Abstract: In this work, the rectilinear flow of a complex fluid (human blood) under a pulsating time-dependent pressure gradient is analyzed. A first approximation of the real case of blood flowing in a vein is described. The pressure gradient simulates the pumping work of the heart while the flow geometry (circular tube) is assumed rigid, smooth and cylindrical. The rheological behavior of blood with different cholesterol levels is modelled using the Bautista-Manero-Puig (BMP) constitutive equation. According to the analytical solution, a flow enhancement is predicted to first order which represents the optimum pumping work of the heart which governs the flow of blood in the entire body. This work is a contribution to the understanding of the complex rheology involved in the discontinuous pressure driven flow of blood in the human body.