Stabilities of natural and mixed convective flow of fluids of low and moderate Prandtl numbers in an annulus between two concentric horizontal cylinders are studied numerically. Firstly, two-dimensional natural convection of a fluid of low Prandtl number (Pr = 0.02) is investigated in a wide range of gap widths. At low Grashof numbers, a steady unicellular convection is obtained. Above a transition Grashof number which depends on the gap width, a steady bicellular flow occurs. With further increase of the Grashof number, steady or time-periodic multicellular convection occurs and, finally, complex unsteady convective flow appears. A plot is presented which predicts the type of flow patterns for various combinations of gap widths and Grashof numbers. Secondly, three-dimensional linear stability analysis of natural convective flow in an annulus between two concentric cylinders is investigated for a fluid of Prandtl number 0.71 (air). Vector potential formulation is adopted. For the medium-sized annulus with relative inverse gap ratio (ratio of inner diameter to gap width) between 2.1 and 10.0, the basic two-dimensional flow is found to be unstable with respect to three-dimensional disturbances. Critical Rayleigh numbers above which the two-dimensional basic flow is unstable show good agreement with experimental results. The disturbance velocity distribution obtained by the stability analysis suggests that the instability is mainly caused by the buoyance effects. Thirdly, three-dimensional linear stability analysis of mixed convective flow in an annulus between two concentric cylinders is investigated for a fluid of Prandtl number 0.71. The same numerical scheme with that of the above natural convective flow of Prandtl number 0.71 is adopted. The critical stability curves in (Re$^2$, Gr) is searched and the results show good agreement with those of three-dimensional numerical computations. For the inverse relative gap width of 3, the followings are found: for buoyancy-dominated mixed convection ($\gamma > 0.33$) the critical mode of disturbance is stationary and the effects of rotation stabilize the flow, for rotation-dominated fow ($\gamma < 0.16$) the critical mode of disturbance is also stationary and critical Reynolds number increases with Grashof number, in the range $0.16 < \gamma < 0.33$, where both rotation and buoyancy are important, the most unstable mode of disturbance is oscillatory, i.e., the principle of exchange of stability is upset by competition between destabilizing mechanisms.
