# Application of Mathematical Models for Blood Flow in Aorta and Right Coronary Artery
## 2 Materials and Methods
### Mathematical Models  
1. Newtonian Model: Assumes a constant viscosity irrespective of the shear rate. This model is suitable for large-caliber vessels like the aorta, where shear rates are high, and blood behaves nearly like a Newtonian fluid [2; 3; 4]. The Newtonian model simplifies calculations, making it computationally efficient but less accurate for small or highly deformable vessels. 2. Non-Newtonian Shear-Thinning Models:
* If \(\dot{\gamma}\ll\frac{1}{\lambda}\), the fluid behaves as a Newtonian fluid. - If \(\dot{\gamma}\gg\frac{1}{\lambda}\), the fluid exhibits power-law fluid behavior. The parameter values of the Carreau Model can be considered as follows (as reported in [PERSON] et al. [6]): \[\mu_{\infty}=0.0035\,\mathrm{Pa}\cdot\mathrm{s},\quad\mu_{0}=0.056\,\mathrm{ Pa}\cdot\mathrm{s},\quad\lambda=3.313\,\mathrm{s},\quad n=0.3568.\]
* Casson Model: Incorporates a yield stress (\(\tau_{y}\)) below which the fluid does not flow. It is particularly suited for blood, considering the aggregation of red blood cells at low shear rates. Additionally, it predicts viscosity changes due to hematocrit (Hct) variations. The model is described by the following equations: \[\mu=\frac{\tau_{y}}{\dot{\gamma}}+\frac{2\sqrt{\tau_{y}}\,N_{\infty}}{\sqrt{ \dot{\gamma}}}+N_{\infty}^{2}\] \[N_{\infty}=\frac{\sqrt{\mu_{p}}}{8\sqrt{1-\mathrm{Hct}}}\] where \(\mu\): dynamic viscosity (in \(\mathrm{Pa}\cdot\mathrm{s}\)), \(\tau_{y}\): yield stress, the minimum stress required for the fluid to flow (in \(\mathrm{Pa}\)), \(\dot{\gamma}\): shear rate (in \(\mathrm{s}^{-1}\)), \(N_{\infty}\): a parameter related to viscosity changes due to hematocrit variations (dimensionless), \(\mu_{p}\): plasma viscosity, a constant property of the plasma (in \(\mathrm{Pa}\cdot\mathrm{s}\)), and Hct: hematocrit, the volume fraction of red blood cells in the blood (dimensionless, typically expressed as a percentage). The parameter values for this model, as reported in [PERSON] et al. [6], are as follows: \[\tau_{y}=0.625\,\mathrm{Hct},\quad\mathrm{Hct}=0.4,\quad\mu_{p}=0.00145\, \mathrm{Pa}\cdot\mathrm{s}.\] Here, \(\tau_{y}\) is linearly proportional to Hct and the Hct value is 40% of blood volume. This model effectively captures the non-Newtonian behavior of blood and accounts for the effects of both yield stress and hematocrit on viscosity. * Viscoelastic Models [6]:
* Oldroyd-B Model: The model describes blood as a viscoelastic fluid, combining a Newtonian solvent with elastic polymer-like components. Despite its computational simplicity, it lacks precision in capturing shear-thinning effects. The governing equation is given as follows: \[\tau+\lambda_{1}\bigg{(}\frac{\partial\tau}{\partial t}\bigg{)}=\eta_{0}\bigg{(} \dot{\gamma}+\lambda_{2}\frac{\partial\dot{\gamma}}{\partial t}\bigg{)}\] where \(\tau\) represents the stress tensor, \(\lambda_{1}\) is the relaxation time, \(\lambda_{2}\) is the retardation time, \(\eta_{0}\) is the zero-shear viscosity, and \(\dot{\gamma}\) denotes the shear rate. This model provides a basic framework for studying viscoelastic behaviors in complex fluids such as blood. * Giesekus Model: A nonlinear extension of the Oldroyd-B model, the Giesekus model incorporates a mobility parameter (\(\alpha\)) to account for shear-thinning and elastic behaviors, making it more realistic for describing blood flow in geometrically complex vessels. The governing equation is expressed as follows: \[\tau+\lambda_{1}\bigg{(}\frac{\partial\tau}{\partial t}-\mathbf{L}\cdot\tau- \tau\cdot\mathbf{L}^{T}\bigg{)}+\frac{\alpha\lambda_{1}}{\eta_{0}}\tau\cdot \tau=\eta_{0}\dot{\gamma},\] where \(\alpha\) is the mobility parameter (\(0\leq\alpha\leq 1\)), \(\mathbf{L}\) represents the velocity gradient, \(\tau\) is the stress tensor, \(\lambda_{1}\) is the relaxation time, \(\eta_{0}\) is the zero-shear viscosity, and \(\dot{\gamma}\) denotes the shear rate.