## 2 Methodology
### Mathematical Formulation
#### 2.1.2 Multiphase Mixture Model  
This model represents the flow by two phases, the first phase is water, and the second phase is a homogeneous Newtonian fluid, the properties of which are equivalent to those of a suspension of nanoparticles in water. In this second phase, the coupling between the motion of the nanoparticles and the fluid motion surrounding them is assumed to be strong, so that the nanoparticles move with their neighboring fluid in the second phase. Each phase has its own velocity vector field. The first and second phases are immiscible and the second phase is modelled as spherical bubbles dispersed in the first phase continuum, in a bubbly flow regime [41]. The portion of space occupied by the second phase bubbles at any spatial location is defined by the volume fraction. The first phase moves the second phase by way of a drag force and the second phase affects the first phase because of a reduction  
Figure 1: (**a**) Isometric schematic of the rectangular minichannel heat sink and (**b**) front view of the minichannel heat sink. The heat flux direction is indicated by the up-pointing arrows. in the first phase mean momentum. The coolant flow is modelled as a two-phase mixture based on the following assumptions:  
1. The flow is steady, laminar, incompressible, as well as three-dimensional. 2. The second phase comprises spherical bubbles of prescribed uniform size [42]. 3. All of the phases share a single pressure [22]. 4. The working fluid thermal properties are volume fraction dependent [22]. 5. Gravitational effects are modelled, and radiative heat transfer is disregarded [14]. 6. The volume fraction of nanoparticles in the mixture is sufficiently low so that collision and heat conduction by contact between particles are negligible [42]. 7. The diameter of the nanoparticles is smaller than the diameter of the bubbles so that the nanoparticles are uniformly dispersed in the second phase [19], so that the volume fraction \(\Phi\) of the nanoparticles in the second phase has one average value and that the spherical shape of the bubbles is unaffected by the presence of the nanoparticles in it. 8. The diameter of the bubbles is smaller than the minichannel hydraulic diameter so that a local equilibrium is established over short spatial length scales [42]. The continuity equation in relation to the mixture is:  
\[\
abla\cdot\left(\rho_{m}\overset{\rightarrow}{u}_{m}\right)=0 \tag{5}\]  
where \(\overset{\rightarrow}{u}_{m}\) is the mass-averaged mixture velocity  
\[\overset{\rightarrow}{u}_{m}=\frac{\alpha_{bf}\rho\rho_{bf}\overset{ \rightarrow}{u}_{bf}+\alpha_{nf}\rho_{nf}\overset{\rightarrow}{u}_{nf}}{ \rho_{m}} \tag{6}\]  
and \(\rho_{m}\) is the mixture density  
\[\rho_{m}=\alpha_{bf}\rho_{bf}+\alpha_{nf}\rho_{nf} \tag{7}\]  
The momentum equation for the mixture can be shown as:  
\[ \begin{array}{l}\
abla\cdot\left(\rho_{m}\overset{\rightarrow}{u}_{m} \overset{\rightarrow}{u}_{m}\right)=-\
abla p+\mu_{m}\
abla^{2}\overset{ \rightarrow}{u}_{m}+\rho_{m}\overset{\rightarrow}{S}+\\ \
abla\cdot\left(\alpha_{bf}\rho_{bf}\rho_{bf}\overset{\rightarrow}{u}_{dr,bf }\overset{\rightarrow}{u}_{dr,bf}+\alpha_{nf}\rho_{nf}\overset{ \rightarrow}{u}_{dr,nf}\overset{\rightarrow}{u}_{dr,nf}\right)\end{array}  \tag{8}\]  
The molecular viscosity of the mixture \(\mu_{m}\) is  
\[\mu_{m}=\alpha_{bf}\mu_{bf}+\alpha_{nf}\mu_{nf} \tag{9}\]  
where \(\mu_{bf}\) and \(\mu_{nf}\) are the molecular viscosities of the first and of the second phases, respectively. The drift velocity of the second phase \(\overset{\rightarrow}{u}_{dr,nf}\) is defined as the phase dispersion velocity relative to the mixture velocity:  
\[\overset{\rightarrow}{u}_{dr,nf}=\overset{\rightarrow}{u}_{nf~{}bf~{}bf~{} bf}-\overset{\rightarrow}{u}_{m} \tag{10}\]  
\[\overset{\rightarrow}{u}_{nf~{}bf~{}bf~{}bf~{}bf~{}bf~{}bf~{}bf~{}bf~{}bf~{} bf}=\frac{\left(\rho_{nf}-\rho_{m}\right)d_{nf}^{2}}{18\mu_{mf}d_{mag}}\overset{ \rightarrow}{a} \tag{11}\]  
where \(d_{nf}\) is the uniform diameter of the second phase bubbles.