Comments on the solutions of boundary value problems in non-Newtonian fluid mechanics. (English) Zbl 0859.76006
Summary: The problem of the availability of boundary conditions in non-Newtonian fluid mechanics is briefly reviewed. The flow with suction over a flat plate of differential, integral, and hierarchy types of fluids is studied. It is concluded that while differential and integral models, with physically motivated conditions, give unique solutions, the hierarchy model gives rise to non-uniqueness. An alternate method which provides a better approximate solution for a hierarchy model is suggested.
MSC:
| 76A10 | Viscoelastic fluids |
References:
| [1] | Brennan, M., Jones, R. S., Walters, K.: Some mathematical problems arising in modern developments in non-Newtonian fluid mechanics. Lecture Notes in Physics249, 409-421 (1985). · Zbl 0602.76004 · doi:10.1007/BFb0016408 |
| [2] | Hassager, O., Armstrong, R. C., Bird, R. B.: Limitation on the use of the retarded motion expansion. J. Non-Newtonian Fluid Mech.34, 241-245 (1990). · doi:10.1016/0377-0257(90)80020-Z |
| [3] | Crochet, M. J., Davies, A. R., Walters, K.: Numerical simulation of non-Newtonian flow, p. 88. Amsterdam: Elsevier 1984. · Zbl 0583.76002 |
| [4] | Denn, M. M., Petrie, C. J. S., Avenas, P.: Mechanics of steady spinning of a viscoelastic liquid. A. I. Ch. E. J.21, 791-799 (1975). |
| [5] | Sarpkaya, T., Rainey, P. G.: Stagnation point flow of a second order viscoelastic fluid. Acta Mech.11, 237-246 (1971). · Zbl 0228.76007 · doi:10.1007/BF01176558 |
| [6] | Ji, Z., Rajagopal, K. R., Szeri, A. Z.: Multiplicity of solutions in von Karman flows of viscoelastic fluids. J. Non-Newtonian Fluid Mech.36, 1-25 (1990). · Zbl 0708.76009 · doi:10.1016/0377-0257(90)85001-F |
| [7] | Renardy, M.: Inflow boundary conditions for steady flows of vicoelastic fluids with differential constitutive equations; Errata. Rocky Mt. J. Math.18, 445-453;19, 561 (1989). · Zbl 0646.76009 · doi:10.1216/RMJ-1988-18-2-445 |
| [8] | Renardy, M.: An alternative approach to inflow boundary conditions for Maxwell fluids in three space dimensions. J. Non-Newtonian Fluid Mech.36, 419-425 (1990). · Zbl 0709.76014 · doi:10.1016/0377-0257(90)85022-Q |
| [9] | Frater, K. R.: On the solution of some boundary-value problems arising in elastico-viscous fluid mechanics. ZAMP21, 134-137 (1970). · Zbl 0185.54102 · doi:10.1007/BF01594990 |
| [10] | White, F. M.: Viscous fluid flow, p. 150. New York: McGraw-Hill 1974. · Zbl 0356.76003 |
| [11] | Lighthill, M. J.: A technique for rendering approximate solutions to physical problems uniformly valid. Phil. Mag.40, 1179-1201 (1949). · Zbl 0035.20504 |
| [12] | Coleman, B. D., Duffin, R. J., Mizel, V. J.: Instability, uniqueness, and non-existence theorems for the equationu t=u xx ?u xtx on a strip. Arch. Rat. Mech. Anal.19, 100-116 (1965). · Zbl 0292.35016 · doi:10.1007/BF00282277 |
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