## 2 Governing Principles of Biomechanics
### Fluid Mechanics of Blood Flow  
For the study of fluids, it is more practical to implement an Eulerian formulation of the governing equations. This formulation is obtained by applying the Reynolds transport theorem to the equations for mass and momentum balance. Thus, this formulation solves the relation between flow driving forces, flow velocity, and deformation rates (Figure 2). A simplifying assumption applicable to biological systems is the incompressibility of the fluids, as most of them are either liquids or gases moving at subsonic velocities. Additionally, it is convenient to decompose the stress tensor into a spherical tensor representing the hydrostatic pressure and a deviatoric stress tensor. With this decomposition, constitutive equations can be designed to specifically relate the deviatoric stress components to the viscous dissipation of momentum.  
Blood flow is generally assumed to be laminar throughout the circulatory system. The main arguments for this assumption are the pulsatile nature of the flow, the reduced dimensions of the vessels, and relatively low velocities, each contributing to the viscous effects overcoming the inertial forces and preventing turbulent random motion. However, it has been argued that transition to turbulent flows could be achieved locally in stenotic arteries. The use of a laminar model to study those cases could lead to an underestimation of wall shear stress, and stress oscillation [58, 59]. Unlike most conventional engineering flows, blood flow is pulsatile and contained by compliant conduits of complex geometry. Since the 1950s, [PERSON] [60], [PERSON] [61], [PERSON] [62], [PERSON] [63], and others, developed analytical and experimental studies of pulsatile flow in mammals, identifying the most relevant parameters and features of this type of flow, thus setting the bases for modern hemodynamics.  
Besides the intricacies of pulsatile flow in distensible conduits, the blood itself is a complex fluid. Blood consists of a suspension of cells in an aqueous solution of proteins and minerals called plasma. Plasma occupies approximately 55% of the blood volume, the rest being mainly occupied by red blood cells, white blood cells, and platelets. The rheological behavior of blood depends on how its constituents interact with each other and with the vessel walls, in consequence, this behavior is non-linear and highly dependent on the volumetric composition of blood, the flow conditions, and vessel dimensions. Modeling the complex interactions of blood constituents is a challenging statistical mechanics problem [64]. Some researchers have shown that cell aggregation and disaggregation are relevant to accurately describing blood rheology, especially in capillary flows where the cell size is comparable to the vessel diameter. Multiscale approaches have been successful in coupling the behavior of single cells as elastic entities with the transport equations of fluid flow, which are relevant for the study of clotting, aggregation, and platelet activation [65, 66]. These approaches are computationally expensive, making them impractical for the study of large vessels.