# Bladeless Heart Pump Design: Modeling and Numerical Analysis
## 2 Theoretical Analysis of Heart Pump Parameters
\[\frac{\partial\rho}{\partial t}+\frac{1}{r}\frac{\partial}{\partial r}(\rho rv _{r})+\frac{1}{r}\frac{\partial}{\partial\theta}(\rho v_{\theta})+\frac{ \partial}{\partial z}(\rho v_{z})=0 \tag{2}\]  
\[\rho\frac{\partial v_{r}}{\partial t}+\rho\Bigg{(}v_{r}\frac{\partial v_{r}}{ \partial r}+\frac{v_{\theta}}{r}\frac{\partial v_{\theta}}{\partial\theta}- \frac{v_{\theta}^{2}}{r}+v_{z}\frac{\partial v_{r}}{\partial z}\Bigg{)}=\rho f _{r}-\frac{\partial p}{\partial r}+\mu\bigg{[}\frac{1}{r}\frac{\partial}{ \partial r}(rv_{r})\bigg{)}+\frac{1}{r^{2}}\frac{\partial^{2}v_{r}}{\partial \theta^{2}}+\frac{\partial^{2}v_{r}}{\partial z^{2}}-\frac{2}{r^{2}}\frac{ \partial v_{\theta}}{\partial\theta}\bigg{]} \tag{3}\]  
\[\rho\frac{\partial v_{\theta}}{\partial t}+\rho\Bigg{(}v_{r}\frac{\partial v_{ \theta}}{\partial r}+\frac{v_{\theta}}{r}\frac{\partial v_{\theta}}{\partial \theta}+\frac{v_{r}v_{\theta}}{r}+v_{z}\frac{\partial v_{\theta}}{\partial z} \Bigg{)}=\rho f_{\theta}-\frac{1}{r}\frac{\partial p}{\partial\theta}+\mu \bigg{[}\frac{\partial}{\partial r}\bigg{(}\frac{1}{r}\frac{\partial}{ \partial r}(rv_{\theta})\bigg{)}+\frac{1}{r^{2}}\frac{\partial^{2}v_{\theta}}{ \partial\theta^{2}}+\frac{\partial^{2}v_{\theta}}{\partial z^{2}}+\frac{2}{r^ {2}}\frac{\partial v_{r}}{\partial\theta}\bigg{]} \tag{4}\]  
\[\rho\frac{\partial v_{z}}{\partial t}+\rho\bigg{(}v_{r}\frac{\partial v_{z}}{ \partial r}+\frac{v_{\theta}}{r}\frac{\partial v_{z}}{\partial\theta}+v_{z} \frac{\partial v_{z}}{\partial z}\bigg{)}=\rho f_{z}-\frac{\partial p}{\partial z }+\mu\bigg{[}\frac{1}{r}\frac{\partial}{\partial r}\bigg{(}r\frac{\partial v_ {z}}{\partial r}\bigg{)}+\frac{1}{r^{2}}\frac{\partial^{2}v_{z}}{\partial \theta^{2}}+\frac{\partial^{2}v_{z}}{\partial z^{2}}\bigg{]} \tag{5}\]  
Considering the assumptions above, the following expressions in Equations (6) and (7) are derived by solving Equations (2)-(5). \[v_{r}=0,\hskip 28.452756 pt\rho\frac{v_{\theta}^{2}}{r}=\frac{\partial p}{ \partial r}\,\hskip 28.452756 pt\frac{\partial p}{\partial z}=0 \tag{6}\]  
\[v_{\theta}(z)=\frac{1}{3\mu}\frac{\Delta p}{r\pi}\cdot\Big{[}z^{2}-h\cdot z \Big{]}+\omega r \tag{7}\]The pressure field Equation (8) is defined by implementation of Equation (7) in the \(r\) component of the Navier-Stokes Equation (6). \[\frac{\partial p}{\partial r}=\frac{\rho}{r}\gamma_{\theta}^{2}=\frac{\rho}{r} \bigg{(}\frac{\Delta p^{2}}{9\pi^{2}\mu^{2}r^{2}}\ \ \ \Big{(}z^{2}-h\cdot z\Big{)}^{2}+\omega^{2}r^{2}+\frac{2\Delta p\omega}{3\pi\mu} \Big{(}z^{2}-h\cdot z\Big{)}\bigg{)} \tag{8}\]  
The rate of change of pressure with respect to radius depends on the variables \(z\) and \(r\) as shown in Equation (8) and it is calculated in three positions, Equations (9)-(11):  
\[z=0\qquad\frac{\partial p}{\partial r}=\frac{\rho}{r}\omega^{2}r^{2}\ \ \ \ \ \rightarrow\qquad p=\rho\frac{\omega^{2}r^{2}}{2}+\mathrm{C} \tag{9}\]  
\[z=h\qquad\frac{\partial p}{\partial r}=\frac{\rho}{r}\omega^{2}r^{2}\ \ \ \ \ \ \ \rightarrow\qquad p=\rho\frac{\omega^{2}r^{2}}{2}+\mathrm{C} \tag{10}\]  
\[\frac{\partial\rho}{\partial t}+\frac{1}{r}\frac{\partial}{\partial r}(\rho rv _{r})+\frac{1}{r}\frac{\partial}{\partial\theta}(\rho v_{\theta})+\frac{ \partial}{\partial z}(\rho v_{z})=0 \tag{2}\]0  
The rate of change of pressure with respect to the radius in the space between the disks Equation (11) depends on three variables \(\alpha\), \(\beta\), \(\gamma\). The magnitude of each individual variable is calculated for a typical working condition of the heart pump: \(\Delta p=65\) mmHg, \(\mu=3.5\times 10^{-3}\) Pa-s, \(h=1\) mm, \(r=1.5\) cm, \(\omega=628.319\) s\({}^{-1}=6000\) rpm. From the numerical results, term \(\alpha\) in Equations (9) and (11) is eleven times smaller than the other two terms, so it can be neglected.