Taylor-Couette flows in a horizontal annular gap between finite coaxial cylinders in rotor-stator configuration are numerically investigated. The inner cylinder (rotor) rotates at a constant angular velocity while the outer cylinder (stator) is at rest. They are limited at their extremities by two fixed walls that prevent axial fluid flow. In addition, a heat transfer is generated by an imposed temperature difference, with the rotor hotter than the stator while the end-walls are adiabatic. The fluid physical properties are temperature dependent. This non-linear physics problem, with a strong coupling of the conservation equations and boundary conditions, is solved by a finite volume method with numerical schemes of second order space and time accuracies. The radius and aspect ratios and the Taylor, Grashof and Prandt numbers are the control parameters. The developed numerical code has been tested for different meshes and perfectly validated. Extensive calculations have been made in large ranges of the Taylor and Grashof numbers to analyze the Taylor-Couette flow in convection modes. The results highlight the dynamic and thermal instabilities generated in the Taylor Couette flow from the appearance of Ekman cells to the Taylor vortex propagation in the entire annulus. The combined effect of these vortices with the secondary flow improves the heat transfer. Furthermore, the influence of the physical properties in the radial direction is more marked in the vicinity of the walls. Finally, we propose an empirical correlation of the Nusselt number in the studied parameter ranges.