## 4 Results and Discussion  
### Flow Uniformity  
The effect of obstacles is surveyed for its contribution on the flow uniformity. However, the edge flow accelerated from the blockage of the main central one effects on the unnecessary fluctuation near the side wall of diffuser. In Figure 9, the effect of boundary layers, or the shaded area neighboring to the side walls must be neglected to make a fair comparison. The boundary layer on a flat plate is calculated with Blasius theory to simplify Equations (1) and (2) to the following incompressible boundary-layer equations in the two-dimension for the region of boundary layer near the side wall [20,22,24]:  
\[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0 \tag{11}\]  
\[u\frac{\partial u}{\partial x}+v\frac{\partial v}{\partial y}=-\frac{1}{\rho} \frac{dp}{dx}+v\frac{\partial}{\partial y}\bigg{(}\frac{\partial v}{\partial y }\bigg{)}^{n} \tag{12}\]  
where the kinematic viscosity is simply defined as \(v=\mu/\rho\) just like Newtonian fluid, which does not lose the generality even for such non-Newtonian fluids. The stream function, \(\psi\) is defined as, satisfying Equation (11),  
\[u=\frac{\partial\psi}{\partial x},v=-\frac{\partial\psi}{\partial y} \tag{13}\] \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Coefficients**} & \(C_{DP}\) & \(C_{DF}\) & \(C_{D}\) & \(C_{DP}\) & \(C_{DF}\) & \(C_{D}\) & \(C_{DP}\) & \(C_{DF}\) & \(C_{D}\) \\ \cline{2-11}  & & \(n=1\) & & & \(n=0.6\) & & & \(n=0.2\) & \\ \hline Present & 0.9457 & 1.222 & 2.168 & 1.030 & 1.030 & 2.060 & 1.161 & 0.6817 & 1.843 \\ [PERSON], [PERSON], et al. [16] & 0.8893 & 1.173 & 2.063 & 0.9900 & 1.013 & 2.003 & 1.184 & 0.7020 & 1.886 \\ Error (\%) & 6.0 & 4.0 & 4.9 & 3.9 & 1.6 & 2.8 & 2.0 & 3.0 & 2.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Numerical values of drag coefficients with power-law analysis around an elliptic obstacle. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Coefficients**} & \(C_{DP}\) & \(C_{DF}\) & \(C_{D}\) & \(C_{DP}\) & \(C_{DF}\) & \(C_{D}\) & \(C_{DP}\) & \(C_{DF}\) & \(C_{D}\) \\ \cline{2-11}  & & \(n=1\) & & & \(n=0.6\) & & & \(n=0.2\) & \\ \hline Present & 1.274 & 0.8362 & 2.111 & 1.355 & 0.6272 & 1.982 & 1.439 & 0.3372 & 1.776 \\ [PERSON], [PERSON], et al. [16] & 1.201 & 0.8047 & 2.005 & 1.304 & 0.6137 & 1.918 & 1.484 & 0.3394 & 1.824 \\ Error (\%) & 5.8 & 3.8 & 5.0 & 3.7 & 2.2 & 3.2 & 3.1 & 0.7 & 2.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Numerical values of drag coefficients with power-law analysis around a circular obstacle. Figure 9: The schematic diagram for the cut regions near side walls. A dimensionless distance and the stream function are also defined as, just like those of [PERSON],  
\[\eta=y\Big{(}\frac{U_{\infty}^{2-n}}{v\chi}\Big{)}^{\frac{1}{n+1}} \tag{14}\]  
\[\psi(\eta)=\Big{(}vxU_{\infty}^{2n-1}\Big{)}^{\frac{1}{n+1}}f(\eta) \tag{15}\]  
Substituting Equation (15) into Equation (13), from a simple calculus, we get the velocity components such as  
\[u=U_{\infty}f \tag{16}\]  
\[v=\frac{1}{n+1}\Bigg{(}v\frac{U_{\infty}^{2n-1}}{x^{nt}}\Bigg{)}^{\frac{1}{n+1 }}(\eta f^{\prime}-f) \tag{17}\]  
Note that there is no change in Equation (16) from the Newtonian fluid, and Equation (17) also falls down to Newtonian if \(n=1\). As the outer flow from a flat plate, \(dU_{\infty}/dx\approx 0\), and the pressure gradient term in Equation (12) is erased out. Equations (16) and (17) are substituted to Equation (12), the final ordinary equation is written down as  
\[n(n+1)(f^{\prime\prime})^{n-1}f^{\prime\prime\prime}+ff^{\prime\prime}=0 \tag{18}\]  
The boundary condition of Equation (18) is also specified from the no-slip condition at the wall (\(u=v=0\), or \(f=f=0\) at \(\eta=0\)) and the inviscid condition at the outer edge of boundary layer (\(u=U_{\infty}\), or \(f=1\) at \(\eta\rightarrow\infty\)).