At very low Reynolds numbers, we calculate the drag force exerted on a circular cylinder in cross flow fixed midway between two parallel plane walls which are fixed while the fluid experiences a Poiseuille profile at upstream and downstream. The drag wall correction factor is numerically investigated from a very weak interaction to the lubrication regime. The Navier-Stokes and continuity equations are expressed in the stream function and vorticity formulation and are rewritten in an orthogonal system of curvilinear co-ordinates. These equations are solved with using a finite differences method. The generation of the grid was carried out by the singularities method. We calculated the separate contributions of the pressure and viscous forces numerically. At very weak interactions, our numerical results are in good agreement with those obtained analytically by Harrison (Trans. Camb. Phil. Soc. 23 (1924) 7 1) and Faxen (Proc. Roy. Swed. Acad. Eng. Sci. 187 (1946) 1). In the lubrication regime these numerical calculations are in very good agreement with those we carried out by asymptotic expansion. So that, the accuracy of the numerical code is tested. This analysis allowed us to show how that the pressure term prevails over the viscosity term in the lubrication regime. At very weak interaction, these forces have the same value. (C) 2004 Published by Elsevier Ltd.