Numerical methodology
The flow of blood has been described as a laminar flow, steady state, and 2D incompressible viscous blood, Newtonian fluid, and unidirectional flow. The present set of simulations has been performed for different sizes of the blood vessels diameters (100-500µm). The length of the tube is assumed to be large enough compared to its diameter. The permeability of the porous medium has been assumed to vary in function of the distance from the centre of the vessel in the radial direction. The governing mass and momentum conservation equations for solving isothermal fluid flow inside the blood vessels are given by [20]:
\(\frac{\text{Dρ}}{\text{Dt}}+\rho\nabla.\mathbf{v=}0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)(1)
\(\frac{D\mathbf{v}}{\text{Dt}}=-\frac{1}{\rho}\nabla P+\frac{1}{\rho}\nabla.\mathbf{\tau}\mathbf{+}S\mathbf{\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }}\ \)(2)
Whit\(\ D,\ \ \rho,\ \ t,\ \ \mathbf{v},\ \ P,\ \ \mathbf{\tau},\ \ S\ \)being the fluid parameters essential derivative, density, time, velocity vector, pressure of the fluid, shear stress and volumetric source term, respectively.
Meanwhile the blood inside the porous region has been described using the Darcy– Brinkman equation as shown by Eq. (3). The terms in Eq. (3) carries the viscous and the form drag interactions between the fluid and the walls.
\(\frac{1}{\varepsilon}\frac{\partial\mathbf{v}}{\partial t}+\frac{1}{\varepsilon^{2}}\mathbf{v}.\left(\nabla^{2}\mathbf{v}\right)=\frac{1}{\rho}\nabla P+\frac{\nu}{\varepsilon}\nabla^{2}\mathbf{v}+\frac{\mu}{K}\mathbf{v}\mathbf{\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }}\)(3)
Where μ is the dynamic viscosity \(\ (m^{2}s^{-1})\),\(\varepsilon\) is the porosity of the porous medium and\(\ K(m^{-2})\), being the permeability. The axial velocity gradient is presented along the axis of symmetry. The arterial wall is considered as fixed.