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Boundary layer equations and stretching sheet solutions for the modified second grade fluid. (English) Zbl 1213.76060
Summary: A modified second grade non-Newtonian fluid model is considered. The model is a combination of power-law and second grade fluids in which the fluid may exhibit normal stresses, shear thinning or shear thickening behaviors. The equations of motion are derived for two dimensional incompressible flows. The boundary layer equations are derived from the equations. Symmetries of the boundary layer equations are calculated using Lie Group theory. For a special power law index of $$m = - 1$$, the principal Lie algebra extends. Using one of the symmetries, the partial differential system is transferred to an ordinary differential system. The ordinary differential equations are numerically integrated for the stretching sheet boundary conditions. Effects of power-law index and second grade coefficient on the boundary layers are shown and solutions are contrasted with the usual second grade fluid solutions. The shear stress on the boundary is also calculated.

##### MSC:
 76D10 Boundary-layer theory, separation and reattachment, higher-order effects 76A05 Non-Newtonian fluids 76M60 Symmetry analysis, Lie group and Lie algebra methods applied to problems in fluid mechanics
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##### References:
 [1] Man, C.S.; Sun, Q.X., J. glaciol., 33, 268-273, (1987) [2] Man, C.S., Arch. rat. mech. anal., 119, 35-57, (1992) [3] Franchi, H.; Straughan, B., J. math. anal. appl., 180, 122-137, (1993) [4] Gupta, G.; Massoudi, M., Acta mech., 99, 21-33, (1993) [5] Massoudi, M.; Phuoc, T.X., Acta mech., 150, 23-37, (2001) [6] Massoudi, M.; Phuoc, T.X., Continuum mech. thermodyn., 16, 529-538, (2004) [7] Hayat, T.; Khan, M., Nonlinear dyn., 42, 395-405, (2005) [8] Dunn, J.E.; Fosdick, R.L., Arch. rat. mech. anal., 56, 191-252, (1974) [9] Rajagopal, K.R.; Fosdick, R.L., Proc. R. soc. lond. A, 339, 351-377, (1980) [10] Dunn, J.E.; Rajagopal, K.R., Int. J. eng. sci., 33, 689-729, (1995) [11] Bluman, G.W.; Kumei, S., Symmetries and differential equations, (1989), Springer New York · Zbl 0718.35003 [12] Stephani, H., Differential equations: their solution using symmetries, (1989), Cambridge University Press New York · Zbl 0704.34001 [13] McLeod, J.B.; Rajagopal, K.R., Arch. rat. mech. anal., 98, 385-393, (1987) [14] Rajagopal, K.R.; Na, T.Y.; Gupta, A.S., Rheol. acta, 23, 213-215, (1984) [15] Troy, W.C.; Overman, E.A.; Ermentrout, G.B., Quart. appl. math., 44, 753-755, (1987) [16] Vajravelu, K.; Rollins, D., J. math. anal. appl., 158, 241-255, (1991) [17] Pakdemirli, M.; Şuhubi, E.S., Int. J. eng. sci., 30, 523-532, (1992) [18] Pakdemirli, M., Int. J. non-linear mech., 27, 785-793, (1992) [19] Pakdemirli, M., Int. J. eng. sci., 32, 141-154, (1994) [20] Head, A.K., Comput. phys. commun., 71, 241-248, (1993) [21] Rajagopal, K.R.; Szeri, A.Z.; Troy, W., Int. J. non-linear mech., 21, 279-289, (1986) · Zbl 0599.76013 [22] Pakdemirli, M., IMA J. appl. math., 50, 133-148, (1993)
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