## 3 Results and Discussion
The pressure drop values for the plate fin heat sink with different inlet velocities and construction parameters are obtained with numerical simulations in the current paper. There are eight different velocities applied to make the numerical simulations for the plate fin heat sink with impinging air flow, the corresponding velocity values are as follows: 0.3 m/s, 0.5 m/s, 0.6 m/s, 0.7 m/s, 0.8 m/s, 1.0 m/s, 1.5 m/s, and 2.0 m/s. The maximum value of Reynolds number is 1270, which belongs to laminar flow regime. Figure 7 shows the comparison of the streamlines in Y-Z symmetry plane for the plate fin heat sink subjected to impinging air flow without elliptic bottom profiles and that with elliptic bottom profiles in the condition of different velocities for \(s=10\%\)\(L\), and the comparison of velocity contours in Y-Z symmetry plane for the plate fin heat sink with and without elliptic bottom profiles for \(s=10\%\)\(L\). The figures (left: a, b, c) show the streamlines in Y-Z symmetry plane for the plate fin heat sink with elliptic bottom profiles and  
bottom profiles on the bottom of the channels. As we can see, the flow impinging to the heat sink creates one vortex which occurs because of the jet impingement with the heat sink base. Relatively higher velocity results in a larger vortex for the heat sink with the same impinging inlet width. There are two vortices in Y-Z symmetry plane when the velocity increases to 2.0 m/s. For higher velocity, the generated vortex is strengthened because of the decreasing viscous force. As the elliptic bottom profiles are added to the channels, the jet vortex becomes smaller. This approach leads to smooth flow in the channels and the vortex because of impingement shifts to the direction of the outlet, away from the symmetry plane. Figures 8-11 compare the pressure drop for the number 1 to number 4 plate fin heat sinks subjected to impinging air flow for different inlet widths obtained from the present model and the numerical simulation results with the experimental data acquired by [PERSON] and [PERSON] [3]. Results for the pressure drop obtained from the experiments, the present model and the present numerical simulations match quite well and the maximum error is no more than 25%. The pressure drop increases with a decrease in impingement inlet width for the same flow rate. The pressure drop of the plate fin heat sink under impinging air flow increases with the inlet velocity especially obvious for larger inlet width. The change rate of pressure drop increases with the decreasing inlet width. As the inlet width increases, the pressure drop in the Z-direction slightly decreases, which is caused by the decrease in the averaged velocity. In the same plate fin heat sink flow path, increasing the velocity directly leads to the collision between the fluid molecules, and the fluid molecules exchange momentum with each other to generate more energy loss, and the quantity of surrounding fluid induced by the jet impingement increases with the increased speed. The motion causes a greater viscous force and more intense shear stress, and momentum exchange occurs between the fluids at the boundary at the two sides of the jet impingement, which in turn causes an increase in maximum pressure drop. Figure 12 shows the comparison of the pressure drop for plate fin heat sinks with elliptic bottom profiles and without elliptic bottom profiles on the bottom of the channels. It can be observed that the pressure drop for the plate fin heat sink without elliptic bottom profiles is larger than that for plate fin heat sink with elliptic bottom profiles especially for the smaller inlet width. The differences of the pressure drop for the heat sink with or without elliptic bottom profiles on the bottom of the channels are not very obvious for the larger inlet width. The contact resistance is decreased with the presence of elliptic bottom profiles due to the additional substrate materials adding at the bottom of fins, which makes channels smoother.