Fundamentals of Fluid Mechanics A Assignment 3 — Mass Conservation  
Question #1
Starting from the principle of the conservation of mass, do the following:
(a)  Derive the mass conservation equation in differential form. $$ \frac{\partial \rho}{\partial t} + \frac{\partial }{\partial x}(\rho u) + \frac{\partial }{\partial y}(\rho v) + \frac{\partial }{\partial z}(\rho w) = 0 $$
(b)  Derive the mass conservation equation in integral form. $$ \frac{{\rm d}}{{\rm d} t} \int_V \rho \,{\rm d} V + \int_S \rho (\vec{v} \cdot \vec{n})\, {\rm d} S=0 $$
Question #2
Consider a duct with height $H$ in which water enters at station 1 and in which water exits at station 2:
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Knowing that the velocity of the water at station 1 is: $$ \vec{v}_1=\frac{y q_{\rm ref}}{H} \vec{i} + \frac{y^2 q_{\rm ref}}{H^2} \vec{j}$$ determine, at steady-state, the mass flow rate of water exiting the duct ($\dot{m}_{\rm out}$). Take the density of water as 1000 kg/m$^3$, the reference velocity $q_{\rm ref}$ as 20 km/hr and the duct height $H$ as 10 cm.
Question #3
Consider the following two-tank system:
figure7.png  ./download/file.php?id=5360&sid=cc2efe867c54cfcf7372e02f1bb6aae7  ./download/file.php?id=5360&t=1&sid=cc2efe867c54cfcf7372e02f1bb6aae7
Knowing that the mass flow rate of water going in is of $\dot{m}_{\rm win}=2$ kg/s, that the mass flow rate of air coming in is of $\dot{m}_{\rm air}=0.02$ kg/s, that the volume of the water in tank 2 increases at the rate of $\rm 0.005~m^3/s$, find the mass flow rate of the water coming out $\dot{m}_{\rm wout}$. Also, find the ratio between the mass flow rate of air entering tank 2 and the mass flow rate of air entering the system $(\dot{m}_{\rm air2}/\dot{m}_{\rm air})$. Start from the control-volume form of the mass conservation equation and outline clearly where the control volume(s) is(are) located. Take the density of water as $\rho_{\rm w}=1000$ $\rm kg/m^3$ and the density of air as $\rho_{\rm air}=1~{\rm kg/m^3}$.
NOTE: You must solve this problem starting from the mass conservation in control volume form only.
Question #4
Consider the following flow in a corner:
figure3.png  ./download/file.php?id=5361&sid=cc2efe867c54cfcf7372e02f1bb6aae7  ./download/file.php?id=5361&t=1&sid=cc2efe867c54cfcf7372e02f1bb6aae7
The flow velocity in the $x$-direction is described by the expression: $$ u=K\left(x^{(3+a)} - 3 x^{(1+a)} y^2 \right) $$ where $K$ and $a$ are constants. Knowing that the flow is constant density and two-dimensional, do the following:
(a)  determine an expression for $v$, the flow velocity in the $y$-direction
(b)  determine the angle $\theta$
(c)  find a way to verify that the angle $\theta$ found in (b) is correct.
Question #5
Consider the following streamtraces of air flowing around an airfoil:
Knowing that $q_1=300$ km/hr, that $\theta=10^\circ$, that $H_1=10~$cm and that $H_2=7$ cm, find $q_2$.
Due on September 24th at 11:00. Do questions #2, #3, and #4 only.
3.  22 kg/s.
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