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.

Appendix:: Recalls on fractional calculus

In this section, a brief introduction to the fundamentals of fractional calculus will be given.

Consider the function f(x), $x\in \mathbb {R}$, the left the right Riemann-Liouville (RL) fractional integral are defined as [25]:

$$ \left( I_{+}^{\alpha} f\right)(x)=\frac{1}{{\Gamma}(\alpha)}{\int}_{-\infty}^{x} \frac{f(\xi)}{(x-\xi)^{1-\alpha}}d\xi $$

(23a)

$$ \left( I_{-}^{\alpha} f\right)(x)=\frac{1}{{\Gamma}(\alpha)}{\int}_{x}^{\infty} \frac{f(\xi)}{(\xi-x)^{1-\alpha}}d\xi $$

(23b)

while the RL fractional derivative are defined as

$$ \left( D_{+}^{\alpha} f\right)(x)=\frac{1}{{\Gamma}(1-\alpha)}\frac{d}{dx}{\int}_{-\infty}^{x} \frac{f(\xi)}{(x-\xi)^{\alpha}}d\xi $$

(24a)

$$ \left( D_{-}^{\alpha} f\right)(x)=-\frac{1}{{\Gamma}(1-\alpha)}\frac{d}{dx}{\int}_{x}^{\infty} \frac{f(\xi)}{(\xi-x)^{\alpha}}d\xi $$

(24b)

where $\alpha \in \mathbb {R}$, 0 ≤ α ≤ 1 and Γ(⋅) is the Euler gamma function. If f(x) is a continuous function with continuous first derivative, the left and right RL fractional derivatives are coincident with the Marchaud fractional derivatives, that may be written as follows:

$$ \left( \textbf{D}_{+}^{\alpha} f\right)(x)=\frac{\alpha}{{\Gamma}(1-\alpha)}{\int}_{-\infty}^{x} \frac{f(x)-f(\xi)}{(x-\xi)^{\alpha}}d\xi $$

(25a)

$$ \left( \textbf{D}_{-}^{\alpha} f\right)(x)=\frac{\alpha}{{\Gamma}(1-\alpha)}{\int}_{x}^{\infty} \frac{f(x)-f(\xi)}{(\xi-x)^{\alpha}}d\xi $$

(25b)

The Marchaud fractional derivatives may be defined also for a bounded domain 0 ≤ x ≤ L as

$$ \begin{array}{@{}rcl@{}} \left( \textbf{D}_{0^{+}}^{\alpha} f\right)(x)\!&=&\!\frac{\alpha}{{\Gamma}(1 - \alpha)}{{\int}_{0}^{x}} \frac{f(x) - f(\xi)}{(x - \xi)^{1+\alpha}}d\xi + \frac{f(x)}{{\Gamma}(1 - \alpha)x^{\alpha}} \end{array} $$

(26a)

$$ \begin{array}{@{}rcl@{}} \left( \textbf{D}_{L^{-}}^{\alpha} f\right)(x)\!&=&\!\frac{\alpha}{{\Gamma}(1 - \alpha)}{{\int}_{x}^{L}} \frac{f(x) - f(\xi)}{(\xi - x)^{1+\alpha}}d\xi + \frac{f(x)}{{\Gamma}(1 - \alpha)(L - x)^{\alpha}} \\ \end{array} $$

(26b)

The definitions of Marchaud fractional derivatives related to a single-variable scalar function may be extended to a multi-variable scalar function. The extension is more readable if referred to the Riesz fractional operators. Then it is necessary to introduce Riesz fractional integral $\left (\bar {I}^{\alpha } f\right )(x)$ and derivative $\left (\bar {D}^{\alpha } f\right )(x)$:

$$ \begin{array}{@{}rcl@{}} \left( \bar I^{\alpha} f\right)(x)&=&\!\nu(\alpha){\int}_{-\infty}^{\infty} \frac{f(\xi)}{|x-\xi|^{1-\alpha}}d\xi \\&=&\!\nu(\alpha)\left[\left( I_{+}^{\alpha} f\right)(x)+\left( I_{-}^{\alpha} f\right)(x)\right] \end{array} $$

(27a)

$$ \begin{array}{@{}rcl@{}} \left( \bar D^{\alpha} f\right)(x)&=&\!\nu(-\alpha){\int}_{-\infty}^{\infty} \frac{f(x-\xi)-f(x)}{|\xi|^{1+\alpha}}d\xi\\ &=&\!{\Gamma}(1 - \alpha)\nu(-\alpha)\left[\left( \textbf{D}_{+}^{\alpha} f\right)(x) + \left( \textbf{D}_{-}^{\alpha} f\right)(x)\right] \\ \end{array} $$

(27b)

where ν(±α) = [2 cos(απ/2)Γ(±α)]^− 1. The Riesz fractional operator may be generalized to multivariate scalar function f(x), with $\textbf {x}\in \mathbb {R}^{n}$:

$$ \begin{array}{@{}rcl@{}} \left( \bar D^{\alpha} f\right)(\textbf{x})&=&\frac{1}{d_{n,\bar l}(\bar\alpha)}{\int}_{\mathbb{R}^{n}} \frac{f({\boldsymbol \xi})-f(\textbf{x})}{||{\boldsymbol \xi}-\textbf{x}||^{n+\alpha}}d{\boldsymbol \xi}\\&=& \frac{\chi(\bar\alpha)}{d_{n,\bar l}(\bar\alpha)}\left[\left( \textbf{D}_{+}^{\alpha} f\right)(\textbf{x})+\left( \textbf{D}_{-}^{\alpha} f\right)({\boldsymbol x})\right] \end{array} $$

(28)

where

$$ d_{n,l}(\alpha)=\beta_{n}(\alpha)\frac{A_{l}(\alpha)}{\sin(\alpha\pi/2)} $$

(29a)

$$ \beta_{n}(\alpha)=\frac{\pi^{1+n/2}}{2^{\alpha} {\Gamma}(1+\alpha/2){\Gamma}(n+\alpha/2)} $$

(29b)

$$ A_{l}(\alpha)=\sum\limits_{k = 0}^{l}(-1)^{k-1}{l\choose k}k^{\alpha} $$

(29c)

and χ(α) = −A_l(α)Γ(α), $\bar \alpha =n-1+\alpha$, $\bar l=n-1+l$, l = {α} + 1 and {α} is the integer part of α. The complete demonstration of Eq. 28 is omitted here for the sake of brevity; more information can be found in [26].

Finally, we briefly introduce the n-dimensional CMFD as

$$ \left( \textbf{D}_{-}^{\alpha} f\right)({\boldsymbol x})=\frac{\alpha}{{\Gamma}(1-\alpha)}{\int}_{\mathbb{R}^{n}} \frac{f({\boldsymbol x})-f({\boldsymbol \xi})}{({\boldsymbol \xi}-{\boldsymbol x})^{n+\alpha}}{\boldsymbol J}_{kj}d{\boldsymbol \xi} $$

(30)

where J_kj = i_ki_j is a Jacoby directional tensor, being i_k the unit vector associated with the direction x −ξ. In the specific problem treated in this paper, the governing equation written in polar coordinates and in axial-symmetric conditions is basically a scalar governing equation, then the Jacoby tensor reduce to unity; this is equivalent to say that the attenuation function, that is responsible for the appearance of fractional operator, reduces in this case to a scalar function. As a consequence, in the governing equation in Eq. 21, the integral term may be recognized as the integral part of the Marchaud fractional derivative defined in bounded domain and reported in Eq. 21. More details can be found in [30].