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Similarity of the two-dimensional and axisymmetric boundary-layer flows for purely viscous non-Newtonian fluids

Published online by Cambridge University Press:  28 March 2006

Nisiki Hayasi
Nisiki Hayasi*
Affiliation:
NASA-Ames Research Center, Moffett Field, California
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Extract

Conditions for the existence of similar solutions for the two-dimensional and axisymmetric boundary layers are obtained for the steady or unsteady flow for purely viscous non-Newtonian fluids, where the shear stress is proportional to the nth power of the velocity gradient and n > 0. These conditions are shown to be a generalization of the similarity conditions for Newtonian fluids. In particular it is found that the velocity at the outer edge of the boundary layer must be proportional to {R(x) + A}m or exp [AR(x)] for steady flows and to (t + A)m, (dx + A)/(t + B) or exp [At] for unsteady flows. Here x is the distance along the wall from the forward stagnation point, t is the time, R(x) = x for two-dimensional flows and for axisymmetric flows, r is the distance from the axis to the wall, and A, B, d, m are constants.

Several examples of the similar solutions are calculated analytically for both steady and unsteady pseudoplastic flows (n < 1). In these solutions the velocity in the boundary layer tends to the outside velocity in such a manner that the difference tends to zero as an inverse power of the distance from the wall, whereas such a difference tends to zero exponentially for the corresponding flow in Newtonian fluids.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 23 , Issue 2 , October 1965 , pp. 293 - 303
Copyright
Copyright © Cambridge University Press 1965

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References

Acrivos, A., Shah, M. J. & Petersen, E. E. 1960 Momentum and heat transfer in laminary boundary-layer flows of non-Newtonian fluids past external surfaces. Amer. Inst. Chem. Engrs. J. 6, 312–317.CrossRefGoogle Scholar
Bird, R. B. 1959 Unsteady pseudoplastic flow near a moving wall. Amer. Inst. Chem. Engrs. J. 5, 565 and 6D.CrossRefGoogle Scholar
Blasius, H. 1908 Grenzschichten in Flüssigkeiten mit kleiner Reibung. Z. f. Math. u. Phys. 56, 1–37; also Nat. Adv. Com. Aero. Tech. Memo. no. 1256.Google Scholar
Curle, N. 1962 The Laminar Boundary Layer Equations. Oxford: Clarendon Press.Google Scholar
Goldstein, S. 1939 A note on the boundary layer equations. Proc. Camb. Phil. Soc. 35, 338–340.CrossRefGoogle Scholar
Hartree, D. R. 1937 On an equation occurring in Falkner and Skan's approximate treatment of the equations of the boundary layer. Proc. Camb. Phil. Soc. 33, 223–239.CrossRefGoogle Scholar
Hayasi, N. 1960 On similar solutions of the steady quasi-two-dimensional incompressible laminar boundary-layer equations. J. Phys. Soc. Japan, 15, 522–527; 16, 2329.CrossRefGoogle Scholar
Hayasi, N. 1961 On similar solutions of the unsteady quasi-two-dimensional incompressible laminar boundary-layer equations. J. Phys. Soc. Japan, 16, 2316–2328; 17, 252.CrossRefGoogle Scholar
Hayasi, N. 1962 On semi-similar solutions of the unsteady quasi-two-dimensional incompressible laminar boundary-layer equations. J. Phys. Soc. Japan, 17, 194–203.CrossRefGoogle Scholar
Hayasi, N. 1965 Correlation of two-dimensional and axisymmetric boundary-layer flows for purely viscous non-Newtonian fluids. To be published in Amer. Inst. Aero Astron. J.CrossRefGoogle Scholar
Schlichting, H. 1960 Boundary Layer Theory, 4th ed. New York: McGraw-Hill.Google Scholar
Schowalter, W. R. 1960 The application of boundary-layer theory to power-law pseudoplastic fluids: similar solutions. Amer. Inst. Chem. Engrs. J. 6, 24–28.CrossRefGoogle Scholar
Schuh, H. 1955 Über die ‘ähnlichen’ Lösungen der instationären laminaren Grezschichtgleichung in inkompressibler Strömung. 50 Jahre Grenzschichtforschung, p. 147–152. Braunschweig: Friedr. Vieweg Sohn.CrossRefGoogle Scholar
Wells, C. S. 1964a Unsteady boundary-layer flow of a non-Newtonian fluid on a flat plate. Amer. Inst. Aero. Astron. J. 2, 951–952.Google Scholar
Wells, C. S. 1964b Similar solutions of the boundary layer equations for purely viscous non-Newtonian fluids. Nat. Aero. Space Adm. Tech. Note D-2262.Google Scholar