Fluid Mechanics Assignment 6 — Viscous Fluid Flow  
$\xi$ is a parameter related to your student ID, with $\xi_1$ corresponding to the last digit, $\xi_2$ to the last two digits, $\xi_3$ to the last three digits, etc. For instance, if your ID is 199225962, then $\xi_1=2$, $\xi_2=62$, $\xi_3=962$, $\xi_4=5962$, etc. Keep a copy of the assignment — the assignment will not be handed back to you. You must be capable of remembering the solutions you hand in.
Question #1
Starting from the $y$-component of Newton's law:
$$ \sum F_y= m \frac{{\rm d} v}{{\rm d} t}$$ and from the shear stresses: $$ \tau_{xy}=\mu\frac{\partial v}{\partial x}~~~~\tau_{yy}=\mu\frac{\partial v}{\partial y}~~~~\tau_{zy}=\mu\frac{\partial v}{\partial z}$$ Prove the $y$-component of the momentum equations for a viscous fluid: $$ \rho \left(\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} + w \frac{\partial v}{\partial z} \right) = -\frac{\partial P}{\partial y} + \mu \frac{\partial^2 v}{\partial x^2} + \mu \frac{\partial^2 v}{\partial y^2} + \mu \frac{\partial^2 v}{\partial z^2} +B_y $$ Outline all assumptions.
Question #2
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=1376&sid=af9ce545b24af3d818d953b92cd617d8  ./download/file.php?id=1376&t=1&sid=af9ce545b24af3d818d953b92cd617d8
The two planes are rigid and the problem is at steady-state.
Question #3
Two equally big circular plates rotate very close to each other in a viscous fluid as follows:
figure3.png  ./download/file.php?id=1375&sid=af9ce545b24af3d818d953b92cd617d8  ./download/file.php?id=1375&t=1&sid=af9ce545b24af3d818d953b92cd617d8
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 $H=(3-0.2\times\xi_1)$ mm, determine the amount of power ${\cal P}_{\rm L}$ needed to sustain $\omega_{\rm L}=3000~$rpm.
Question #4
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=1394&sid=af9ce545b24af3d818d953b92cd617d8  ./download/file.php?id=1394&t=1&sid=af9ce545b24af3d818d953b92cd617d8
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 $(3600+100\times \xi_2)$ 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 #5
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=1378&sid=af9ce545b24af3d818d953b92cd617d8  ./download/file.php?id=1378&t=1&sid=af9ce545b24af3d818d953b92cd617d8
The kinematic viscosity of the oil is $(65+\xi_2)$ 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 #6
Use the viscous momentum equations to calculate the velocity distribution, the pressure distribution in the $y$-direction and the mass flow rate per unit depth between the two plates located at $y=0$ and $y=H$:
figure6.png  ./download/file.php?id=1379&sid=af9ce545b24af3d818d953b92cd617d8  ./download/file.php?id=1379&t=1&sid=af9ce545b24af3d818d953b92cd617d8
The velocity is in the positive $x$-direction. The velocity is only varying in the $y$ direction. The gravity is in the negative $y$ direction and is the only body force present. The plane at $y=H$ is moving in the $x$-direction with the velocity $q_{\rm top}$. The pressure is assumed to be independent of $x$.
Due on Wednesday November 19th
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