A study is made of the stability of the steady periodic regime that arises in a horizontal layer of fluid in the presence of spatial modulation of of the temperature on the solid bottom boundary. The upper free boundary of the layer is in contact with the atmosphere. The fundamental resonance values of the wave number of the modulation are found; there are five of them. If the temperature of the lower boundary of the layer is constant, and the temperature gradient is not too large, the fluid is in equilibrium. When the temperature gradient passes through the critical value, the equilibrium ceases to be stable, and steady convection develops in the fluid . In the presence of spatial modulation of the temperature on the lower boundary of the layer the fluid cannot be in equilibrium, and a spatially periodic steady regime is established in it. The aim of the present paper is to find the critical values of the temperature gradient at which this fundamental steady regime becomes unstable and a secondary steady regime develops in the fluid. An analogous problem for the case when both boundaries of the layer are free surfaces and without allowance for the influence of the atmosphere has been solved by Vozovoi and Nepomnyashchii .
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G. Z. Gershuni and E. M. Zhukhovitskii, Convective Stability of an Incompressible Fluid [in Russian], Nauka, Moscow (1972), p. 392.
L. P. Vozovoi and A. A. Nepomnyashchii, “Convection in a horizontal layer in the presence of spatial modulation of the temperature on the boundaries,” in: Hydrodynamics, No. 7 [in Russian], Perm' (1974), pp. 105–117.
V. Kh. Izakson and V. I. Yudovich, “On the development of convection in a layer of fluid with free boundary,” Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 4, 23 (1968).
Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 128–132, May–June, 1983.
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Batishchev, V.A., Kolesov, V.V., Slitinskaya, S.K. et al. Influence of spatial modulation of the temperature field on the stability of two-dimensional steady flow in a horizontal layer of fluid. Fluid Dyn 18, 442–445 (1983). https://doi.org/10.1007/BF01090566
- Free Surface
- Temperature Gradient
- Lower Boundary