A small flexural wall rigidity brings unique features to cross-sectional shapes and blood flow within veins, which are characterised by a non-uniform hemodynamical environment acting upon endothelial cells. Velocity fields and related wall shear stress were numerically determined for a large number of conditions, assuming a fully developed, steady, incompressible laminar flow through an uniform smooth pipe with a constant cross-section. It was shown that the flatness greatly influences 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 a constant oval-shaped tube, with a lumen perimeter of the order of 5 × 10−2 meter, the maximum wall shear stress is found at about 2 Pascal where the local curvature is minimal. On the other hand, the minimal wall shear stress of the order of 1 Pascal is found where the local curvature is maximal. Clear indications have been reported showing that the hemodynamical wall shear stress does alter endothelial cell morphology and orientation. These results are being used for developing an experimental set-up in order to locally map out the characteristic shear stresses looking for endothelial shape modifications whenever a viscous fluid flow is applied.