In the present study, a general method for obtaining the optimal geometry of vascular systems is developed. For this purpose, a Y-shaped configuration that consists of the mother branch and two daughter branches is used. The total energy loss which is the sum of the viscous dissipation and metabolic loss is minimized. Minimizing the total energy loss, results in the optimal distribution of WSS. Using optimal distribution of WSS and the continuity equation at the junction of bifurcation, a relationship between diameters of mother and daughter vessels is obtained. Also, the optimal distribution of WSS leads to viscous loss that is proportional to the volume of conduit. Therefore, minimization of the volume of bifurcation results in the optimal bifurcation angle. Moreover, it is shown that the optimal WSS provides minimal flow resistance. The optimal relationship between diameters of the mother and daughter vessels is obtained for both Newtonian and non-Newtonian fluids. Also, turbulent flows are considered using, the non-dimensional friction factor as a function of Reynolds number. It is shown that for both Newtonian and non-Newtonian fluids the optimal relationship between the mother and daughter branches are the same as Murray’s law. However, for turbulent flows the relationship between diameters of the mother and daughter branches are a function of the tube roughness. Moreover, it is shown that the optimal WSS is obtained a constant for laminar flows. However, the optimal WSS in turbulent flows is a function of tube roughness and diameters. Based on the arbitrary distribution of WSS a general formulation for optimal bifurcation angle is proposed. For the derivation of the optimal bifurcation angle, the power law model is used. It is observed that increasing the WSS of daughter branches compared to the value of mother branch leads to smaller bifurcation angle. In order to investigate the effect of the Reynolds number on the analytical results numerical simulations have been performed. Using non-Newtonian power law and Carreau models, analytical results have been tested. For the simulation a Y-shaped configuration with a mother branch and two daughter branches has been chosen. The flow resistance has been computed for different ratios of the diameter of the mother branch to the diameter of the daughter branches. The results show that by increasing Re, the optimal diameters ratio changes form 0.8, (predicted by analytical formulations) to one. Also, the increase of Re leads to the smaller optimal bifurcation angles rather than 75° which is predicted analytically. In other words, when the inertia effect becomes dominant smaller bifurcation angles leads to smaller energy dissipation in the bifurcation. Key words : Vascular System, Murray's law, Flow Resistance, Constructal Theory
