Effect of Boundary Conditions on the Onset of Convection

Abstract

The effect of various boundary conditions on the onset of convection in a semi-infinite fluid which has a time-dependent temperature profile is investigated. The linearized hydrodynamic and thermodynamic equations are solved by expanding the variables in a series of orthonormal functions satisfying the boundary conditions and integrating the resulting set of coupled ordinary differential equations numerically. For a semi-infinite fluid the initial behavior of the system is found to depend mainly on the Prandtl number and the manner in which the fluid is cooled or heated. The various combinations of rigid, free, conducting, and insulating boundary conditions only slightly affect the time of onset of manifest convective behavior and the horizontal wavenumber of the disturbances which are amplified the most at this time.