# Modeling Open Channel Flows of a Viscous Fluid: Critical Transition and Apparent Bottom
## 2 The Navier-Stokes and the Shallow Water Models
### Shallow Water Model (SWM)  
When \(\rho_{1}\) is large compared to \(\rho_{2}\), which we assume here, the vertical integration of the Navier-Stokes equations over the heavy layer leads to the Shallow Water equations. We define the heavier fluid depth \(h\) and the mean longitudinal velocity \(U\) as:  
\[h(t,x)=Z_{s}(t,x)-Z(x) \tag{13}\]  
and  
\[U(t,x)=\frac{1}{h(t,x)}\int_{Z(x)}^{Z_{s}(t,x)}u(t,x,z)dz. \tag{14}\]  
The pressure is assumed to be hydrostatic,  
\[p(t,x,z)=p_{2}+\rho_{1}g\cos\theta(Z_{s}(t,x)-z) \tag{15}\]  
and \(p_{2}\), the pressure of the lighter fluid, is constant. Integrating Equations (5)-(7) on the vertical axis \(z\) and taking into account the surface boundary conditions, we obtain the SWM  
\[\frac{\partial h}{\partial t}+\frac{\partial}{\partial x}(Uh)=0\, \tag{16}\]  
\[\frac{\partial}{\partial t}(Uh)+\frac{\partial}{\partial x}\bigg{(}\beta U^{2 }h+g\cos\theta\frac{h^{2}}{2}\bigg{)}=-gh\cos\theta\frac{\partial Z}{\partial x }+gh\sin\theta-3\
u\lambda\frac{U}{h}\, \tag{17}\]  
where \(\
u=\mu_{1}/\rho_{1}\) is the kinematic viscosity, \(\beta\), which measures the variance of the velocity profile, is defined by:  
\[\beta(t,x)=\frac{1}{U^{2}(t,x)h(t,x)}\int_{Z(x)}^{Z_{s}(t,x)}u^{2}(t,x,z)dz\, \tag{18}\]and \(\lambda\), which measures this momentum dissipation, is defined by:  
\[\lambda(t,x)=-\frac{1}{3}\frac{h}{U}\int_{Z(x)}^{Z_{s}(t,x)}\left[\frac{\partial^ {2}u}{\partial x^{2}}(t,x,z)+\frac{\partial^{2}u}{\partial z^{2}}(t,x,z)\right] \,dz. \tag{19}\]  
In the case of a laminar flow on a flat bottom, one obtains \(\beta=1.2\) and \(\lambda=1\). The departure from these values can be due to the adjustment of the flow to a non vanishing topography \(Z(x)\).  
For a given mass flux \(q\), we, respectively, denote \(h_{n}\) and \(U_{n}\), the height and the mean velocity of the normal flow defined by the relations  
\[\left\{\begin{array}{l}q=h_{n}U_{n}\\ 0=gh_{n}\sin\theta-3v\frac{U_{n}}{h_{n}}.\end{array}\right. \tag{20}\]  
This solution corresponds to the steady flow on an infinite inclined plane, characterized by a balance between gravitational acceleration and viscous dissipation.  
The local Froude number \(F(t,x)\) is defined by  
\[F=\frac{U}{\sqrt{gh\cos\theta}}. \tag{21}\]  
When \(\beta(t,x)\) and \(\lambda(t,x)\) are prescribed, the Shallow Water Equations (16) and (17) are hyperbolic and the two wave velocities read  
\[V_{\pm}=\beta U\left(\frac{1+\beta}{2}\pm\sqrt{\frac{1-\beta}{2}+\frac{1}{F^{ 2}}}\right). \tag{22}\]  
Steady solutions are such that \(h(x)\), \(U(x)\)\(\beta(x)\) and \(\lambda(x)\) satisfy the ordinary differential equation  
\[\left(1-\beta\,F^{2}\right)\frac{dh}{dx}=\tan\theta-\frac{dZ}{dx}-F^{2}\left( \frac{d\beta}{dx}+\frac{3v\,\lambda}{q}\right)\qquad\text{with $q=h\,U$ constant}. \tag{23}\]  
The critical condition is thus obtained for \(F=\beta^{-1/2}\) for both wave propagation and backwater curves. In the present SW simulations both the \(\beta\) and \(\lambda\) parameters are considered constant and equal to \(1\). Since the flow under investigation is laminar, it can be argued that this choice for the \(\beta\) parameter is incorrect. Nevertheless, this is due to the fact that a \(\beta\
eq 1\) will allow the SW to be Galilean invariant.