Scaling laws and thermodynamic analysis for vascular branching of microvessels

When blood flows through small vessels, the two-phase nature of blood as a suspension of red cells (erythrocytes) in plasma cannot be neglected, and with decreasing vessel size, a homogeneous continuum model become less adequate in describing blood flow. Following the Haynes’ marginal zone theory, and viewing the flow as the result of concentric laminae of fluid moving axially, the present work provides models for fluid flow in dichotomous branching composed by larger and smaller vessels, respectively. Expressions for the branching sizes of parent and daughter vessels, that provides easier flow access, are obtained by means of a constrained optimization approach using the Lagrange multipliers. This study shows that when blood behaves as a Newtonian fluid, Hess – Murray law that states that the daughters-to-parent diameter ratio must equal to 2^(-1/3) is valid. However, when the nature of blood as a suspension becomes important, the expression for optimum branching diameters of vessels is dependent on the separation phase lengths. It is also shown that the same effect occurs for the relative lengths of daughters and parent vessels. For smaller vessels (e. g., arterioles and capillaries), it is found that the daughters-to-parent diameter ratio may varies from 0,741 to 0,849, and the daughters-to-parent length ratio varies from 0,260 to 2,42. For larger vessels (e. g., arteries), the daughters-to-parent diameter ratio and the daughters-to-parent length ratio range from 0,458 to 0,819, and from 0,100 to 6,27, respectively. In this paper, it is also demonstrated that the entropy generated when blood behaves as a single phase fluid (i. e., continuum viscous fluid) is greater than the entropy generated when the nature of blood as a suspension becomes important. Another important finding is that the manifestation of the particulate nature of blood in small vessels reduces entropy generation due to fluid friction, thereby maintaining the flow through dichotomous branching vessels at a relatively lower cost.