Stability of unstably stratified shear flow in a channel under non-Boussinesq conditions. (English) Zbl 0868.76029
The stratified plane Poiseuille flow of an ideal gas between two horizontal plates is subject to a vertical temperature gradient. The temperature variations are very high. The linear stability of this thermodynamic phenomenon is investigated numerically on the bases of the equations of mass, momentum and energy conservation containing four real parameters. The coefficients of thermal conductivity and dynamic viscosity depend on the temperature according to the Sutherland law. The perturbation equations are deduced by a Laplace and Fourier transform. The obtained eigenvalue problem is shown to have only two types of eigensolutions. The Squire’s theorem is shown to hold. Finally, the authors solve numerically the generalized Orr-Sommerfeld equations by means of a Chebyshev pseudo-spectral method. The comparison with the Boussinesq case is done.
Reviewer: A.Georgescu (Bucureşti)
MSC:
| 76E05 | Parallel shear flows in hydrodynamic stability |
| 76V05 | Reaction effects in flows |
| 76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |
| 80A20 | Heat and mass transfer, heat flow (MSC2010) |
Keywords:
plane Poiseuille flow; ideal gas; vertical temperature gradient; linear stability; thermal conductivity; dynamic viscosity; Sutherland law; Squire’s theorem; generalized Orr-Sommerfeld equations; Chebyshev pseudo-spectral methodSoftware:
IMSL Numerical LibrariesReferences:
| [1] | Jensen, K. F., Einset, E. O., Fotiadis, D. I.: Flow phenomena in chemical vapor deposition of thin films. Annu. Rev. Fluid Mech.23, 197-232 (1991). · doi:10.1146/annurev.fl.23.010191.001213 |
| [2] | Paolucci, S.: On the filtering of sound from the Navier-Stokes equations. Sandia National Laboratories Rep. SAND82-8257 (1982). |
| [3] | Deardorff, J. W.: Gravitational instability between horizontal plates with shear. Phys. Fluids8, 1027-1030 (1965). · doi:10.1063/1.1761351 |
| [4] | Gage, K. S., Reid, W. H.: The stability of thermal stratified plane Poiseuille flow. J. Fluid Mech.33, 21-32 (1968). · Zbl 0155.55702 · doi:10.1017/S0022112068002326 |
| [5] | Fujimura, K., Kelly, R. E.: Stability of unstably stratified shear flow between parallel plates. Fluid Dyn. Res.2, 281-282 (1988). · doi:10.1016/0169-5983(88)90006-8 |
| [6] | Vanderbrock, G., Platten, J. K.: Approximate (variational) and exact (numerical) solutions of B?nard type problems with temperature dependent material properties. Int. J. Eng. Sci.12, 897-917 (1974). · Zbl 0287.76059 · doi:10.1016/0020-7225(74)90008-1 |
| [7] | White, F. M.: Viscous fluid flow. New York: McGraw-Hill 1974. · Zbl 0356.76003 |
| [8] | Chenoweth, D. R., Paolucci, S.: Natural convection in an enclosed vertical air layer with large horizontal temperature differences. J. Fluid Mech.169, 173-210 (1986). · Zbl 0623.76097 · doi:10.1017/S0022112086000587 |
| [9] | Chiu, K.-C., Rosenberger, F.: Mixed convection between horizontal plates-entrance effects. Int. J. Heat Mass Transfer30, 1645-1654 (1987). · doi:10.1016/0017-9310(87)90310-3 |
| [10] | Hatziavramidis, D., Ku, H.-C.: An integral Chebyshev expansion method for boundary-value problems of O.D.E. type. Comp. Ath. Appl.11, 581-586 (1985). · Zbl 0596.65054 · doi:10.1016/0898-1221(85)90040-9 |
| [11] | IMSL Mathematical Library, Version 1.1 1989. |
| [12] | Ku, H.-C., Hatziavramidis, D.: Chebyshev expansion methods for the solution of the extended Graetz problem. J. Comp. Phys.56, 495-512 (1984). · Zbl 0572.76084 · doi:10.1016/0021-9991(84)90109-8 |
| [13] | Zebib, A.: A Chebyshev method for the solution of boundary value problems. J. Comp. Phys.53, 443-455 (1984). · Zbl 0541.76036 · doi:10.1016/0021-9991(84)90070-6 |
| [14] | Brenier, B., Roux, B., Bontoux, P.: Comparison des methodes Tau-Chebyshev et Galerkin dans l’etude de stabilite des mouvements de convection naturelle. Probleme des valeurs propres parasites. J. Mec. Theor. Appl.5, 95-119 (1986). · Zbl 0595.76045 |
| [15] | Platten, J. K., Legros, J. C.: Convection in liquids. Berlin Heidelberg New York Tokyo: Springer 1984. · Zbl 0545.76048 |
| [16] | Chandrasekhar, S.: Hydrodynamic and hydromagnetic stability. Oxford: Oxford University Press 1961. · Zbl 0142.44103 |
| [17] | Busse, F. H.: The stability of finite amplitude cellular convection and its relation to an extremum principle. J. Fluid Mech.30, 625-649 (1967). · Zbl 0159.28202 · doi:10.1017/S0022112067001661 |
| [18] | Paolucci, S., Chenoweth, D. R.: Departures from the Boussinesq approximation in laminar B?nard convection. Phys. Fluids30, 1561-1564 (1987). · doi:10.1063/1.866218 |
| [19] | Fr?hlich, J.: R?solution num?rique des ?quations de Navier-Stokes ? faible nombre de Mach par m?thode spectrale. Ph. D. Thesis, University of Nice, France 1990. |
| [20] | Evans, G., Greif, R.: A study of traveling wave instabilities in a horizontal channel flow with applications to chemical vapor deposition. Int. J. Heat Mass Transfer32, 895-911 (1989). · doi:10.1016/0017-9310(89)90239-1 |
| [21] | Ahlers, G.: Effect of departure from the Oberbeck-Boussinesq approximation on the heat transport of horizontal convecting fluid layers. J. Fluid Mech.98, 137-148 (1980). · doi:10.1017/S0022112080000067 |
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