A small flexural wall rigidity brings unique features to cross-sectional shapes and bloodflow within veins, which are characterised by a non-uniform hemodynamical environment actingupon endothelial cells. Velocity fields and related wall shear stress were numerically determinedfor a large number of conditions, assuming a fully developed, steady, incompressible laminar flowthrough an uniform smooth pipe with a constant cross-section. It was shown that the flatness greatlyinfluences the resulting distribution of the wall shear stresses along the lumen perimeter. For instance, under a steady longitudinal pressure gradient at about 500 Pascal per meter inside aconstant oval-shaped tube, with a lumen perimeter of the order of 5 × 10−2 meter, themaximum wall shear stress is found at about 2 Pascal where the local curvature is minimal. Onthe other hand, the minimal wall shear stress of the order of 1 Pascal is found where the localcurvature is maximal. Clear indications have been reported showing that the hemodynamical wall shear stress does alterendothelial cell morphology and orientation. These results are being used for developing anexperimental set-up in order to locally map out the characteristic shear stresses looking for endothelial shape modifications whenever a viscous fluid flow is applied.