - - Numerical solution (50 time increments); Numerical solution (100 time increments); n Values taken from ref. [I].
will show that the development of the velocity profile given by the Szymanat all with that given by Balmer and ski’s solution do not correspond Fiorina. Figure 1 represents the ratio of centerline velocity to steady state centertime, T, for the line velocity ( +cL /( +cL Atea+ State) versus the dimensionless start-up flow of a Newtonian fluid *. Reported in this figure, are the analytical solution [2,3], our numerical solutions and values of the numerical solution of Balmer and Fiorina, as a function of the dimensionless time. We can see that considerable deviations exist between the analytical solution and the solution of Balmer and Fiorina, although they claimed an agreement within 1%. In the worst case, the deviation is approximately 15%, which is absolutely inacceptable. We determined the evolution of the centerline velocity with time, using a numerical solution based on an alternating direction implicit method. Because the problem is unidirectional, an appropriate choice of the number of time increments gave us a satisfactory result. In Fig. 1, we show two results: using 50 time increments (maximum error: 3%) and using 100 time increments (maximum error: C 1%). Not only the results reported by Balmer and Fiorina [l] for the Newtonian case are erroneous, but also their results for the non-Newtonian cases with the power-law index (n) values of 0.25, 0.5, 1.5 and 2.5. We estimated * We use the same nomenclature as in ref. [I]. ~=pr/R*p radius of circular tube, p: density and t: time.
with ~1: Newtonian
viscosity;
R:
375
that the deviations were of the same order of magnitude as for the Newtonian case. We do not report here our calculated results since the deviations between our results and those of Balmer and Fiorina are of the same magnitude as shown in Fig. 1. Finally we solved numerically the start-up flow problem with a superimposed oscillating pressure gradient and our results showed the same differences when compared to those of Balmer and Fiofina. Since these problems are relatively important, we thought that it was necessary to point out these inaccuracies. Further work is done on these problems using more adequate rheological models than the power-law model, whose principal disadvantage consists in predicting an infinite viscosity at zero-shear rate. Acknowledgments
This work has been carried out during a graduate course in Applied Mathematics in our department. We wish to thank Professor P.J. Carreau for this thoughtful comments. References I R.T. Balmer and M.A. Fiorina, J. Non-Newtonian Fluid Mech., 7 (1980) 189-198. 2 P. Szymanski, J. Math. Pure Appl., Serie 9, 11 (1932) 67- 107. 3 R.B. Bird, W.E. Stewart and E.N. Lightfoot, Transport Phenomena, John Wiley and Sons, New York, 1960, pp. 126- 130.