Natural convection in horizontal cylinders with differentially heated ends is of practical interest for many applications. A knowledge of the flow pattern and heat transfer within the cylinder is important for optimising processes. Numerical solutions of the three-dimensional equations for buoyancy driven flows in a horizontal differentially heated cylinder are presented. The governing non-linear coupled equations (vorticity-vector potential) are approximated using finite differences. The energy and vorticity transport equations are solved using the Samarskii-Andreyev ADI scheme. A fast Fourier transform algorithm is used to solve the elliptic partial differential equations. Solutions are presented for an aspect ratio (length to radius) of 10, Prandtl number (Pr=0.73) and Rayleigh number (based on the radius) 100 ≤ Ra ≤ 20000. The ratio of the heated length to the length of cylinder is varied from 0.125 to 0.5.