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.