## 2 Materials and Methods  
### Mathematical Models  
The level set function coupled with heat transfer, continuity and Navier-Stokes equations were discretized and solved numerically using the finite element method (FEM). The blood and cold plasma plume in the mixing chamber domain were tracked with the level set method using the level set variable (\(\phi\)). The standard heat equation was used to model the heat transfer between the two fluids within the mixing chamber. The present theoretical model consisted of two main parts: (1) fluid dynamics with level set interface module; and (2) heat transfer module, which were coupled to each other. The results were obtained using parallel computing. In the present work, the level set method was used to capture the mixing and evolution of blood in cold plasma in two different mixing chamber designs in three-dimensional coordinate system, which was a continuation of our previous work in which the level set method was used to simulate cold plasma plume propagation in a cell buffer medium [29]. The level set method has been extensively used in simulation of two-phase fluid systems in which a liquid-gas interface is present [33; 34; 35; 36]. Applications of 3D computational fluid dynamics (CFD) simulations were aimed at providing realistic information and data [37] and at demonstrating the versatility and capability of the model over simplified models such as 2D axisymmetric models for cylindrical chambers. The present model was based on our previously developed theory and here it was modified for the present two-phase fluid systems [29]. The plasma carrier gas was assumed to be helium (He) and its evolution in the mixing chamber that got filled with the injected blood was explained using the level set function as  
\[\frac{\partial\phi}{\partial t}+u\cdot\
abla\phi=\gamma\
abla\cdot\left( \varepsilon\
abla\phi-\phi(1-\phi)\frac{\
abla\phi}{|\
abla\phi|}\right) \tag{1}\]  
where \(\phi\) was the level set function, \(\varepsilon\) was the interface thickness equivalent to the largest mesh element size mainly to allow a better interface reconstruction between the two fluids [29] and its value here was \(4\times 10^{-4}\) m. Oscillations in the level set function were reduced by the parameter \(\gamma\) to keep the interface thickness constant to achieve better numerical stability, and \(\gamma\) was set to be the largest initial velocity in the system to effectively decrease the oscillations in the level set function [29], which in this case was 0.1. The continuity equation defined in the mixing chamber involving blood and cold plasma is given by  
\[\frac{\partial\rho}{\partial t}+\
abla\cdot(\rho u)=0 \tag{2}\]  
The Navier-Stokes momentum equation is given by  
\[\rho\frac{\partial u}{\partial t}+\rho u\cdot\
abla u=-\
abla P+\
abla\cdot \left[\mu\Big{(}\
abla u+(\
abla u)^{T}\Big{)}-\frac{2}{3}\mu(\
abla\cdot u) I\right]+F \tag{3}\]  
where \(\rho\) is density, \(P\) is pressure, \(u\) is the velocity of the flow, \(F\) is given by \(F=\rho g+F_{st}+F_{vf}\), \(\rho g\) is the body force, \(F_{st}\) is the surface tension and \(F_{vf}\) is the volume force. Furthermore, the heat transfer in the mixing chamber is explained using the heat equation, with the use of the thermophysical properties of blood and plasma carrier gas:  
\[\rho C_{p}\frac{\partial T}{\partial t}+\rho C_{p}u\cdot\
abla T=-\
abla\cdot (-k\
abla T)+Q \tag{4}\]  
where \(C_{p}\) is the specific heat capacity, \(\rho\) represents the density, \(k\) is the thermal conductivity and \(Q\) represents any external source of heat energy, which can be evaluated as heat energy generated per unit volume. In the present work, instead of defining an external heat source \(Q\), the initial temperaturesof the fluids at the inlet boundaries of the nozzles were chosen.