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The stability of plane Couette flow with viscous heating

Published online by Cambridge University Press:  29 March 2006

Peter C. Sukanek ,
Charles A. Goldstein  and
Robert L. Laurence
Peter C. Sukanek
Affiliation:
Department of Chemical Engineering, University of Massachusetts, Amherst
Charles A. Goldstein
Affiliation:
Department of Chemical Engineering, Princeton University
Robert L. Laurence
Affiliation:
Department of Chemical Engineering, University of Massachusetts, Amherst
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Abstract

An investigation of the stability of plane Couette flow with viscous heating of a Navier–Stokes–Pourier fluid with an exponential dependence of viscosity upon temperature is presented. Using classical small perturbation theory, the stability of the flow can be described by a sixth-order set of coupled ordinary differential equations. Using Galerkin's method, these equations are reduced to an algebraic eigenvalue problem. An eigenvalue with a negative real part means that the flow is unstable.

Neutral stability curves are determined at Brinkman numbers of 15, 19, 25, 30,40,80 and 600 for Prandtl numbers of 1, s and 50. A Brinkman number of 19 corresponds approximately to the maximum shear stress which can be applied to the system.

The results indicate that four different modes of instability occur: one termed an inviscid mode, arising from an inflexion point in the primary flow; a viscous mode, due to the stratification of viscosity in the flow field and an associated diffusive mechanism; a coupling mode, resulting from the convective and viscous dissipation terms in the energy equation; and finally a purely thermal mode.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 57 , Issue 4 , 6 March 1973 , pp. 651 - 670
Copyright
© 1973 Cambridge University Press

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References

Ames, W. F. 1965 Non-Linear Partial Differential Equatiorw in Engineering. Academic.
Betchov, R. & Criminale, W. O. 1967 Stability of Parallel Flows. Academic.
Chandrasephar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.
Craip, A. D. D. 1969 J. Fluid Mech. 36, 685.
Craik, A. D. D. & Smith, F. I. P. 1968 J. Fluid Mech. 34, 393.
Deardorff, J. W. 1963 J. Fluid Mech. 15, 623.
Dunn, D. W. & Lin, C. C. 1953 J. Aero. Sci. 19, 491.
Esch, R. E. 1957 J. Fluid Mech. 3, 289.
Finlayson, B. A. 1968 J. Fluid Mech. 33, 201.
Finlayson, B. A. & Scriven, L. E. 1968 Proc. Roy. Soc. A 310, 183.
Francis, J. G. 1961 Computer J. 4, 265, 332.
Gallagher, A. P. 1969 Siam J. Appl. Math. 4, 3.
Gallauher, A. P. & Mercer, A. 1962 J. Fluid Mech. 13, 91.
Gallagher, A. P. & Mercer, A. 1964 J. Fluid Mech. 15, 350.
Gallauher, A. P. & Mercer, A. 1965 Proc. Roy. Soc. A 286, 117.
Gavis, J. & Laurence, R. L. 1968 Ind. Engng Chem. Fund. 7, 232.
Gebhart, B. 1971. Heat Transfer. McGraw-Hill.
Gill, A. E. 1965 J. Fluid Mech. 21, 503.
Goldstein, C. 1968 M.S. thesis, The Johns Hopkins University.
Ingersoll, A. P. 1966 Phys. Fluids, 9, 682.
Joseph, D. D. 1964 Phys. Fluids, 7, 1761.
Joseph, D. D. 1965 Phys. Fluids, 8, 2195.
Knowles, C. P. & Gebhart, B. 1968 J. Fluid Mech. 34, 657.
Lees, L. & Lin, C. C. 1946 N.A.S.A. Tech. Note, no. 1115.
Muller, U. 1968 Acta Mech. 6, 78.
Nachtsheim, P. R. 1963 N.A.S.A. Tech. Note, no. D-2089.
Nahme, R. 1940 Ingr. Arch. 11, 19.
Petrov, G. I. 1940 Prikl. Math. Mech. 4, 3.
Ponomarenko, Iu. B. 1968 J. Appl. Math. Mech. 32, 627.
Reid, W. IS. & Harris, D. L. 1958 Astrophys. J. Suppl. Ser. 3, 429, 448.
Southwell, R. V. & Chitty, L. 1930 Phil. Trans. A 229, 205.
Squire, H. B. 1933 Proc. Roy. Soc. A 142, 621.
Sukanek, P. C. 1970 M.S. thesis, University of Massachusetts.
Wilde, D. J. & Beightler, C. S. 1967 Foundations of Optimization. Prentice-Hall.
Yih, C. S. 1967 J. Fluid Mech. 27, 337.
Yih, C. S. 1969 Fluid Mechanics. MeGraw-Hill.