Fundamentals of Fluid Mechanics Assignment 4 — Momentum Conservation
 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:
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:
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:
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.
 2. $-1625$ N, 735 N, 410 N, $-172$ N. 4. 16 m/s.
 $\pi$