Starting from the left-hand side of the equation, we have:

$\rho \frac{D}{D t}\left(\frac{q_i q_i}{2}\right)=\rho \frac{d}{d t}\left(\frac{q_i q_i}{2}\right)+\rho \frac{\partial}{\partial x_j}\left(u_j \frac{q_i q_i}{2}\right)$

where we have used the material derivative and the fact that $\rho u_i$ is the momentum density, which is related to the velocity by $u_i=q_i/\rho$. Now we can expand the first term on the right-hand side using the chain rule:

$\begin{aligned} \rho \frac{d}{d t}\left(\frac{q_i q_i}{2}\right) & =\rho \frac{\partial}{\partial t}\left(\frac{q_i q_i}{2}\right)+\rho u_j \frac{\partial}{\partial x_j}\left(\frac{q_i q_i}{2}\right) \ & =\rho \frac{\partial}{\partial t}\left(\frac{q_i q_i}{2}\right)+\rho q_i \frac{\partial u_j}{\partial x_j}+\rho u_j q_i \frac{\partial}{\partial x_j}\left(q_i\right) \ & =\rho \frac{\partial}{\partial t}\left(\frac{q_i q_i}{2}\right)+\rho q_i f_i+\rho u_j q_i \frac{\partial}{\partial x_j}\left(q_i\right)\end{aligned}$

where we have used the continuity equation $\partial \rho/\partial t + \partial(\rho u_j)/\partial x_j = 0$ and the equation of motion $\rho D u_i/Dt = \rho f_i – \partial \sigma_{ij}/\partial x_j$ to simplify the second term. Using the identity $\partial (q_i^2)/\partial x_j=2q_i\partial q_i/\partial x_j$ and the Einstein summation convention, we can rewrite the last term as $\partial(q_i q_j)/\partial x_j=q_i f_i+\partial(\sigma_{ij} q_i)/\partial x_j-\sigma_{ij}\partial q_i/\partial x_j$. Putting everything together, we get:

$\rho \frac{D}{D t}\left(\frac{q_i q_i}{2}\right)=\rho f_i q_i+\frac{\partial\left(\sigma_{i j} q_i\right)}{\partial x_j}-\sigma_{i j} \frac{\partial q_i}{\partial x_j}$

as desired.

In two-dimensional flow, the stress tensor $\sigma_{ij}$ has only three non-zero components: $\sigma_{xx}$, $\sigma_{yy}$, and $\sigma_{xy}=\sigma_{yx}$. Therefore, we can write

$\Phi=\sigma_{x x} \frac{\partial q}{\partial x}+\sigma_{y y} \frac{\partial p}{\partial y}+2 \sigma_{x y} \frac{\partial q}{\partial y}$

where $q$ and $p$ are the velocity potential and stream function, respectively. In two-dimensional flow, the velocity components can be written as $u=\frac{\partial \phi}{\partial y}$ and $v=-\frac{\partial \phi}{\partial x}$, where $\phi=q+ip$ is the complex potential.

Using the Cauchy-Riemann equations, we can write $\frac{\partial q}{\partial x}=\frac{\partial p}{\partial y}$ and $\frac{\partial q}{\partial y}=-\frac{\partial p}{\partial x}$. Substituting these relations and using the fact that $\sigma_{xx}=2\mu \frac{\partial u}{\partial x}$, $\sigma_{yy}=2\mu \frac{\partial v}{\partial y}$, and $\sigma_{xy}=\sigma_{yx}= \mu \left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)$ (where $\mu$ is the dynamic viscosity), we get

$\Phi=2 \mu\left(\frac{\partial u}{\partial x} \frac{\partial p}{\partial y}+\frac{\partial v}{\partial y} \frac{\partial p}{\partial x}\right)+2 \mu\left(\frac{\partial u}{\partial y} \frac{\partial q}{\partial y}-\frac{\partial v}{\partial x} \frac{\partial q}{\partial y}\right)$

Using the definition of the complex potential, we can write

$\frac{\partial u}{\partial x}=\frac{\partial}{\partial x}\left(\frac{\partial \phi}{\partial y}\right)=\frac{\partial^2 \phi}{\partial x \partial y}, \quad \frac{\partial u}{\partial y}=-\frac{\partial}{\partial y}\left(\frac{\partial \phi}{\partial x}\right)=-\frac{\partial^2 \phi}{\partial x \partial y}$

and

$\frac{\partial v}{\partial x}=-\frac{\partial}{\partial x}\left(\frac{\partial \phi}{\partial x}\right)=-\frac{\partial^2 \phi}{\partial x^2}, \quad \frac{\partial v}{\partial y}=\frac{\partial}{\partial y}\left(\frac{\partial \phi}{\partial y}\right)=\frac{\partial^2 \phi}{\partial y^2}$

The control volume is defined by two cross-sectional areas, $A_1$ and $A_2$, and a length, $\Delta x$. The volume of fluid entering the control volume at $x$ per unit time is $q(x,t)$, and the volume leaving the control volume at $x+\Delta x$ per unit time is $q(x+\Delta x,t)$. The change in volume of the fluid within the control volume per unit time is $\frac{\partial h}{\partial t} \Delta x$, where $h(x,t)$ is the height of the fluid at $x$. By the principle of conservation of mass, we have:

$\frac{\partial h}{\partial t} \Delta x+q(x, t)-q(x+\Delta x, t)=0$

Dividing by $\Delta x$ and taking the limit as $\Delta x \rightarrow 0$, we obtain:

$\frac{\partial h}{\partial t}+\frac{\partial q}{\partial x}=0$

where $q(x,t)$ is the discharge rate at $x$ and time $t$, defined as:

$q(x, t)=\int_0^{h(x, t)} u(x, y, t) d y$

where $u(x,y,t)$ is the velocity profile of the fluid.

For the case of uniform flow, the velocity profile is constant in the $y$ direction, and we have:

$u(x, y, t)=\frac{Q}{h(x, t)}$

where $Q$ is the discharge per unit width of the channel. Substituting this into the definition of $q(x,t)$, we have:

$q(x, t)=\int_0^{h(x, t)} \frac{Q}{h(x, t)} d y=Q$

Therefore, for uniform flow, the discharge rate is constant along the channel. Substituting this into the mass conservation equation, we obtain:

$\frac{\partial h}{\partial t}+\frac{\partial Q}{\partial x}=0$

To obtain a set of equations for $h(x,t)$ and $h_o(x,t)$, we need an additional equation that relates the depth of the fluid to the width of the channel. For a rectangular channel of width $b$, the area of the cross-section is $A(x,t) = b h(x,t)$, and the wetted perimeter is $P(x,t) = b + 2h(x,t)$. Using these relations, we can derive the equation of motion for $h(x,t)$ by applying the principle of conservation of momentum to the control volume shown in the figure above.

Assuming the flow is steady and inviscid, the momentum equation reduces to:

$\frac{\partial}{\partial x}\left(\frac{h^2}{2}+g h+\frac{Q^2}{2 g b^2}\right)=0$

where $g$ is the acceleration due to gravity. This equation can be integrated to obtain:

$\frac{h^2}{2}+g h+\frac{Q^2}{2 g b^2}=C(x)$

where $C(x)$ is the integration constant. The value of $C(x)$ can be determined from the boundary conditions. For example, if the depth of the fluid is $