## 3 Results and Discussions  
### Validation of IM  
The validation of the accuracy and effectiveness of IM are demonstrated by a comparison with SM. The two methods are both free of the probability distributions of the uncertain variables. A deviation of 10% in solid volume fraction is considered, which means \(\xi_{\phi}^{I}=\) [4.5%, 5.5%]. Comparisons between the results of IM and SM are summarized in Table 4. A good agreement can be observed in Table 4, which proves the accuracy of IM. Wherein, LB and UB denote the lower bound and the upper bound, respectively. There are 100 samples used in the prediction procedure by employing SM while only five samples are used by using IM, which proves the effectiveness of IM. The variation of temperature and velocity of nanofluid on a featured line (\(y\) = 15 \(\upmu\)m) are shown in Figure 3. In general, due to  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Grids** & \(\mathbf{T_{out,y}=15\ \upmu\text{m/K}}\) & **Difference/\%** & \(\mathbf{u_{out,(y=15\ \upmu\text{m/m}\cdot s^{-1})}}\) & **Difference/\%** & \(\mathbf{Nu_{m}}\) & **Difference/\%** \\ \hline
400 \(\times\) 20 & 296.644 & 0.065 & 2.025 & \(-\)3.094 & 3.775 & 0.568 \\
600 \(\times\) 30 & 296.507 & 0.019 & 2.070 & \(-\)0.953 & 3.750 & \(-\)0.076 \\
800 \(\times\) 40 & 296.470 & 0.006 & 2.082 & \(-\)0.384 & 3.750 & \(-\)0.088 \\
1000 \(\times\) 50 & 296.451 & Ref & 2.090 & Ref & 3.753 & Ref \\ \hline \hline \end{tabular}
\end{table}
Table 3: Grid Independence study at \(\varphi\) = 0.05, Re = 25.  
Figure 2: The average Nusselt number at \(\varphi\) = 0.  
the forced convection, the temperature increases with the x-coordinate. The velocity decreases as the height of the microchannel increases and vice versa. The interval variable in solid volume fraction hardly changes the trend of temperature and velocity, and obvious bounds in temperature and velocity can be observed. Similarly, obvious bounds in the local Nusselt number of the walls can be observed in Figure 4. The local Nusselt number decreases at the grooves and the width of the bounds of the local Nusselt number is relatively small when the nanofluid enters and leaves the grooves.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Item**} & \multirow{2}{*}{**Bound**} & \multicolumn{2}{c}{**SM (100 Samples)**} & \multicolumn{2}{c}{**IM (5 Samples)**} \\ \cline{3-5}  & & **Value** & **Value** & **Error** \\ \hline \multirow{3}{*}{Temperature/K} & LB & 296.378 & 296.377 & \(-2.558\times 10^{-06}\) \\  & UB & 296.568 & 296.572 & \(1.411\times 10^{-05}\) \\ \multirow{3}{*}{Velocity/m\(\cdot\)s\({}^{-1}\)} & LB & 1.989 & 1.989 & \(1.407\times 10^{-05}\) \\  & UB & 2.180 & 2.180 & \(-2.026\times 10^{-05}\) \\ \multirow{3}{*}{Nu\({}_{m}\)} & LB & 3.635 & 3.635 & \(-6.301\times 10^{-05}\) \\  & UB & 3.835 & 3.835 & \(-7.172\times 10^{-06}\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Bounds of variables with 10% deviation in \(\varphi\).  
Figure 4: Bounds of local Nusselt number along the \(x\)-axis with a 10% deviation in \(\varphi\) at \(\delta\) = 10 \(\upmu\)m and Re = 25: (**a**) The upper wall; (**b**) the lower wall.  
Figure 3: Bounds of temperature and velocity along \(x\)-axis with 10% deviation in \(\varphi\) at \(y\) = 15 \(\upmu\)m, \(\delta\) = 10 \(\upmu\)m and Re = 25: (**a**) Temperature; (**b**) velocity.