The paper focuses on the flow and heat transfer flow between two concentric cylinders. It investigates the Taylor-Couette system with a rotating inner cylinder and an imposed radial temperature gradient. Taylor number, aspect ratio, and Grashof number were discussed to determine the temperature gradient in the Taylor-Couette flow. In the mixed-convection region, a distorted form of the Taylor cells appeared. Hence, the choice of different aspect ratios and Grashof to determine the maximum heat transports mechanism at a fixed Taylor number. These results in a particularly compact and simple formulation to propose the axisymmetric energy equation in the cylindrical coordinate system. It is rearranged in the Cartesian coordinate system with extra terms. Simulations of several complex buoyancy-driven thermal flows and including swilling effects in cylindrical geometries using the axisymmetric cascaded LB schemes show good agreement with prior benchmark results for the structures of the velocity and thermal fields as well as the heat transfer rates given in terms of the Nusselt numbers. Furthermore, the governing equations were solved by the lattice Boltzmann method (LBM), allowing us to understand the mechanisms of heat transport.