2016 Viscous Flow Midterm Exam
The midterm exam will take place on Monday October 31st from 18:30 till 20:30. I'll announce soon the room number in this thread.
 10.19.16
The midterm will take place in room 9302.
Monday October 31st 2016
18:30 — 20:30

NO NOTES OR BOOKS; USE VISCOUS FLOW TABLES THAT WERE DISTRIBUTED; ANSWER ALL 4 QUESTIONS; ALL QUESTIONS HAVE EQUAL VALUE.
 10.25.16
 Question #1
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 a viscous fluid corresponds to: $$\frac{\partial \rho v}{\partial t} + \frac{\partial \rho u v}{\partial x} + \frac{\partial \rho v^2}{\partial y} + \frac{\partial \rho w v}{\partial z} = -\frac{\partial P}{\partial y} + \frac{\partial \tau_{xy}}{\partial x} + \frac{\partial \tau_{yy}}{\partial y} + \frac{\partial \tau_{zy}}{\partial z}$$ with $P$ the pressure and $\tau_{ij}$ the shear stress vector along $j$ acting on the faces perpendicular to $i$.
 Question #2
Starting from the angular distortion and volume expansion of a fluid element, show that the shear stresses for a fluid with a linear stress-strain relationship become: $$\tau_{xx}=2 \mu \frac{\partial u}{\partial x}\\ \tau_{yy}=2 \mu \frac{\partial v}{\partial y}\\ \tau_{zz}=2 \mu \frac{\partial w}{\partial z}$$ and $$\tau_{xy}=\tau_{yx}=\mu\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x} \right)\\ \tau_{xz}=\tau_{zx}=\mu\left(\frac{\partial u}{\partial z}+\frac{\partial w}{\partial x} \right)\\ \tau_{yz}=\tau_{zy}=\mu\left(\frac{\partial v}{\partial z}+\frac{\partial w}{\partial y} \right)$$ where $\mu$ is the viscosity of the fluid.
 Question #3
Consider the following piston-cylinder assembly:
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 #4
Consider air flowing on top of a flat plate as follows:
It is known that the length of the plate is $L=19$ cm, that the thickness of the boundary layer at the plate exit at $x=L$ is of $\delta=3$ mm, that the air viscosity is of $\mu=2\times 10^{-5}$ kg/ms, that the air density is of $\rm 1~kg/m^3$. Also, although the freestream velocity is not known precisely, it is certain that it is higher than 12 m/s: $$u_{\infty} \gt 12~{\rm m/s}$$ Knowing the latter, do the following:
 (a) Determine whether the flow is laminar or turbulent at $x=L$. (b) Find the freestream velocity $u_{\infty}$. (c) Find total drag force acting on the plate per unit depth in N/m due to friction effects.
 $\pi$