## 5 Results and Discussion  
### Algebraic Solution of the Navier-Stokes Equation for a Duct Flow  
In dealing with the liquid flows in minichannels and microchannels of the characteristic length-scale above 1 um, in the absence of any wall surface effects such as the electrokinetic or electroosmotic forces, the flow is not expected to experience any fundamental changes from the continuum approximation employed in macrofluidic applications [50]. In this case, the fully developed flow hydrodynamics of a fluid can be described by the Navier-Stokes equation (NSE) or Boltzmann equation (BE) with the same relevance. However, we have to remember that the traditional NS model of fluid flows with no-slip boundary conditions works only for a certain range of the governing parameters. (1) The fluid is a continuum, which is satisfied when more than 1 million molecules are located in the smallest volume in which appreciable macroscopic changes take place. This is the molecular chaos restriction. (2) The flow is not too far from thermodynamic equilibrium, which is satisfied if there is a sufficient number of molecular encounters during a small time period, compared to the smallest time-scale for flow changes. During this time period, the average molecule would have moved a small distance compared to the smallest flow length scale [51]. In the free-molecule flow regime, or when the slip flow must be considered (Kn > 0.001), the alternative continuum equations (e.g., Burnett) are needed to close the NE, while the BE is still applicable. [PERSON] and [PERSON] (S&M) proposed an algebraic solution of the NSE for the fully developed laminar flow of Newtonian fluids in the rectangular duct, which was derived using the finite Fourier transform [52]:  
\[\frac{u(x^{\prime},y^{\prime})}{P}=v(x^{\prime},y^{\prime})=\frac{16\beta^{2}}{ \pi^{4}}\sum_{i\_odd}^{\infty}\sum_{j\_odd}^{\infty}\frac{sin\left(i\pi\frac{ \zeta}{a}\right)sin\left(j\pi\frac{\eta}{b}\right)}{ij(\beta^{2}i^{2}+j^{2})} \tag{1}\]  
where  
\[P=\frac{a^{2}}{\mu}\left(\frac{-dp}{d\zeta}+\rho g\zeta\right) \tag{2}\]  
and \(a\) and \(b\) are the total channel width and height, respectively, along the \(\zeta\) and \(\eta\) axes, while pressure drop dp is along the \(\zeta\) axis (Figure 12a). \(\mu\) is the fluid dynamic viscosity. The dimensionless coordinates are \(x^{\prime}=\zeta/a\) (\(0\leq x^{\prime}\leq 1\)) and \(y^{\prime}=\eta/a\) (\(0\leq y^{\prime}\leq\beta\), where \(\beta\) is the aspect ratio \(b/a\)). Hence, the average velocity \(u_{m}\) over the domain cross-section perpendicular to the direction of fluid flow \(\zeta\) can be expressed as:  
\[\frac{u_{m}}{P}=v_{m}=\frac{64}{\pi^{6}}\sum_{i\_odd}^{\infty}\sum_{j\_odd}^ {\infty}\frac{1}{i^{2}j^{2}\left(i^{2}+\frac{j^{2}}{\beta^{2}}\right)} \tag{3}\]  
The commonly used simplified Purday's model (PM) [53]:  
\[\frac{u(x,y)}{u_{max}}=\left[1-\left(\frac{y}{b/2}\right)^{n}\right]\left[1- \left(\frac{x}{a/2}\right)^{m}\right] \tag{4}\]approximates the velocity profile of fluid flowing in the z-direction of the rectangular duct (Figure 12b), with the aspect ratio \(\beta\leq 0.5\), and where \(n=2\) and m takes on values from 2.37 for \(\beta=0.5\) to \(\infty\) for \(\beta=0\). The integration of Equation (4) over the duct cross-section yields:  
\[\frac{u(x,y)}{u_{m}}=\bigg{(}\frac{m+1}{m}\bigg{)}\bigg{(}\frac{n+1}{n}\bigg{)} \bigg{[}1-\bigg{(}\frac{y}{b/2}\bigg{)}^{n}\bigg{]}\bigg{[}1-\bigg{(}\frac{x}{ a/2}\bigg{)}^{m}\bigg{]} \tag{5}\]  
and  
\[\frac{u_{max}}{u_{m}}=\bigg{(}\frac{m+1}{m}\bigg{)}\bigg{(}\frac{n+1}{n}\bigg{)} \tag{6}\]  
where \(u_{m}\) is the average velocity in the channel and \(u_{max}\) is the maximal fluid velocity at the middle of the channel's height. For \(\beta=0\), the \(u_{max}/u_{m}=1.5\), which is the well-known value for the laminar flow between two parallel plates [54]. [PERSON] and [PERSON] [55] proposed the empirical equation for \(m\) and \(n\) calculation:  
\[m=1.7+0.5\beta^{-1.4} \tag{7}\]  
and  
\[n=\bigg{\{}