2.1. Governing Equations for Hemodynamic Flow

OpenFOAM v1912 is software used to solve fluid flow problems by numerically solving the governing equations for blood flow using the SIMPLE and PIMPLE algorithms. These algorithms can be used to perform steady-state and transient simulations, respectively, and are capable of handling 3D laminar, turbulent, steady, and unsteady flows and compressible and incompressible flows. When calculating blood flow in OpenFOAM, the software uses the incompressible 3D Navier–Stokes formula (Equation (1)) to model the flow of the fluid.

ρ—density of blood, kg/m³;

υi—i component of velocity, m³/s;

Vi,t—velocity derivative with respect to time;

p—pressure, Pa;

τij—stress tensor (viscous portion).

OpenFOAM uses the finite volume method (FVM) and SIMPLE/SIMPLEC/PISO pressure correction schemes to solve the Navier–Stokes equations for incompressible fluid flow. The domain of the equation is defined as Ω in three dimensions, and the boundary conditions are specified as Γ=ΓD∪ΓN, where ΓD represents Dirichlet boundary conditions and ΓN represents Neumann boundary conditions. The domain is discretized using nel linear, unstructured tetrahedral elements,Ω−e. The final form of the equation is given by Equation (3).

where w∈Wnk and q∈Phk.