Exact sine series solution for oscillatory convection in a binary fluid

The linear equations governing the onset of convection in a binary fluid subject to the Soret and Dufour effects are solved exactly for rigid boundary conditions by expanding the velocity and temperature as Fourier sine series. The velocity and temperature distributions are found to be those of the free-boundary problem, plus small admixtures of higher harmonics. Exact expressions for the Rayleigh number and oscillation frequency are obtained in terms of infinite series. For the special case of negligible Dufour effect our results agree with exact numerical calculations. Results for the general case are compared with previous approximate methods of solution.