In this paper constitutive hydrodynamic equations are numerically solved for a nematic liquid crystal in a cylindrical sample rotating around its axis at a constant angular velocity and subject to a magnetic field. The computational solution of the problem is achieved in three dimensions by uncoupling the velocity and the director fields, and assuming the velocity field of a classical Newtonian fluid. The influence of initial and boundary conditions is considered. The formation and time evolution of patterns is calculated and discussed in terms of average behaviour and spatial distributions, which give evidence of the complex local behaviour of the nematic director.