A fractional nonlocal approach to nonlinear blood flow in small-lumen arterial vessels

The behavior of human blood flowing in arteries is still an open topic for its multi-phase nature and heterogeneity. In large arterial vessels the well-known Hagen-Poisueille law, which main assumption is that the blood is Newtonian, is considered acceptable. In small arterial vessels, instead, this law does not reproduce experimental results that show non-parabolic profiles of velocity across the vessel diameter. For capillary vessels the Casson model of fluids that is nonlinear is used in place the Newton law, resulting in nonlinear governing equations and difficulties in mathematical manipulation. For these reasons an alternative approach is proposed in this paper. Starting from the micro-mechanics of blood, the Hagen-Poisueille model is enriched with long-range interactions that simulate the interactions of non-adjacent fluid volume elements due to the presence of red blood cells and other dispersed cells in the plasma. These nonlocal forces are defined as linearly dependent on the product of the volumes of the considered elements and on their relative velocity. Moreover, as the distance between two volume elements increases, the nonlocal forces are scaled through an attenuation function; if this function is chosen as a power law of real order of the distance between the volume elements, an operator related to the fractional derivative of relative velocity appears in the resulting governing equation. It is shown that the fractional Hagen-Poisueille law is able to reproduce experimentally measured profiles of velocity with a great accuracy, moreover as the dimension of the vessel increases, nonlocal forces become negligible and the proposed model reverts to the classical Hagen-Poisueille model.