A numerical treatment is developed and implemented to solve the constitutive equations of nematodynamics according to the Leslie-Ericksen formulation for a cylindrical geometry. Nematic director equations in three dimensions are coupled to a Navier-Stokes description of the velocity. This model is used to study the influence of the sample size for an experimental set-up corresponding to an combined electron spin resonance (ESR) rheological experiment in which the nematic sample is subject to a constant rotational velocity and to an aligning magnetic field perpendicular to the axis of rotation. The results of the simulations are employed to interpret experimental findings for a series of measurements of the ESR signal of tempone-N-14 dissolved in capillaries of different sizes containing the nematic liquid crystal ZLI 1083, with diameters ranging from 0.1 to 0.5 mm.