×

Concave solutions of a general self-similar boundary layer problem for power-law fluids. (English) Zbl 1312.34069

Summary: We consider the following third-order nonlinear boundary value problem \[ \begin{gathered} n|f''|^{n-1} f'''+\lambda ff''- f'h(f')= 0,\quad \eta> 0,\\ f(0)= 0,\quad f'(0)= 1,\quad f'(+\infty)= 0,\end{gathered} \] where \(n\) and \(\lambda\) are two positive numbers and the given function \(h\) is continuous on some interval containing \([0,1]\).
We establish the uniqueness, existence and nonexistence of (normal) concave solutions or generalized concave solutions to the problem, and obtain some results about boundedness and asymptotic behavior of the (normal) concave solution or the generalized concave solution.

MSC:

34B60 Applications of boundary value problems involving ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34B40 Boundary value problems on infinite intervals for ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ali, M. E., On thermal boundary layer on a power-law stretched surface with suction or injection, Int. J. Heat and Fluid Flow, 16, 4, 280-290 (1995)
[2] Altan, T.; Oh, S.; Gegel, H., Metal Forming Fundamentals and Applications (1979), American Society of Metals Park: American Society of Metals Park OH
[3] Fisher, E. G., Extrusion of Plastics (1976), Wiley: Wiley New York
[4] Tadmor, Z.; Klein, I., Engineering principles of plasticating extrusion, (Polymer Science and Engineering Series (1970), Van Nostrand-Reinhold: Van Nostrand-Reinhold New York)
[5] Sakiadis, B. C., Boundary-layer behavior on continuous solid surface: I. Boundary-layer equations for two-dimensional and axisymmetric flow, AICHE J., 7, 26-28 (1961)
[6] Sakiadis, B. C., Boundary-layer behavior on continuous solid surface: II. The boundary-layer equations on a continuous flat surface, AICHE J., 7, 221-225 (1961)
[7] Crane, L. J., Flow paste a stretching plane, ZAMP, 21, 645-647 (1970)
[8] Prasad, K. V.; Pal, Dulal; Datti, P. S., MHD power-law fluid flow and heat transfer over a non-isothermal stretching sheet, Commun. Nonlinear Sci. Numer. Simulat., 14, 2178-2189 (2009)
[9] Banks, W. H.H., Similarity solutions of the boundary layer equations for a stretching wall, J. Mech. Theor. Appl., 2, 375-392 (1983) · Zbl 0538.76039
[10] Chen, C. K.; Char, M. I., Heat transfer of a continuous surface with suction or blowing, J. Math. Anal. Appl., 135, 568-580 (1988) · Zbl 0652.76062
[11] Magyari, E.; Keller, B., Exact solutions for self-similar boundary-layer flows induced by permeable stretching walls, Eur. J. Mech. B-Fluids, 19, 109-122 (2000) · Zbl 0976.76021
[12] Andersson, H. I.; Dandapat, B. S., Flow of a power-law fluid over a stretching sheet, Stability Appl. Anal. Continuous Media, 1, 339-347 (1991)
[13] Andersson, H. I.; Kumaran, V., On sheet-driven motion of power-law fluids, Int. J. Nonlinear Mech., 41, 1228-1234 (2006) · Zbl 1160.76302
[14] Andersson, H. I.; Bech, K. H.; Dandapat, B. S., Magnetohydrodynamic flow of a power-law fluid over a stretching sheet, Int. J. Nonlinear Mech., 27, 929-936 (1992) · Zbl 0775.76216
[15] Liao, S. J., On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids over a stretching sheet, J. Fluid Mech., 488, 189-212 (2003) · Zbl 1063.76671
[16] Zhang, Z.; Wang, J., On the similarity solutions of Magnetohydrodynamic flows of power-law fluids over a Stretching sheet, J. Math. Anal. Appl., 330, 207-220 (2007) · Zbl 1119.76072
[17] Oleinik, O. A.; Samokhin, V. N., Mathematical Model in Boundary Layer Theory, Differential Equations (1999), Chapman & Hall/CRC · Zbl 0928.76002
[18] Belhachmi, Z.; Brighi, B.; Taous, K., On a family of differential equations for boundary layer approximations in porous media, European J. Appl., 12, 4, 513-528 (2001) · Zbl 0991.76084
[19] Brighi, B.; Hoernel, J. D., On a general similarity boundary layer equation, Acta Math. Univ. Comenianae, LXXVII, 1, 9-22 (2008) · Zbl 1164.34006
[20] Guo, J. S.; Tsai, J. C., The structure of solutions for a third order differential equation in boundary layer theory, Japan J. Indust. Appl. Math., 22, 311-351 (2005) · Zbl 1095.34006
[21] Cortell, R., A note on magnetohydrodynamic flow of a power-law fluid over a stretching sheet, Appl. Math. Comput., 168, 557-566 (2005) · Zbl 1081.76059
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.