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.
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=ac7ea59d1e7bcc99055a8ad9151f298d  ./download/file.php?id=5363&t=1&sid=ac7ea59d1e7bcc99055a8ad9151f298d
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=ac7ea59d1e7bcc99055a8ad9151f298d  ./download/file.php?id=5364&t=1&sid=ac7ea59d1e7bcc99055a8ad9151f298d
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=ac7ea59d1e7bcc99055a8ad9151f298d  ./download/file.php?id=5367&t=1&sid=ac7ea59d1e7bcc99055a8ad9151f298d
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.
Due on Thursday October 3rd at 11:00. Do Questions #1, #2, and #4. Question #3 is optional.
For problem #2, the answers given are correct (including the signs). Note that in 2b, you need to find the force needed to keep the elbow in place to sustain the atmospheric pressure, not the force that the atmospheric pressure exerts on the structure.
PDF 1✕1 2✕1 2✕2