# Hemodynamic Analysis on the Anastomosis Angle in Arteriovenous Graft Using Multiphase Blood Model
## 3 Numerical Details  
### Governing Equations  
The multiphase blood model was used to consider the shear thinning effect of blood according to the hematocrit [21; 23]. The multiphase blood model calculates the mass, momentum, and energy exchange between two phase substances by assuming that red blood cells are solid in plasma. The multiphase blood model equation is as follows. Continuity equation: The continuity equation is expressed as Equation (1). Here, the subscript k denotes plasma and red blood cell phases. The subscript \(\varepsilon\) denotes the volume fraction of each phase.  
\[\frac{\partial\left(\rho_{\mathrm{k}}\varepsilon_{\mathrm{k}}\right)}{ \partial\mathrm{t}}+\
abla\cdot\left(\rho_{\mathrm{k}}\varepsilon_{\mathrm{k}} \vec{v_{\mathrm{k}}}\right)=0 \tag{1}\]  
\[\sum_{\mathrm{k}=1}^{\mathrm{np}}\varepsilon_{\mathrm{k}}=1.0 \tag{2}\]Blood mixture density (3).  
\[\mathrm{P_{mix}}=\epsilon_{\mathrm{plasma}}\mathrm{\rho_{plasma}}+\epsilon_{ \mathrm{RBC}}\mathrm{\rho_{RBC}} \tag{3}\]  
Momentum equation: momentum equation is expressed as Equation (2).  
\[\begin{array}{c}\frac{\partial\left(\rho_{\mathrm{k}}\epsilon_{\mathrm{k}} \right)}{\partial t}+\
abla\cdot\left(\rho_{\mathrm{k}}\epsilon_{\mathrm{k}} \overrightarrow{\mathrm{v}}_{\mathrm{k}}\overrightarrow{\mathrm{v}}_{ \mathrm{k}}\right)=\\ -\epsilon_{\mathrm{k}}\
abla\rho+\
abla\overline{\tau}_{\mathrm{k}}+\epsilon_{ \mathrm{k}}\rho_{\mathrm{k}}\overrightarrow{\mathrm{g}}+\sum\beta_{\mathrm{ k}}(\overrightarrow{\mathrm{v}_{\mathrm{1}}}-\overrightarrow{\mathrm{v}}_{ \mathrm{k}})+\overrightarrow{\mathrm{F}}_{\mathrm{k}}\\ 1=\mathrm{l},\mathrm{m},\mathrm{n}\\ 1\
eq\mathrm{k}\end{array} \tag{4}\]  
Plasma stress tensor (5).  
\[\overline{\tau}_{\mathrm{plasma}}=\epsilon\mathrm{\mu_{plasma}}(\
abla \overrightarrow{\mathrm{v}}+(\
abla\overrightarrow{\mathrm{v}})^{T})+ \epsilon(\mathrm{K}-\frac{2}{3})\
abla\overrightarrow{\mathrm{v}} \overrightarrow{\mathrm{I}} \tag{5}\]  
RBC stress tensor (6).  
\[\overline{\tau}_{\mathrm{RBC}}=-\mathrm{p}\delta+\epsilon\mathrm{\mu_{RBC}}( \
abla\overrightarrow{\mathrm{v}}+(\
abla\overrightarrow{\mathrm{v}})^{T})+ \epsilon(\mathrm{K}-\frac{2}{3})\
abla\overrightarrow{\mathrm{v}} \overrightarrow{\mathrm{I}} \tag{6}\]  
The viscosity of the mixture fluid is determined by the volume fraction of each phase. Because of these characteristics, the multiphase blood model has advantages for depicting flow characteristics where the hematocrit varies locally, such as in hemodialysis. The relative viscosity of the mixture fluid is the same as Equation (7).  
\[\eta=\frac{\epsilon_{\mathrm{RBC}}\mathrm{\mu_{RBC}}+\epsilon_{\mathrm{plasma }}\mathrm{\mu_{plasma}}}{\mathrm{\mu_{plasma}}}=\mathrm{m}[1+(\lambda \dot{\gamma})^{2}]^{(n-1)/2} \tag{7}\]  
In this model, the percentage of red blood cells is assumed as the hematocrit.  
For \(\dot{\gamma}\) greater than 6.  
\[\mathrm{n}=0.8092\epsilon_{\mathrm{RBC}}^{2}+0.8246\epsilon_{\mathrm{RBC}}^{2} -0.3503\epsilon_{\mathrm{RBC}}+1\mathrm{m}=122.28\epsilon_{\mathrm{RBC}}^{2} -51.2136\epsilon_{\mathrm{RBC}}^{2}-16.305\epsilon_{\mathrm{RBC}}+1\]  
For \(\dot{\gamma}\) less than 6.  
\[\frac{\mathrm{n}-1}{\mathrm{k}_{0}}=-0.8913\epsilon_{\mathrm{RBC}}^{2}+2.0679 \epsilon_{\mathrm{RBC}}^{2}-1.7814\epsilon_{\mathrm{RBC}}\mathrm{m}=70.782 \epsilon_{\mathrm{RBC}}^{2}-22.454\epsilon_{\mathrm{RBC}}^{2}-9.7193\epsilon_{ \mathrm{RBC}}+1\]  
where \(\mathrm{k}_{0}=\mathrm{ln}\left(\ln\overline{\dot{\gamma}}\right)/\ln\overline {\dot{\gamma}}\).  
Figure 2 shows the viscosity calculated according to the hematocrit of the multiphase blood model used in this study. Viscosity shows shear thinning characteristics that decrease with increasing shear rate. In addition, as the hematocrit increases, the viscosity increases relatively significantly. These characteristics are expected to have a significant effect on WSS on the venous side, where high hematocrit flows during hemodialysis.