A Fractional Approach to Non-Newtonian Blood Rheology in Capillary Vessels

In small arterial vessels, fluid mechanics involving linear viscous fluid does not reproduce experimental results that correspond to non-parabolic profiles of velocity across the vessel diameter. In this paper, an alternative approach is pursued introducing long-range interactions that describe the interactions of non-adjacent fluid volume elements due to the presence of red blood cells and other dispersed cells in plasma. These non-local 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 non-local forces decay with a material distance-decaying function. Assuming that decaying function belongs to a power-law functional class of real order, a fractional operator of the relative velocity appears in the resulting governing equation. It is shown that the mesoscale approach involving Hagen-Poiseuille law is able to reproduce experimentally measured profiles of velocity with a great accuracy. Additionally as the dimension of the vessel increases, non-local forces become negligible and the proposed model reverts to the classical Hagen-Poiseuille model.