Problem 1 (50 points)
For the two-dimensional element shown, use the momentum equation in the integral form given by
$\vec{F} = \frac{\partial}{\partial t} \int_{cv} \rho \vec{V} \, dV + \int_{cs} \vec{V} (\rho \vec{V} \cdot \vec{dA})$,
and show that the corresponding momentum equation in the differential form in $x$ and $y$ directions are given
by
$f_x = \frac{\partial(\rho u)}{\partial t} + \frac{\partial(\rho u^2)}{\partial x} + \frac{\partial(\rho uv)}{\partial y}$,
$f_y = \frac{\partial(\rho v)}{\partial t} + \frac{\partial(\rho uw)}{\partial x} + \frac{\partial(\rho v^2)}{\partial y}$.
(1)
(2)
In the above equations $\vec{V} = ui + vj$ is the velocity vector, $V$ is volume per unit of width, and $\vec{f} = f_x \hat{i} + f_y \hat{j}$,
is the force $\vec{F}$ per unit volume.
