Abstract
The velocity distribution between two sidewalls is M-shaped for the MHD channel, flows with rectangular cross section and thin conducting walls in a strong transverse magnetic field. Assume that the dimensionless numbersR m ≪1,M, N≫ 1, and σ* and that the distance between two perpendicular walls is very long in comparison with the distance between two sidewalls. First, the equation for steady flow is established, and the solution of M-shaped velocity distribution is given. Then, an equation for stability of small disturbances is derived based on the velocity distribution obtained. Finally, it is proved that the stability equation for sidewall flow can be transformed into the famous Orr-Sommerfeld equation, in addition, the following theorems are also proved, namely, the analogy theorem, the generalized Rayleigh's theorem, the generalized Fjørtoft's theorem and the generalized Joseph's theorems.
Similar content being viewed by others
References
Walker JE. Magnetohydrodynamic flows in rectangular ducts with thin conducting walls, Part 1.J de Méchanique, 1981, 20(1): 89–112
Sidorenkov S, Shishko A. Variational method for MHD flow calculation in conducting walls slotted channel.Magnetohydrodynamics, 1991, (4): 87–96
Reed CB, Picologlou BF. Sidewall flow instabilities in liquid metal MHD flows under blanket relevant conditions.Fusion Technology, 1989, 15: 705–715
Moreau R. Magnetohydrodynamics. Dordrecht: Kluwer Academic Publishers, 1990
Drazin PG, Reid WH. Hydrodynamic Stability. London: Cambridge University Press, 1981
Joseph DD. Eigenvalue bounds for the Orr-Sommerfeld equation, Part 1.J Fluid Mech, 1968, 33(3): 617–621
Joseph DD. Eigenvalue bounds for the Orr-Sommerfeld equation, Part 2.J Fluid Mech, 1969, 36(4): 721–734
Georgescu A. Hydrodynamic Stability Theory. Dordrecht: Martinus Nijhoff Publishers, 1985
Author information
Authors and Affiliations
Additional information
The project supported by the National Natural Science Foundation of China
Rights and permissions
About this article
Cite this article
Fu, X. Sidewall flow stability in the MHD channel flows. Acta Mech Sinica 12, 97–103 (1996). https://doi.org/10.1007/BF02486788
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02486788