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Symmetry reductions of unsteady three-dimensional boundary layers of some non-Newtonian fluids. (English) Zbl 0904.76006
Summary: Three-dimensional, unsteady, laminar boundary layer equations of a general model of non-Newtonian fluids are treated. In this model, the shear stresses are considered to be arbitrary functions of velocity gradients. Using Lie group analysis, we calculate the infinitesimal generators accepted by the equations in the arbitrary shear stress case. The extension of Lie algebra, for the case of Newtonian fluids, is also presented. Then we consider a general boundary value problem modeling the flow over a moving surface with suction or injection, and calculate the restrictions imposed by the boundary conditions on the generators. Assuming all flow quantities to be independent of the $$z$$-direction, the three-independent-variable partial differential system is converted into a two-independent-variable system by using two different subgroups of the general group. Lie group analysis is further applied to the resulting equations, and final reductions to ordinary differential systems are obtained.

##### MSC:
 76A05 Non-Newtonian fluids 35Q35 PDEs in connection with fluid mechanics 35A30 Geometric theory, characteristics, transformations in context of PDEs
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##### References:
 [1] Acrivos, A.; Shah, M.J.; Petersen, E.E., A.i.ch.e.j., 6, 312, (1960) [2] Schowalter, W.R., A.i.ch.e.j., 6, 25, (1960) [3] Bizzell, G.D.; Slattery, J.C., Chemical engineering science, 17, 777, (1962) [4] Pascal, H.; Pascal, F., International journal of engineering science, 23, 571, (1985) · Zbl 0564.76093 [5] Hayasi, N., Journal of fluid mechanics, 23, 293, (1965) [6] Na, T.Y.; Hansen, A.G., International journal of non-linear mechanics, 2, 373, (1967) · Zbl 0153.28402 [7] Pakdemirli, M., IMA journal of applied mathematics, 50, 133, (1993) [8] Srivastava, A.C., Zamp, 9, 80, (1958) [9] Rajeswari, G.K.; Rathna, S.L., Zamp, 13, 43, (1962) [10] Rajagopal, K.R.; Gupta, A.S.; Wineman, A.S., International journal of engineering science, 18, 875, (1980) [11] Pakdemirli, M.; Suhubi, E.S., International journal of engineering science, 30, 523, (1992) · Zbl 0756.76004 [12] Rajagopal, K.R.; Gupta, A.S.; Na, T.Y., International journal of non-linear mechanics, 18, 313, (1983) · Zbl 0527.76010 [13] Garg, V.K.; Rajagopal, K.R., Acta mechanica, 88, 113, (1991) [14] Massoudi, M.; Ramezan, M., International journal of non-linear mechanics, 24, 221, (1989) · Zbl 0693.76003 [15] Garg, V.K.; Rajagopal, K.R., Mechanics research communications, 17, 415, (1990) [16] Pakdemirli, M.; Suhubi, E.S., International journal of engineering science, 30, 611, (1992) · Zbl 0756.76004 [17] Rajagopal, K.R.; Szeri, A.Z.; Troy, W., International journal of non-linear mechanics, 21, 279, (1986) · Zbl 0599.76013 [18] Pakdemirli, M., International journal of non-linear mechanics, 27, 785, (1992) [19] Beard, D.W.; Walters, K., (), 667 [20] Astin, J.; Jones, R.S.; Lockyer, P., Journal mechanique, 12, 527, (1973) [21] Frater, K.R., Zamp, 20, 712, (1969) [22] Hsu, C., Journal of fluid mechanics, 27, 445, (1997) [23] Pakdemirli, M., International journal of engineering science, 32, 141, (1994) [24] Lee, S.Y.; Ames, W.F., A.i.ch.e.j., 12, 700, (1966) [25] Hansen, A.G.; Na, T.Y., Journal of basic engineering, 90, 71, (1968) [26] Timol, M.G.; Kalthia, N.L., International journal of non-linear mechanics, 21, 475, (1986) · Zbl 0605.76004 [27] Pakdemirli, M., International journal of non-linear mechanics, 29, 187, (1994) [28] Pakdemirli, M.; Yurusoy, M.; Kucukbursa, A., International journal of non-linear mechanics, 31, 267, (1996) [29] Yurusoy, M.; Pakdemirli, M., Modern group analysis VI, (), Johannesburgh, Africa · Zbl 1345.76009 [30] Yurusoy, M., Lie group analysis of unsteady boundary layer equations of non-Newtonian fluids, (), (in Turkish) · Zbl 0945.76556 [31] Bluman, G.W.; Kumei, S., Symmetries and differential equations, (1989), Springer New York · Zbl 0718.35003 [32] Stephani, H., Differential equations: their solution using symmetries, (1989), Cambridge University Press Cambridge · Zbl 0704.34001 [33] Ibragimov, N.H., ()
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