## 2 Methodology
### Geometry  
The minichannel heat sink consists of nine parallel rectangular-shaped ducts with a length of 30 mm, a height of 4 mm, a channel width of 1 mm, a fin width of 1 mm, and a channel height of 3 mm. Only one duct is modelled to decrease the complexity and the computational cost. The temperature field is assumed to be symmetric between adjacent ducts, about the sectioned planes shown with dashed lines in Figure 1a. Table 3 shows the thermo-physical properties of the solid domain made from aluminum. The Reynolds number referenced to the inlet conditions ranges from 91 to 455 and it is defined as:  
\[R_{e}=\frac{\rho_{nf}u_{in}D_{h}}{\mu_{nf}} \tag{20}\]  
where \(\rho_{nf}\) is the density of the working fluid evaluated by Equation (16) with \(\phi\) = 1%, \(u_{in}\) is the average fluid velocity at the minichannel inlet, and \(\mu_{nf}\) is the dynamic viscosity of the working fluid evaluated by Equation (19) with \(\phi\) = 1%. The hydraulic diameter (\(D_{h}\)) is  
\[D_{h}=\frac{4A}{P}=\frac{4 ab}{2(a+b)}=\frac{2 ab}{a+b} \tag{21}\]  
where, \(P\), \(a\) and \(b\) are, respectively, the perimeter, width, and thickness of the duct.  
An auxiliary test case is defined to test the model limitations identified in the minichannel heat sink simulations, using a geometry that features greater flow complexity. To this end, a simulation is performed of the flow and heat transport over a microscale backwards  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & \(\rho\) (kg/m\({}^{3}\)) & \(Cp\) (J/kg\(\cdot\)K) & \(k\) (W/m\(\cdot\)K) \\ \hline Aluminum & 2719 & 871 & 202.4 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The thermo-physical properties of aluminum [49].  
facing step, which is uniformly heated over its base, as shown by the two-dimensional schematic of Figure 2. The inlet channel height \(h_{u}=0.4\) mm, the step height \(h_{d}=0.6\) mm, the length of the narrow channel upstream of the step \(L_{u}=100\) mm, and the length of the wide channel downstream of the step \(L_{d}=150\) mm. The geometry is tested at a constant Reynolds number of 280, which is based on the inflow conditions and on the hydraulic diameter \(D_{h}=(h_{u}+h_{d})\). The nanofluid, made by a 1% by volume suspension of Al\({}_{2}\)O\({}_{3}\) nanoparticles in water, flows through the conduit. A constant heat flux \(\dot{q}_{2}=30\) kW/m\({}^{2}\) is applied to the bottom wall, as in [PERSON], et al. [50]. Further details are provided in [PERSON], et al. [51] and [PERSON] [52] where both measurements and numerical predictions are available for comparison purposes.  
Considering the purpose of this auxiliary test case is to address the model limitations in the presence of greater flow complexity, this computation is limited to the flow domain, which removes the requirement of solving a conjugate heat transfer problem through the solid walls. To this end, the single-phase approach of Section 2.1.1 and the two-phase mixture model of Section 2.1.2 are used, together with the nanofluid model of Section 2.2. At the inlet, boundary condition 1 of Section 2.3 is used. At the outlet, boundary condition 2 of Section 2.3 is used. The bottom wall is modelled with the no-slip boundary condition and \(-k_{f}\frac{\partial T_{f}}{\partial x_{2}}=\dot{q}_{2}\). The remaining walls are modelled as no-slip and adiabatic, by which \(u_{1}=u_{2}=0\) and \(\frac{\partial T_{f}}{\partial x_{n}}=0\), where \(n\) is the coordinate normal to the wall.