Viscous Flow Assignment 3 — Couette Flow  
Question #1
Use the momentum equations in viscous form and the mass conservation equation to determine the velocity distribution and the mass flow rate per unit depth between two planes located at $y=0$ and $y=H$ for $x\gg H$:
figure2.png  ./download/file.php?id=2573&sid=c6273fe0cbdab1b059c1d4353e756b43  ./download/file.php?id=2573&t=1&sid=c6273fe0cbdab1b059c1d4353e756b43
The two planes are rigid and the problem is at steady-state.
Question #2
Two equally big circular plates rotate very close to each other in a viscous fluid as follows:
figure3.png  ./download/file.php?id=2574&sid=c6273fe0cbdab1b059c1d4353e756b43  ./download/file.php?id=2574&t=1&sid=c6273fe0cbdab1b059c1d4353e756b43
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 #3
A crankshaft journal bearing in an automobile engine is lubricated by oil with a kinematic viscosity of $10^{-4}$ m$^2$/s and a density of 885 kg/m$^3$:
figure4.png  ./download/file.php?id=2575&sid=c6273fe0cbdab1b059c1d4353e756b43  ./download/file.php?id=2575&t=1&sid=c6273fe0cbdab1b059c1d4353e756b43
The bearing inner diameter $D_{\rm i}$ is of 10 cm, the bearing outer diameter $D_{\rm o}$ is of 11 cm, and the bearing rotates at $7200$ rpm. The bearing is under no load so the clearance is symmetric. Determine the torque per unit depth and the power dissipated per unit depth.
Question #4
A piston with a diameter of $D_{\rm i}=100$ mm and a length $L=150$ mm is moving concentrically in a cylinder with a diameter $D_{\rm o}=100.1$ mm. The gap between the cylinder and the piston is filled with oil:
figure5.png  ./download/file.php?id=2576&sid=c6273fe0cbdab1b059c1d4353e756b43  ./download/file.php?id=2576&t=1&sid=c6273fe0cbdab1b059c1d4353e756b43
The kinematic viscosity of the oil is 65 cSt and the density is 885 kg/m$^3$. How big is the force that has to be applied to move the piston in the axial direction with a speed $q=3$ m/s if only the viscous resistance is considered? Note: 1 cSt=$10^{-6}$ m$^2$/s.
Question #5
Consider the following piston-cylinder assembly:
question5a.png  ./download/file.php?id=3840&sid=c6273fe0cbdab1b059c1d4353e756b43  ./download/file.php?id=3840&t=1&sid=c6273fe0cbdab1b059c1d4353e756b43
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 #6
Consider a journal bearing as follows:
Q7.png  ./download/file.php?id=4474&sid=c6273fe0cbdab1b059c1d4353e756b43  ./download/file.php?id=4474&t=1&sid=c6273fe0cbdab1b059c1d4353e756b43
Starting from the Navier-Stokes equations in cylindrical coordinates, derive an expression for $v_\theta$ as a function of $r$. It is known that $R_2-R_1=\frac{1}{4} R_1$. Indicate clearly the assumptions (including terms dropped) and explain why this is valid in this case.
2.  $\omega_{\rm L}/2$, 51.7 W.
3.  10.48 N, 7903 W/m.
5.  6.02 m/s.
6.  $\omega R_1 \ln(r/R_2) \left/ \ln(R_1/R_2)\right.$
Due on Wednesday October 17th at 9:00. Do problems #2, #5, and #6 only.
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