This work is concerned with a numerical study of the laminar flow of a viscous fluid between two coaxial cones, with the inner one rotating on which a constant axial flow is imposed. The two cones have the same apex angle value, so that the gap width between them is constant. Two configurations of the same volume are studied: (i) divergent flow passage (Div) and (ii) convergent flow passage (Cov), which represents geometrically the reverse case of (i). Analysis with the aid of the boundary layer assumptions is carried out by the use of an implicit finite difference method. A coordinate transformation is applied to the governing equations in order to remove the explicit effect of the apex angle from the calculation process for both cases, so that the boundary conditions can be simplified. The swirl characteristics are studied in both configurations with regard to the same inlet flux and the same rotation speed of the inner cone. The competition between centrifugal and axially-transporting effects in both cases is discussed. As a result, it is found that the swirl increases in Div from the entrance for a fixed value of the apex angle, while the swirl evolution in Cov depends strongly on the apex angle value.