The stability of natural convection in a vertical fluid layer

The stability of natural convection in fluid between two parallel vertical plates is investigated theoretically. The two plates are maintained at different temperatures and a uniform stable temperature gradient β is present in the vertical direction. The Prandtl number of the fluid is fixed at 7.5. An orthonormalization method is used in numerical integrations of the disturbance equations. It is shown how the critical Grashof number varies with β for both stationary and travelling disturbances. It is found that for β ≤ 7.1 × 10⁻³ the convection is unstable to stationary disturbances and for β ≥ 7.1 × 10⁻³ it is unstable to travelling disturbances. The critical Grashof number is given by \[ G_c = \left\{\begin{array}{@{}l@{\quad}c@{\quad}l@{}} 500 & {\rm for} & \beta < 1.0\times 10^{-3},\\ 1.3\times 10^6\beta^3 & {\rm for} &\beta > 4.1\times 10^{-2}, \end{array}\right. \] and even for intermediate values of β the variation of G_c is rather simple but not monotonic.