# On the Accuracy of Finite Element and Finite Difference Predictions of Non-Newtonian Slot Pressures for a Maxwell Fluid

## Abstract

Plane slow flow of a Maxwell fluid over a transverse slot is considered. Results are computed by a finite difference method (FDM) using the differential form of the constitutive equation, and by a finite element method (FEM) using the integral form. Even on fine grids, the two methods produce different results, particularly at low *D* _{ e }. However, extrapolation of the results to zero mesh spacing shows excellent agreement between the two methods. Hence both methods are convergent with mesh refinement, but high accuracy would require extremely fine meshes. An explanation is provided for why it is unreasonable to expect either method accurately to obtain the singular limit of *P* _{ e }/*N* _{1} as *D* _{ e } → 0. Also an explanation for the errors at very low *D* _{ e } is offered. If we presume the the second-order fluid (SOE) result holds for very low *D* _{ e } (i. e. *P* _{ e } = *N* _{1}/4), both the FEM and FDM predict only minor deviation from this value for the Maxwell fluid, in the range 0 ≤ *D* _{ e } ≤ 1.

## Keywords

Error Model Mesh Refinement Finite Difference Method Discretization Error Error Indicator## Preview

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## References

- [1].M. F. Webster, The hole pressure problem, Rheologica Acta 23 (1984), 582–590.zbMATHCrossRefGoogle Scholar
- [2].A. C. Pipkin and R. I. Tanner, A survey of theory and experiment in viscometric flows of viscoelastic liquids, in
**Mechanics Today**, Vol. I, S. Nemat-Nasser, ed. Pergamon Press, 1972, 262–321.Google Scholar - [3].K. Higashitani and W. G. Pritchard, A kinematic calculation of intrinsic errors in pressure measurements made with holes, Trans. Soc. Rheology 16 (1972), 687–696.ADSCrossRefGoogle Scholar
- [4].A. S. Lodge and L. de Vargas, Positive hole-pressures and negative exit pressure generated by molten low-density polyethylene flowing through a slit die, Rheologica Acta 22 (1983), 151.CrossRefGoogle Scholar
- [5].T. Cochrane, K. Walters, and M. F. Webster, On Newtonian and non-Newtonian flow in complex geometries, Phil. Trans. Roy. Soc. London A301 (1982), 163–181.ADSGoogle Scholar
- [6].R. D. Pike and D. G. Baird, Evaluation of the Higashitani and Pritchard analysis of the hole pressure using flow birefringence, J. Non-Newtonian Fluid Mech. 16 (1984), 211.CrossRefGoogle Scholar
- [7].G. D. Richards and P. Townsend, A finite element computer model of the hole-pressure problem, Rheologica Acta 20 (1981), 261–269.CrossRefGoogle Scholar
- [8].D. S. Malkus and B. Bernstein, Flow of a Curtiss-Bird fluid over a transverse slot using the finite element drift-function method, J. Non-Newtonian Fluid Mech. 16 (1984), 77–116.zbMATHCrossRefGoogle Scholar
- [9].E. Salathe, Unsteady flows of a third-order viscoelastic fluid, Rheologica Acta 19 (1980), 574–579.zbMATHCrossRefGoogle Scholar
- [10].D. S. Malkus, Finite element methods for viscoelastic flow, in
**Viscoelasticity and Rheology**, A. S. Lodge, M. Renardy, and J. A. Nohel, eds. Academic Press, Orlando, 1985, 391–419.Google Scholar - [11].D. S. Malkus and Minwu Yao, On hole pressures in plane Poiseuille flow over transverse slots, Mathematics Research Center Technical Summary Report #2943, Mathematics Research Center, University of Wisconsin-Madison (1986).Google Scholar
- [12].R. Keunings, On the high Weissenberg number problem, J. Non-Newtonian Fluid Mech., to appear.Google Scholar
- [13].R. I. Tanner, The stability of some numerical schemes for model viscoelastic fluids, J. Non-Newtonian Fluid Mech. 10 (1982), 169–174.CrossRefGoogle Scholar
- [14].A. R. Davies, The high Weissenberg number problem, J. Non-Newtonian Fluid Mech. 16 (1984), 195–209.zbMATHCrossRefGoogle Scholar
- [15].M. F. Webster, E. E. Süli, and K. W. Morton, A numerical study of a non-Newtonian flow problem, Numerical Analysis Report 8/86, Reading University.Google Scholar
- [16].R. B. Bird, O. Hassager, R. C. Armstrong (Vols. I and II), and C. F. Curtiss (Vol. II only),
**Dynamics of Polymeric Liquids**, Wiley, New York, 1977.Google Scholar - [17].B. Bernstein, D. S. Malkus, and E. T. Olsen, A finite element for incompressible plane flows of fluids with memory, Int. J. Num. Meths. Fluids 5 (1985), 43–70.MathSciNetzbMATHCrossRefGoogle Scholar
- [18].P. G. Ciarlet,
**The Finite Element Method for Elliptic Problems**, North-Holland, New York, 1980.Google Scholar - [19].H. Holstein, Jr. and D. J. Paddon, A singular finite difference treatment of re-entrant corner flow. Part I: Newtonian fluids, J. non-Newtonian Fluid Mech. 8 (1981), 81–93.zbMATHCrossRefGoogle Scholar
- [20].S. D. Conte and C. de Boor,
**Elementary Numerical Analysis**, Second Edition, McGraw-Hill, New York, 1972.zbMATHGoogle Scholar - [21].D. S. Malkus and M. F. Webster, Numerical verification of an empiri0063al measurement relation for non-Newtonian slot pressures, Mathematics Research Center Technical Summary Report, to appear.Google Scholar