Convective Heat Transfer Assignment 1 — Mass and Momentum  
Question #1
Starting from the principle of conservation of mass, show that: $$ \frac{\partial \rho}{\partial t} + \frac{\partial \rho u}{\partial x} + \frac{\partial \rho v}{\partial y} + \frac{\partial \rho w}{\partial z} =0$$ with $\rho$ the mass density, and $u,v,w$ the $x,y,z$ components of the velocity vector.
Question #2
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 #3
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=4063&sid=0c72382a150717d8384c59628b083af0  ./download/file.php?id=4063&t=1&sid=0c72382a150717d8384c59628b083af0
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 #4
Consider the following piston-cylinder assembly:
question5a.png  ./download/file.php?id=4083&sid=0c72382a150717d8384c59628b083af0  ./download/file.php?id=4083&t=1&sid=0c72382a150717d8384c59628b083af0
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.
4.  6.02 m/s.
Due on September 21st at 17:00. Do Questions 1, 2, and 4 only.
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