## 2 Materials and Methods
### Model Equations
As a result of the assumptions, the general equation of heat transfer takes the form of an equation for a fluid:  
\[\rho_{f}c_{pf}\left(v_{x}\frac{\partial T_{f}}{\partial x}+v_{y}\frac{ \partial T_{f}}{\partial y}+v_{z}\frac{\partial T_{f}}{\partial z}\right)= \lambda_{f}\left(\frac{\partial^{2}T_{f}}{\partial x^{2}}+\frac{\partial^{2}T_ {f}}{\partial y^{2}}+\frac{\partial^{2}T_{f}}{\partial z^{2}}\right) \tag{6}\]where \(\rho_{f}\), \(c_{pf}\), \(T_{f}\), and \(\lambda_{f}\) are the density, specific heat capacity, temperature, and thermal conductivity of the fluid, respectively. Since a solid is nondeformable, the only equation that describes its behavior is the heat transfer equation. For a solid, the general Equation (5), after using the assumptions of the model, is simplified to the Laplace equation:  
\[\lambda_{s}\left(\frac{\partial^{2}T_{s}}{\partial x^{2}}+\frac{\partial^{2}T _{s}}{\partial y^{2}}+\frac{\partial^{2}T_{s}}{\partial z^{2}}\right)=0 \tag{7}\]  
where \(T_{s}\) and \(\lambda_{s}\) are the temperature and thermal conductivity of the solid, respectively. The final system of equations describing the problem consists of the continuity Equation (2), Navier-Stokes Equation (4), and heat transport Equations (6) and (7). These differential equations require appropriate boundary conditions. At the inlet to the microchannel, the velocity is assumed to be uniform, perpendicular to the surface, and pressure is also uniform:  
\[v_{x}|_{z=0}=v_{y}|_{z=0}=0\quad v_{z}|_{z=0}=v_{0}\quad p|_{z=0}=p_{0} \tag{8}\]  
where \(v_{0}\) and \(p_{0}\) are the inlet fluid velocity and pressure, respectively. Due to the viscosity of the fluid, the velocity of the fluid in contact with the solid is assumed to be zero:  
\[v_{x}|_{sl}=0\quad v_{y}|_{sl}=0\quad v_{z}|_{sl}=0 \tag{9}\]  
where index \(sl\) is the surface between the solid and fluid (solid-liquid interface). The temperature of the inlet fluid is assumed to be uniform, and the temperature of at the bottom of solid is also assumed to be constant:  
\[T_{f}\Big{|}_{z=0}=T_{f0}\quad T_{s}|_{y=0}=T_{s0} \tag{10}\]  
where \(T_{f0}\) and \(T_{s0}\) are the two constant temperatures of fluid and solid, respectively. Heat transfer between a solid and a fluid occurs through conduction, which means that the heat fluxes described by [PERSON]'s law are equal at the solid-liquid interface:  
\[\lambda_{s}\left.\frac{\partial T_{s}}{\partial n}\right|_{sl}=\lambda_{f} \left.\frac{\partial T_{f}}{\partial n}\right|_{sl} \tag{11}\]  
At both ends of the segment and at the boundary between two adjacent segments, no heat conduction is assumed for the solid body and no heat conduction in the fluid and the end of the segment:  
\[\left.\frac{\partial T_{s}}{\partial z}\right|_{z=0}=\left.\frac{\partial T_{ s}}{\partial z}\right|_{z=l}=\left.\frac{\partial T_{s}}{\partial n}\right|_{ ss}=\left.\frac{\partial T_{f}}{\partial z}\right|_{z=l} \tag{12}\]  
where index \(ss\) means the boundary between solid and solid (two adjacent segments). In the upper part of the region, the conditions of symmetry are assumed,  
\[\left.\frac{\partial v_{x}}{\partial y}\right|_{y=h}=\left.\frac{\partial v_{ y}}{\partial y}\right|_{y=h}=\left.\frac{\partial v_{z}}{\partial y}\right|_{y=h}= 0\quad\left.\frac{\partial p}{\partial y}\right|_{y=h}=0\quad\left.\frac{ \partial T_{s}}{\partial y}\right|_{y=h}=\left.\frac{\partial T_{s}}{\partial y }\right|_{y=h}=0 \tag{13}\]