Convective Heat Transfer Assignment 3 — Reynolds Number  
Question #1
Starting from the $x$-component of the momentum equation: $$ \rho \left(\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z} \right) = -\frac{\partial P}{\partial x} + \mu \frac{\partial^2 u}{\partial x^2} + \mu \frac{\partial^2 u}{\partial y^2} + \mu \frac{\partial^2 u}{\partial z^2} $$ and from the mass conservation equation: $$ \frac{\partial \rho}{\partial t} + \frac{\partial }{\partial x}(\rho u) + \frac{\partial }{\partial y}(\rho v) + \frac{\partial }{\partial z}(\rho w) = 0 $$ Show that the skin friction coefficient and the thickness of a laminar boundary layer correspond to: $$ C_f=0.647 \cdot {\rm Re}_x^{-0.5} {\rm ~~~~and~~~~} \delta/x=4.64 \cdot {\rm Re}_x^{-0.5} $$ Outline all assumptions.
Question #2
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.
Question #3
An air stream with a speed of $50$ m/s and density of $\rho=1.0$ kg/m$^3$ flows parallel to a flat plate with a length of 45 cm and a width of 100 cm. Determine the total drag force on the flat plate and calculate the boundary layer thickness 10 and 45 cm from the leading edge. Take the kinematic viscosity as $15\times 10^{-6}$ m$^2$/s.
2.  211 m/s, 13.4 N/m.
3.  0.8 mm, 6.9 mm, 3.52 N.
Due on Wednesday April 11th at 17:00. Do all 3 questions.
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