Fundamentals of Fluid Mechanics B Assignment 5 — Nearly Parallel Viscous Flow  
Question #1
Two equally big circular plates rotate very close to each other in a viscous fluid as follows:
figure3.png  ./download/file.php?id=6030&sid=3be7dafad0336e95e070130f865d71d4  ./download/file.php?id=6030&t=1&sid=3be7dafad0336e95e070130f865d71d4
One of the plates is driven by a constant power ${\cal P}_{\rm L}$ and at a constant angular speed $\omega_{\rm L}$. The other plate is braked with a power ${\cal P}_{\rm R}$. For a plate radius $R$ much larger than the distance between the plates $H$, do the following:
(a)  Determine the angular speed $\omega_{\rm R}$ as a function of $\omega_{\rm L}$ and if the breaking power ${\cal P}_{\rm R}=\frac{1}{2}{\cal P}_{\rm L}$
(b)  If $\omega_{\rm R}=0$, $\mu=10^{-2}$ kg/ms, $\rho=800$ kg/m$^3$, $R=0.1$ m, and $H=3$ mm, determine the amount of power ${\cal P}_{\rm L}$ needed to sustain $\omega_{\rm L}=3000~$rpm.
Question #2
Consider the following piston-cylinder assembly:
question5a.png  ./download/file.php?id=6033&sid=3be7dafad0336e95e070130f865d71d4  ./download/file.php?id=6033&t=1&sid=3be7dafad0336e95e070130f865d71d4
In the latter, the cylinder is fixed while the piston is allowed to move and is subject to a gravity force $mg$. Knowing that the gravitational acceleration is of $g=9.8$ m/s$^2$, that the radius of the piston and of the cylinder are of $R_{\rm p}=10$ cm and $R_{\rm c}=10.3$ cm, respectively, that the height of the piston is of $H=5$ cm, that the density of the piston is of $\rho_{\rm p}=2000$ kg/m$^3$, and that the oil viscosity and density are of $\mu_{\rm oil}=0.5$ kg/ms and $\rho_{\rm oil}=800$ kg/m$^3$, do the following:
(a)  Find the force acting on the piston in the positive $y$ direction due to viscous effects as a function of the piston speed $q$. For simplicity, you can assume that $R_{\rm c}-R_{\rm p}\ll R_{\rm c}$.
(b)  Using the expression derived in (a), find the maximum speed $q$ that the piston would get if it is allowed to fall freely assuming negligible drag on its top and bottom surfaces.
Question #3
Consider fully-developed flow between two plates as follows:
A5Q3.png  ./download/file.php?id=6070&sid=3be7dafad0336e95e070130f865d71d4  ./download/file.php?id=6070&t=1&sid=3be7dafad0336e95e070130f865d71d4
Two fluids are flowing between the two plates and do not mix with one another. Focus on the fully-developed region and assume that the density and viscosity of each fluid is known. Start from the mass and momentum conservation equations and do the following:
(a)  Find the pressure distribution along $y$.
(b)  Find the velocity distribution in fluid A.
(c)  Find the velocity distribution in fluid B.
(d)  Find the shear stress on each wall and add them together to find $\tau_w$.
(e)  Find the mass flow rate of fluid A within the duct.
Question #4
Consider a fluid flow in a pipe system as follows:
A5Q4.png  ./download/file.php?id=6072&sid=3be7dafad0336e95e070130f865d71d4  ./download/file.php?id=6072&t=1&sid=3be7dafad0336e95e070130f865d71d4
Given the pipe lengths $L_1$, $L_2$, and the pipe radii $R_1$ and $R_2$, as well as the fluid density $\rho$ and viscosity $\mu$, derive an expression (starting from the equations listed in the tables) for the pressure difference $P_1-P_3$ needed to entrain a mass flow rate $\dot{m}$. Outline clearly your assumptions.
1.  $\omega_{\rm L}/2$, 51.7 W.
2.  6.02 m/s.
3.  $P=P(x,y)$, $u_{\rm A}=\frac{1}{\mu_{\rm A}}\frac{\partial P}{\partial x} \left( \frac{y^2}{2}-\frac{Hy}{3} \right) + \frac{y \tau_i}{\mu_{\rm A}}$, $\tau_i=\frac{\partial P}{\partial x} \left( \frac{H}{6\mu_{\rm A}}- \frac{2H}{3\mu_{\rm B}} \right) \left/ \left(\frac{1}{\mu_{\rm A}}+ \frac{2}{\mu_{\rm B}} \right)\right.$, $-H\frac{\partial P}{\partial x}$, $\frac{\rho_{\rm A}D}{\mu_{\rm A}}\left( \frac{\tau_i H^2}{18}-\frac{1}{81} \frac{\partial P}{\partial x}\right)$
4.  $\frac{8\mu\dot{m}}{\rho\pi}\left(\frac{L_1}{R_1^4} + \frac{L_2}{R_2^4}\right)$
Due on Tuesday March 24th at 11:00. Do Problems #3 and #4 again and work on them until you get the right answer.
I added the answers to Question #3 (d) and (e). Note that I simplified part (e) to finding the mass flow rate instead of a non-dimensional number.
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