Fundamentals of Fluid Mechanics A Assignment 4 — Momentum Conservation  
Instructions
Write your solutions in single column format, with one statement following another vertically. Write your solutions neatly so that they are easy to read and verify. Don't write one line with two equal signs. Highlight your answers using a box. To solve the problems, you must start using only the equations out of the Tables — you can not use equations form other places including the class notes. Failure to do this will result in a lower score and fewer comments on my part.
Question #1
(a)  Starting from Newton's law $\vec{F}_y=m\frac{dv}{dt}$ and the mass conservation equation show that the $y$-component of the momentum transport equation for an inviscid fluid corresponds to: $$ \frac{\partial \rho v}{\partial t} + \frac{\partial \rho u v}{\partial x} + \frac{\partial \rho v^2}{\partial y} + \frac{\partial \rho w v}{\partial z} = -\frac{\partial P}{\partial y} + B_y $$ with $P$ the pressure, $B_y$ the $y$-component of the body force per unit volume.
(b)  Starting from the conservative form of the momentum equation derived in (a), prove that the integral form of the $y$ momentum equation corresponds to $$ \frac{{\rm d}}{{\rm d} t} \int_V \rho \vec{v}_y \,{\rm d} V + \int_S \rho \vec{v}_y (\vec{v}-\vec{v}_{\rm cv}) \cdot \vec{n}\, {\rm d} S = -\int_S P \vec{n}_y \,{\rm d} S + \int_V \vec{B}_y \, {\rm d} V $$ where $\vec{v}$ is the velocity of the flow and $\vec{v}_{\rm cv}$ is the velocity of the control volume surfaces.
08.01.19
Question #2
Water with a density of 1000 kg/m$^3$ flows steadily through the following elbow:
figure1.png  ./download/file.php?id=5363&sid=8038fe2022f732de3cdeb672cb4a73d8  ./download/file.php?id=5363&t=1&sid=8038fe2022f732de3cdeb672cb4a73d8
At the inlet to the elbow (station 1), the pressure is $5\times 10^5$ Pa and the inner diameter is 80 mm. At the outlet to the elbow (station 2), the inner diameter is 50 mm and the pressure is $4.75 \times 10^5$ Pa. The mass flow rate is 15 kg/s. Around the elbow, the atmospheric pressure is 101300 Pa. Knowing that $\theta=60^\circ$, do the following tasks:
(a)  Determine the $x$ and $y$ components of the force vector $\vec{F}$ that is required to hold the elbow in place.
(b)  How much of this force is due to the atmospheric pressure?
Question #3
Consider a duct with a varying height in which water enters at station 1 and exits at station 2:
figure7.png  ./download/file.php?id=5364&sid=8038fe2022f732de3cdeb672cb4a73d8  ./download/file.php?id=5364&t=1&sid=8038fe2022f732de3cdeb672cb4a73d8
Knowing that the pressure of the fluid at the duct entrance and exit is equal to atmospheric pressure, knowing that the velocity distribution of the water at station 1 is: $$ \vec{v}_1=\frac{y q_{\rm ref}}{H_1} \vec{i} + \frac{y^3 q_{\rm ref}}{H^3_1} \vec{j}$$ and knowing that the velocity distribution at station 2 is uniform and parallel to $\vec{i}$, do the following:
(a)  Find the average speed of the flow at station 2
(b)  Find the forces on the duct along $\vec{i}$ and $\vec{j}$ due to the fluid motion
Take the density of water as 1000 kg/m$^3$, the reference speed $q_{\rm ref}$ as 10 km/hr, the duct entrance height $H_1=1$ m and the duct exit height $H_2=0.5$ m. Note: this is a 2D steady problem without property gradients along $z$.
Question #4
A small cart receives water from a jet as shown:
figure4.png  ./download/file.php?id=5367&sid=8038fe2022f732de3cdeb672cb4a73d8  ./download/file.php?id=5367&t=1&sid=8038fe2022f732de3cdeb672cb4a73d8
The speed and cross-sectional area of the ject are $U_j$ and $A$, respectively. At time $t=0$, the cart is at rest and its mass is $m_0$. Assume the motion is such that aerodynamic forces on the cart are negligible, i.e., ignore viscosity and assume $P=P_{\rm atm}$ at the cart surface. The only force acting on the cart is rolling friction, $\mu m g$ where $\mu$ is the coefficient of rolling friction, $m$ is the instantaneous mass of the cart and $g$ is gravitational acceleration. Do the following:
(a)  Using the translating (and accelerating) control volume indicated by the dash contour, develop differential equations for $dm/dt$ and $dU/dt$.
(b)  Assuming the surface is frictionless so that $\mu=0$, combine the equations developed in Part (a) into a differential equation involving $dU/dm$.
(c)  Solve the equation derived in Part (b) and determine the velocity when $m(t)=5m_0$ and $U_j=20$ m/s.
Answers
2.  $-1625$ N, 735 N, 410 N, $-172$ N.
4.  16 m/s.
Due on Thursday October 1st at 11:00. Do Questions #2, #3, and #4 only.
09.24.20
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