Veins in the cardiovascular system may collapse if the internal pressure is less than the external pressure. Such collapse or buckling will have important consequence in terms of the rate of blood flow. Here a steady, parallel unidirectional flow as an exact solution of the continuity and the Navier Stokes equations is constructed. Various stages of the deformation process of the elastic tube (before contact of opposite sides occurring), from an ellipse to a ;strongly buckled' configuration, are obtained in analytical forms as a by-product of the calculations. The pressure -- area and the pressure -- flow rate diagrams computed numerically from the model agree with the trends measured experimentally.