## 2 Materials and Methods  
To the best knowledge of the authors, the investigations for the effects of adding elliptic bottom profiles (fillet profiles) to a plate fin heat sink are scarce. The objective of this study is to investigate the flow characteristics of a plate fin heat sink with elliptic bottom profiles subjected to impinging air flow. The fillet profile is shaped as semiellipt which has the major semiaxis and minor semiaxis of 1.5 mm and 1.125 mm, respectively. Simulations are carried out by the software Fluent. The model is developed for the low Reynolds number laminar flow in the inter-fin channels of impingement flow plate fin heat sinks as shown in Figure 1. The computational unit of the number 5 plate fin heat sink is shown in Figure 1b, with a fillet profile on the bottom of the channel, which is different from other traditional heat sinks. In this study, five different inlet widths are considered including 10% \(L\), 25% \(L\), 50% \(L\), 75% \(L\), and 100% \(L\). The air enters in the middle top then exits from the sides (TISE). [PERSON] et al. [33] made an experiment to report the flow characteristics for unique \"serpentine\" squared pin fins with TISE arrangement. [PERSON] [34] developed an original laminar flow heat sink model which uses two kinds of fin shapes including triangular and trapezoidal on the inclined planes of the base. In this paper, we choose five kinds of plate fin heat sinks subjected to impinging flow, which are different from each other in geometry. The geometry parameters of the plate fin heat sink is shown in Table 1. According to the table, the plate fin heat sink of #1 and #2 are same in geometrical parameters except the height. One dimensional pressure drop model of plate fin heat sink subjected to impinging flow is developed as follows. We choose one channel as computational domain and only investigated half of the heat sink because of the symmetry for flow and pressure distributions of the other half. This approach can lead to considerable simplification, especially when the original problem requires a great deal of time and computational steps. Then we obtain the simplified model as shown in Figure 2. \(L_{\textit{eff1}}\) and \(L_{\textit{eff2}}\) represent the effective lengths of these two orthogonal channels. This shape of the streamline length can be considered as a right-angled path with orthogonal height of 0.5 \(H\) and orthogonal length of (0.5 \(L\)-0.25 \(s\)). The flow situation at the entrance of the plate fin heat sink is very involved, as well as the right-angled bends. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Dimension** & **\# 1** & **\# 2** & **\# 3** & **\# 4** & **\# 5 (with Elliptic Bottom Profiles)** \\ \hline \(L\) (mm) & 127 & 127 & 127 & 127 & 127 \\ \(W\) (mm) & 122 & 122 & 116 & 116 & 122 \\ \(t_{b}\) (mm) & 12.7 & 12.7 & 12.7 & 12.7 & 12.7 \\ \(t\) (mm) & 1.2 & 1.2 & 1.2 & 1.2 & 1.2 \\ \(b\) (mm) & 2.25 & 2.25 & 4.27 & 4.27 & 2.25 \\ \(H\) (mm) & 26.5 & 50.0 & 34.0 & 50.0 & 26.5 \\ \(N_{f}\) & 36 & 36 & 22 & 22 & 36 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometry parameters of the plate fin heat sinks. Figure 1: The physical model of the plate fin heat sink. (**a**) Schematic diagram of a plate fin heat sink, (**b**) computational unit of #5 plate fin heat sink. Three fundamental assumptions are made in the process of numerical simulations for the pressure drop of plate fin heat sink subjected to impinging flow:  
1. Steady incompressible flow;
2. Negligible buoyancy, viscous heating;
3. Constant fluid properties.