×

Existence and uniqueness results for a nonlinear differential equation arising in viscous flow over a nonlinearly stretching sheet. (English) Zbl 1229.34041

The problem to find a self-similarity solution of the boundary value problem \[ \overline u\overline u_x+ v\overline u_y=\nu\overline u_{yy},\quad u_x= v_y= 0,\quad u= cx^n,\quad v= 0\quad\text{at }y= 0, \]
\[ u\to 0\quad\text{as }y\to\infty \] governing the steady flow over a nonlinearly stretching sheet leads to the boundary value problem for an ordinary differential equation \[ \begin{gathered} f'''+ f''-{2n\over n+1}\, f^{\prime 2}= 0,\\ f(0)= 0,\;f'(0)= 1,\;f'(\eta)\to 0\quad\text{as }\eta\to\infty.\end{gathered}\tag{\(*\)} \] The authors establish the existence of a unique solution \(f\) to \((*)\) for \(-{1\over 3}< n<\infty\) and derive monotonicity properties of \(f\), \(f'\) and \(f''\).

MSC:

34B40 Boundary value problems on infinite intervals for ordinary differential equations
76A10 Viscoelastic fluids
34B08 Parameter dependent boundary value problems for ordinary differential equations
35M32 Boundary value problems for mixed-type systems of PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Sakiadis, B. C., Am. Inst. Chem. Eng. J., 7, 26 (1961)
[2] Erickson, L. E.; Fan, L. T.; Fox, V. G., Ind. Eng. Chem. Fund., 5, 19 (1966)
[3] Crane, L. J.; Angew, Z., Math. Phys., 21, 645 (1970)
[4] Danberg, J. E.; Fansler, K. S., Quart. Appl. Math., 34, 305 (1976)
[5] Gupta, P. S.; Gupta, A. S., Can. J. Chem. Eng., 55, 744 (1977)
[6] Chen, C. K.; Char, M., J. Math. Anal. Appl., 135, 568 (1988)
[7] Fox, V. G.; Erickson, L. E.; Fan, L. T., Am. Inst. Chem. Eng. J., 15, 327 (1969)
[8] Rajagopal, K. R.; Na, T. Y.; Gupta, A. S., Rheol. Acta, 23, 213 (1984)
[9] Coleman, B. D.; Noll, W., Arch. Ration. Mech. Anal., 6, 355 (1960)
[10] Troy, W. C.; Overman II, E. A.; Ermen-Trout, G. B.; Keener, J. P., Quart. Appl. Math., 44, 753 (1987)
[11] Vajravelu, K.; Rollins, D., J. Math. Anal. Appl., 158, 241 (1991)
[12] Vajravelu, K.; Roper, T., Int. J. Nonlinear Mech., 34, 1031 (1999)
[13] Afzal, N.; Varshney, I. S., Warme -und Stoffubertragung, 14, 289 (1980)
[14] Vajravelu, K.; Cannon, J. R., Appl. Math. Comput., 181, 609 (2006)
[15] McLeod, J. B.; Rajagopal, K. R., Arch. Ration. Mech. Anal., 98, 385 (1987)
[16] Liao, S.; Pop, I., Int. J. Heat Mass Transfer, 47, 75 (2004)
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.