Skip to main content Accessibility help
×
Home

Stability of the developing laminar flow in a parallel-plate channel

  • T. S. Chen (a1) (a2) and E. M. Sparrow (a1)

Abstract

The hydrodynamic stability of the developing laminar flow in the entrance region of a parallel-plate channel is investigated using the theory of small disturbances. The stability of the fully developed flow is also re-examined. A wide range of analytical (i.e. asymptotic) and numerical methods are employed in the stability investigation. Among the asymptotic methods, each of three viscous solutions (singular, regular and composite) is used along with the inviscid solution to provide critical Reynolds numbers and complete neutral stability curves. Two numerical methods, finite differences and stepwise integration, are applied to calculate critical Reynolds numbers. The basic flow in the development region is treated from two stand-points: as a channel velocity profile and as a boundary-layer velocity profile. Extensive comparisons among the various methods and flow models disclose their various strengths and ranges of applicability. As a general result, it is found that the critical Reynolds number decreases monotonically with increasing distance from the channel entrance, approaching the fully developed value as a limit.

Copyright

References

Hide All
Chen, T. S. 1966 Hydrodynamic stability of developing flow in a parallel-plate channel. Ph.D. Thesis, Department of Mechanical Engineering, University of Minnesota.
Chen, T. S., Joseph, D. D. & Sparrow, E. M. 1966 Phys. Fluids, 9, 2519.
Fu, T. S. 1967 Viscous instability of asymmetric parallel flows in channels. Ph.D. Thesis Department of Aeronautics and Engineering Mechanics, University of Minnesota.
Gröhne, D. 1954 Z. Angew. Math. Mech. 34, 344.
Hahneman, E., Freeman, J. C. & Finston, M. 1948 J. Aero. Sci. 15, 493.
Heisenberg, W. 1924 Ann. Phys. 74, 577.
Holstein, H. 1950 Z. Angew. Math. Mech. 30, 25.
Kurtz, E. F. & Crandall, S. H. 1962 J. Math. Phys. 41, 264.
Lin, C. C. 1945 Quart. Appl. Math. 3, 117, 213, 277.
Lin, C. C. 1955 The Theory of Hydrodynamic Stability. Cambridge University Press.
Lock, R. C. 1954 Proc. Camb. Phil. Soc. 50, 105.
Miles, J. W. 1960 J. Fluid Mech. 8, 593.
Nachtsheim, P. R. 1964 NASA TN D-2414.
Reid, W. H. 1965 The stability of parallel flows. In Basic Development in Fluid Mechanics. Ed. by M. Holt. New York: Academic Press.
Schlichting, H. 1934 Z. Angew. Math. Mech. 14, 368.
Schlichting, H. 1960 Boundary Layer Theory. New York: McGraw-Hill.
Shen, S. F. 1964 Stability of laminar flows. In Theory of Laminar Flows. Ed. by F. K. Moore, Princeton University Press.
Sparrow, E. M., Lin, S. H. & Lundgren, T. S. 1964 Phys. Fluids, 7, 338.
Stuart, J. T. 1963 Hydrodynamic stability. In Laminar Boundary Layers. Ed. by L. Rosenhead. Oxford University Press.
Thomas, L. H. 1953 Phys. Rev. 91, 780.
Tietjens, O. 1925 Z. Angew. Math. Mech. 5, 200.
Tollmien, W. 1929 Nachr. Ges. Wiss. Göttingen, Math.-Phys. Klasse, p. 21.
Tollmien, W. 1947 Z. Angew. Math. Mech. 25–27, 33, 70.
Tsou, F. K. 1965 Velocity field, hydrodynamic stability, and heat transfer for boundary-layer flow along a continuous moving surface. Ph.D. Thesis, Department of Mechanical Engineering, University of Minnesota.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed