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

Abstract

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.

Alternative fractional derivative operator on non-newtonian calculus and its approaches

“…The fractional integral operator on bi-geometric calculus can be written by substituting ( ) instead of ( ) at (4) which is (Alotta et. al, 2019).…”

“…The expression in the braces at (11) can be considered as the Hadamard integral operator on ( )which is defined as (Alotta et. al, 2019).…”

Advanced materials modelling via fractional calculus: challenges and perspectives

“…A fractional lattice approach has been proposed by Michelitsch et al [98] for n-dimensional periodic and infinite lattices, introducing the concept of centred fractional-order difference operators as a generalization of the second-order centred difference operator appearing in the context of classical lattice models [98]; relations have been found between the fractional-order difference operators in the continuum limit and the classical Riesz fractional Laplacian derivative [98,99]. Non-local spatial fractional operators have been also used to model blood flow in capillary vessels, see [100,101], as well as long-range viscoelastic interactions [102,103].…”

Applications of Distributed-Order Fractional Operators: A Review

“…Unlike integer-order operators, the intrinsic multiscale nature of fractional operators enabled a very unique and effective approach to model historically challenging physical processes involving, as an example, nonlocality or memory effects. Indeed, many of the early applications of FC to physical modeling included viscoelastic effects [ 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 ], nonlocal behavior [ 8 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 ], anomalous and hybrid transport [ 9 , 10 , 11 , 24 , 25 , 26 , 27 , 28 , 29 , 30 ], fractal media [ 12 , 31 , 32 , 33 , 34 , 35 ], and even control theory [ 36 , 37 , 38 , 39 ]. The interested reader is referred to the work in [ 40 ] for a detailed account of the birth and evolution of fractional calculus.…”