The flow behaviour and the molecular orientation of many anisotropic fluids, like nematic liquid crystals, can be described by the Ericksen-Leslie equations. A mapping from the physical domain to a computational domain is needed for the finite-difference solution of the equations. The numerical solution of the equations of motion using an implicit scheme, which is required for stability reasons in the evaluation of the linear momentum equations, is not straightforward due to the complexity of the divergence of the stress tensor in the computational domain. To overcome this difficulty the mapped linear momentum equations are, first, evaluated symbolically using a computer program designed for this purpose and then the analytical results are used in the numerical solution of the equations of motion. A contracted channel flow is taken as a case study and some simulation results are given.