×

Generalized Gear’s method for computing the flow of a viscoelastic fluid. (English) Zbl 0895.76063

Summary: A fourth-order predictor-corrector method is developed for obtaining the numerical solution of a class of singular boundary value problems in which the highest derivative is further multiplied by a parameter which can take arbitrarily small values. First, the method is tested on a prototype linear differential equation. It is then used to compute the two-dimensional stagnation point flow of a viscoelastic fluid. Finally, the flow over a stretching sheet is computed for which an exact solution exists. The comparison of the results shows that the method gives highly accurate results for a moderately sized integration step.

MSC:

76M25 Other numerical methods (fluid mechanics) (MSC2010)
76A10 Viscoelastic fluids
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Beard, D. W.; Walters, K., Elastico-viscous boundary layer flows. I. Two-dimensional flow near a stagnation point, (Proc. Camb. Phil. Soc., 60 (1964)), 667-674 · Zbl 0123.41601
[2] Shrestha, G. M., Laminar elastico-viscous flow through channels with porous walls with different permeability, Appl. Sci. Res., 20, 289-305 (1969)
[3] Mishra, S. P.; Mohapatra, U., Elasticoviscous flow between a rotating and a stationary disk with uniform suction at the stationary disk, J. Appl. Phys., 48, 1515-1521 (1977)
[4] Rajagopal, K. R.; Na, T. Y.; Gupta, A. S., Flow of a viscoelastic fluid over a stretching sheet, Rheol. Acta, 23, 213-215 (1984)
[5] Verma, P. D.; Sharma, P. R.; Ariel, P. D., Applying quasilinearization to the problem of steady laminar flow of a second grade fluid between two rotating porous disks, J. Tribology, Trans. ASME, 106, 448-455 (1984)
[6] Ariel, P. D., A hybrid method for computing the flow of visco-elastic fluids, Int. J. Numer. Methods Fluids, 14, 757-774 (1992) · Zbl 0753.76111
[7] Ariel, P. D., Computation of flow of viscoelastic fluids by parameter differentiation, Int. J. Numer. Methods Fluids, 15, 1295-1312 (1992) · Zbl 0825.76541
[8] Ariel, P. D., Flow of viscoelastic fluids through a porous channel—I, Int. J. Numer. Methods Fluids, 17, 605-633 (1993) · Zbl 0792.76004
[9] Gear, C. W., Numerical Initial Value Problems in Ordinary Differential Equations (1971), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0217.21701
[10] Andrews, L. C., Special Functions for Engineers and Applied Mathematicians (1985), MacMillan Publishing Company: MacMillan Publishing Company New York
[11] Stoer, J.; Bulrisch, R., Introduction to Numerical Analysis (1980), Springer-Verlag: Springer-Verlag New York
[12] Serth, R. W., Solution of a viscoelastic boundary layer equation by orthogonal collocation, J. Engrg. Math., 8, 89-92 (1974) · Zbl 0277.76002
[13] Crane, L. J., Flow past a stretching sheet, ZAMP, 21, 645-647 (1970)
[14] Troy, W. C.; Overmann, E. A.; Eremont-Rout, G. B.; Keener, J. P., Uniqueness of flow of second order fluid past a stretching sheet, Quart. Appl. Math., 44, 753-755 (1987) · Zbl 0613.76006
[15] McLeod, J. B.; Rajagopal, K. R., On the uniqueness of flow of a Navier-Stokes fluid due to stretching boundary, Arch. Rat. Mech. Anal., 98, 385-393 (1987) · Zbl 0631.76021
[16] Chang, Wen-Dong, The nonuniqueness of the flow of a viscoelastic fluid over a stretching sheet, Quart. Appl. Math., 47, 365-366 (1989) · Zbl 0683.76012
[17] Ariel, P. D., On the second solution of flow of viscoelastic fluid over a stretching sheet, Quart. Appl. Math., 53, 629-632 (1995) · Zbl 0841.76006
[18] Ariel, P. D., Stagnation point flow—A free boundary value problem formulation, Int. J. Comput. Math., 49, 123-131 (1993) · Zbl 0798.76015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.