## 2 Materials and Methods
The continuity equation, momentum equations are shown as follows  
\[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{ \partial z}=0 \tag{1}\]  
\[u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u }{\partial z}=-\frac{1}{\rho}\frac{\partial p}{\partial x}+\frac{\mu}{\rho} \
abla^{2}u \tag{2}\]  
\[u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{\partial v }{\partial z}=-\frac{1}{\rho}\frac{\partial p}{\partial y}+\frac{\mu}{\rho} \
abla^{2}v \tag{3}\]  
\[u\frac{\partial w}{\partial x}+v\frac{\partial w}{\partial y}+w\frac{\partial w }{\partial z}=-\frac{1}{\rho}\frac{\partial p}{\partial z}+\frac{\mu}{\rho} \
abla^{2}w \tag{4}\]  
Since the measurement of local pressure along the flow is difficult in microchannels, the measurement of the pressure drop across the inlet and outlet manifolds is only conducted. The resulting pressure drop measurement represents the combined effect of the losses in the bends, entrance and exit losses, developing region effects, and core frictional losses. The pressure drop data are affected by microscale, which should not come as a surprise since values for the macroscale have been obtained empirically from measurements. According to [PERSON]'s equation, the pressure drop model expression is written considering both the frictional and dynamic losses. \[\Delta P=\left[\left(K_{c}+K_{\text{ort}}+4f_{app1}\frac{L_{eff1}}{D_{h_{1}}} \right)\frac{4H^{2}}{s^{2}}+4f_{app2}\frac{L_{eff2}}{D_{h_{2}}}+K_{c}\right] \frac{1}{2}{\rho}{V_{cd_{2}}}^{2} \tag{5}\]  
[PERSON] and [PERSON] proved that in the neighborhood of circular and parallel plate channels (\(x^{+}\leq 0.001\)) inlets, \(f_{app}Re\) is almost independent of channel shape. [PERSON] et al. [35] made numerical simulations of pressure drop for three-dimensional rectangular microchannels. [PERSON] and [PERSON] [36] demonstrated that a relationship between the rate of the dimensionless mean wall shear stress reduction and the rate of the dimensionless mean wall heat flux reduction might exist. [PERSON] et al. [37]  
Figure 2: The simplified model. demonstrated that the appropriate drag coefficient could be related to the Nusselt number. The boundary layer performance in the position of channel entrance is generally identical to that in a thin plate. It clarifies the reason that \(f_{app}Re\) is almost independent of channel shape in the neighborhood of the inlet. The influences of geometry characters become more prominent with the boundary layer developing in downstream (\(x^{+}>0.001\)). The accessorial pressure defect \(\Phi(x^{+})\) in a rectangular channel can be calculated with a model developed by [PERSON] and [PERSON] (2018, 2019) in terms of developing laminar regime:  
\[f_{app}Re_{D_{h}}=\frac{24}{\left(1+(\frac{b}{H})\right)^{2}\left\{1-\frac{192 (\frac{b}{H})}{\pi^{5}}\left[\tanh\left(\frac{\pi}{2(\frac{b}{H})}\right)+ \frac{1}{243}\tanh\left(\frac{3\pi}{2(\frac{b}{H})}\right)\right]\right\}}+\Phi \left(x^{+}\right) \tag{6}\]  
\[\Phi(x^{+})=\frac{1}{3x^{+}\left(1+8 Kn^{*}\right)^{2}}-2\sum_{i=1}^{\infty} \frac{\left(3-\exp(-4\alpha_{i}^{2}x^{+})\right)\exp(-4\alpha_{i}^{2}x^{+})}{ \alpha_{i}x^{+}(1+8 Kn^{*}+4(\alpha_{i}Kn^{*})^{2})} \tag{7}\]  
Figure 1:0  
The friction factor and Reynolds number product of rectangular ducts short asymptote for slip flow is almost independent of the duct shape and therefore can be used to calculate by Equation (7). In order to predict the pressure loss of plate fin heat sinks, an asymptotic limit method is employed here. The eigenvalue \(\alpha_{i}\) is suitable for this equation  
Figure 1:1  
The first kind of order \(\
u\) Bessel function is expressed by \(J_{\
u}(\text{x})\). Only the first two items in the infinite series are really contributed, which are applied in practical engineering limit of \(x^{+}\geq 0.01\). It is concluded that Equation (6) absolutely properly approaches the \(x^{+}\to 0\) asymptote \(2/Kn^{*}\).