## 2 Materials and Methods
### Model Equations  
The entire element considered during the calculations can be divided into two regions: a fluid region inside the microchannel and a solid region outside the microchannel. The phenomena occurring in the fluid region are described by the equations resulting from mass, momentum, and energy balances. The general form of the fluid flow continuity equation resulting from mass balance is:  
\[\frac{\partial\rho_{f}}{\partial t}+\
abla\cdot\left(\rho_{f}\overset{\rightarrow }{v}\right)=0 \tag{1}\]  
where \(\rho_{f}\) is the fluid density and \(\overset{\rightarrow}{v}\) its velocity. For an incompressible fluid, this equation takes the following form:  
\[\frac{\partial v_{x}}{\partial x}+\frac{\partial v_{y}}{\partial y}+\frac{ \partial v_{z}}{\partial z}=0 \tag{2}\]  
where \(v_{x}\), \(v_{y}\), and \(v_{z}\) are the components of the velocity vector in the directions of the coordinate system axes (Figure 1). The general form of the Navier-Stokes equation resulting from momentum balance is as follows:  
\[\rho\left(\frac{\partial\overset{\rightarrow}{v}}{\partial t}+\left( \overset{\rightarrow}{v}\cdot\
abla\right)\cdot\overset{\rightarrow}{v} \right)=-\
abla p+\
abla\cdot\left(\mu\left(\
abla\overset{\rightarrow}{v} +\left(\
abla\overset{\rightarrow}{v}\right)^{T}-\frac{2}{3}\left(\
abla \cdot\overset{\rightarrow}{v}\right)\overset{\leftrightarrow}{I}\right) \right)+\rho\overset{\rightarrow}{g} \tag{3}\]  
where \(\rho_{f}\), \(\overset{\rightarrow}{v}\), \(p\), \(\mu\), \(\overset{\leftrightarrow}{I}\), and \(\overset{\rightarrow}{g}\) are the fluid density, velocity, pressure, dynamic viscosity, identity tensor, and gravity vector, respectively. Assumptions of steady flow, incompressible fluid, and no gravitational force allow us to simplify Equation (3) to the form:  
\[ \begin{array}{l}\rho_{f}\left(v_{x}\frac{\partial v_{x}}{\partial x}+v_{y} \frac{\partial v_{y}}{\partial y}+v_{z}\frac{\partial v_{z}}{\partial z}\right) =-\frac{\partial p}{\partial x}+\left(\frac{\partial}{\partial x} \left(\mu\frac{\partial v_{x}}{\partial x}\right)+\frac{\partial}{ \partial y}\left(\mu\frac{\partial v_{x}}{\partial y}\right)+\frac{\partial}{ \partial z}\left(\mu\frac{\partial v_{x}}{\partial z}\right)\right)\\ \rho_{f}\left(v_{x}\frac{\partial v_{y}}{\partial x}+v_{y}\frac{\partial v_{y} }{\partial y}+v_{z}\frac{\partial v_{z}}{\partial z}\right) =-\frac{\partial p}{\partial y}+\left(\frac{\partial}{\partial x} \left(\mu\frac{\partial v_{x}}{\partial x}\right)+\frac{\partial}{ \partial y}\left(\mu\frac{\partial v_{y}}{\partial y}\right)+\frac{\partial}{ \partial z}\left(\mu\frac{\partial v_{y}}{\partial z}\right)\right)\\ \rho_{f}\left(v_{x}\frac{\partial v_{z}}{\partial x}+v_{y}\frac{\partial v_{z} }{\partial y}+v_{z}\frac{\partial v_{z}}{\partial z}\right) =-\frac{\partial p}{\partial z}+\left(\frac{\partial}{\partial x} \left(\mu\frac{\partial v_{z}}{\partial x}\right)+\frac{\partial}{ \partial y}\left(\mu\frac{\partial v_{z}}{\partial y}\right)+\frac{\partial}{ \partial z}\left(\mu\frac{\partial v_{z}}{\partial z}\right)\right)\end{array}  \tag{4}\]  
where \(\rho_{f}\), \(p\), and \(\mu\) are the density, pressure, and viscosity of the fluid, respectively. The general heat transfer equation of the single-phase medium resulting from energy balance is:  
\[\rho c_{p}\left(\frac{\partial T}{\partial t}+\overset{\rightarrow}{v}\cdot \
abla T\right)=\
abla\cdot\left(\lambda\
abla T\right)+\overset{\leftrightarrow }{\sigma}\cdot\overset{\leftrightarrow}{d}+\rho r \tag{5}\]  
where \(c_{p}\), \(T\), \(\overset{\rightarrow}{v}\), \(\lambda\), \(\overset{\leftrightarrow}{\sigma}\), \(\overset{\leftrightarrow}{d}\), and \(r\) are the specific heat capacity, temperature, velocity, thermal conductivity, stress tensor, strain rate tensor, and heat source, respectively.