Existence and estimate of positive solutions to a nonlinear singular boundary value problem in the theory of dilatant non-Newtonian fluids. (English) Zbl 1170.76006

Zheng, Liancun; Zhang, Xinxin; He, Jicheng

Math. Comput. Modelling 45, No. 3-4, 387-393 (2007).

Summary: We present a rigorous proof of existence and uniqueness of solutions to laminar boundary layer flow of power-law non-Newtonian fluid. A theoretical estimate for skin friction coefficient is given, which is characterized by a power-law exponent. The reliability and efficiency of the proposed estimate are verified by numerical results with a good agreement. The estimate formula can be successfully applied to give the value of skin friction coefficient.

MSC:

76A05	Non-Newtonian fluids
76M55	Dimensional analysis and similarity applied to problems in fluid mechanics
35Q35	PDEs in connection with fluid mechanics

Keywords:

similarity solution; boundary layer; power-law fluid; skin friction coefficient

References:

[1]	Schlichting, H., Boundary layer theory, (1979), McGraw-Hill New York
[2]	Callegari, A.J.; Friedman, M.B., An analytical solution of a nonlinear, singular boundary value problem in the theory of viscous fluids, J. math. anal. appl., 21, 510-529, (1968) · Zbl 0172.26802
[3]	Callegari, A.J.; Nachman, A., Some singular, non-linear differential equations arising in boundary layer theory, J. math. anal. appl., 64, 96-105, (1978) · Zbl 0386.34026
[4]	Vajravelu, K.; Mohapatra, R.N., On fluid dynamic drag reduction in some boundary layer flows, Acta mech., 81, 59-68, (1990) · Zbl 0698.76045
[5]	Vajravelu, K.; Soewono, E.; Mohapatra, R.N., On solutions of some singular, non-linear differential equations arising in boundary layer theory, J. math. anal. appl., 155, 499-512, (1991) · Zbl 0729.35103
[6]	Schowalter, W.R., The application of boundary-layer theory to power-law pseudoplastic fluids: similar solutions, Aiche j., 6, 24-28, (1960)
[7]	Acrivos, A.; Shah, A.; Petersen, E.E., Momentum and heat transfer in laminar boundary-layer flows of non-Newtonian fluids past external surfaces, Aiche j., 6, 312-317, (1960)
[8]	Nachman, A.; Callegari, A., A nonlinear singular boundary value problem in the theory of pseudoplastic fluids, SIAM J. appl. math., 38, 275-281, (1980) · Zbl 0453.76002
[9]	Howell, T.G.; Jeng, D.R.; De Witt, K.J., Momentum and heat transfer on a continuous moving surface in power law fluid, Int. J. heat mass transfer, 40, 1853-1861, (1997) · Zbl 0915.76005
[10]	Rao, J.H.; Jeng, D.R.; De Witt, K.J., Momentum and heat transfer in a power-law fluid with arbitrary injection/suction at a moving wall, Int. J. heat mass transfer, 42, 2837-2847, (1999) · Zbl 0977.76004
[11]	Akcay, M.; Adil Ykselen, M., Drag reduction of a non-Newtonian fluid by fluid injection on a moving wall, Arch. appl. mech., 69, 215-225, (1999) · Zbl 0933.76004
[12]	Zheng, L.; He, J., Existence and non-uniqueness of positive solutions to a non-linear boundary value problems in the theory of viscous fluids, Dynam. systems appl., 8, 133-145, (1999) · Zbl 0926.34012
[13]	Zheng, L.; Ma, L.; He, J., Bifurcation solutions to a boundary layer problem arising in the theory of power law fluids, Acta math. sci., 20, 19-26, (2000) · Zbl 0961.34008
[14]	Lu, C.; Zheng, L., Similarity solutions of a boundary layer problem in power law fluids through a moving flat plate, Int. J. pure appl. math., 13, 143-166, (2004) · Zbl 1059.34018
[15]	Bernfeld, S.R.; Lakshmikantham, V., An introduction to nonlinear boundary value problems, (1974), Academic Press New York · Zbl 0286.34018
[16]	Agarwal, R.P.; O’Regan, D.; Wong, P.J.Y., Positive solutions of differential, difference and integral equations, (1999), Kluwer Academic Publishers Netherlands · Zbl 0923.39002
[17]	Granas, A.; Guenther, R.B.; Lee, J.W., Some general existence principles in the caratheodory theory of nonlinear differential systems, J. math. pure. appl., 70, 153-196, (1991) · Zbl 0687.34009