Thermal convection in a fluid contained between two rigid walls with different mean temperatures is considered when either spatially periodic temperatures are prescribed at the walls or surface corrugations exist. The amplitudes of the spatial non-uniformities are assumed to be small, and the wavelength is set equal to the critical wavelength for the onset of Rayleigh-Bénard convection. For values of the mean Rayleigh number below the classical critical value, the mean Nusselt number and the mean flow are found as functions of Rayleigh number, Prandtl number, and modulation amplitude. For values of the Rayleigh number close to the classical critical value, the effects of the non-uniformities are greatly amplified, and the amplitude of convection is then governed by a cubic equation. This equation yields three supercritical states, but only the state linked to a subcritical state is found to be stable.

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