Fundamentals of Fluid Mechanics Questions and Answers  
Question by AME536A Student
For problem #1 on homework #7, we are asked to solve for the pressure inside a satellite as a function of time. You gave us the speed and mass flux in terms of the molecules, so we can solve with the energy equations. We are a bit confused on the starting point. Do we start from the isentropic relation equation and solve in conjunction with the momentum equation (like we found compressible stagnation pressure), $$ \frac{P}{\rho^{\gamma}_{}} = constant $$ or do we start the solution from the energy transport equation? $$ \frac{d}{dt}\int_{}^{}\rho EdV + \int_{}^{} \left(\rho H \overrightarrow{\rm v} \right) \cdot \overrightarrow{\rm n} ds = \int_{}^{} \dot{Q} dV $$
Start from the total energy transport equation. Find the answer without assuming an isentropic path. If you solve the problem assuming an isentropic path, would you get the same answer? Why or why not?
Question by AME536A Student
For problem #1 on Homework #7 I am facing a problem in finding the exact expression for the pressure. Starting from the energy transport equation $\frac{d}{dt}\int_{}^{}\rho EdV + \int_{}^{} \left(\rho H \overrightarrow{\rm v} \right) \cdot \overrightarrow{\rm n} ds = 0 $ with $H=h+0.5u_{exit}^2$ where $u_{exit}=0.25\sqrt{\frac{8RT}{\pi}}$ and $E=e$ because the kinetic energy is considerable only near the hole I got $T(t)=T_i \left(\frac{\rho(t)}{\rho_i}\right)^{\frac{4\pi+1}{10\pi}}$. By substituting to the mass conservation equation I got an expression without $\gamma$. Is my process correct or I am missing something?
Yes you are missing a term! Don't make an assumption and remove terms unless you have no other option. Keep your expression as complete as possible.
Question by AME536A Student
For Assignment 7, #1: We have gotten to this point, and despite our best efforts have not been able to get to the solution provided at the end of the assignment. Would it be possible to go over this question in class and/or receive an extension (since many of us have had at least one midterm this week)? $$ [1 - \frac{1}{20} \frac{A}{V} \sqrt{\frac{8RT}{\pi}}(\frac{8}{32\pi}+1)]^{\frac{2\gamma - 2}{\gamma}}_{} = \frac{P}{P^{}_{i}} $$
I'll give you an extension until Tuesday 11:00. The quiz will take place on Tuesday, not tomorrow.
Question by AME536A Student
For Assignment 7, #1: I got the following as a solution: $$\frac{P}{P_i}=\left( 1-\frac{bA}{8V} t \sqrt{\frac{8RT_i}{\pi}} \right)^{\frac{2-2b}{b}}$$ However, the solution is : $$\frac{P}{P_i}=\left( 1-\frac{bA}{8V} t \sqrt{3RT_i} \right)^{\frac{2-2b}{b}}$$ Is there any hint for what I miss in solving this question ?
You have the right answer: there was a typo and it is now fixed.
Question by AME536A Student
While studying for the midterm I face a problem concerning the integral form of the energy equation in a moving control volume. I derived that equation and it's the following:
$\frac{d}{dt} \int_{V}^{} \rho E dV +\int_{S}^{} \rho H (\vec{v}-\vec{v_{cv}}) \cdot \vec{n} dS +\int_{S}^{} P (\vec{v_{cv}} \cdot \vec{n}) dS=\dot Q$
I think that the pressure term should vanish but I can't understand why. Is it possible to explain that?
I am not sure why you think the pressure term should vanish. You should explain your reasoning.
Question by AME536A Student
I solved Question #3 Assignment #6 by using the above equation without the pressure term and I got the correct result. If I understand well that term is the work per time done by pressure, but we have already taken into account this.
The work due to pressure in the ground frame scales with the product between pressure and velocity of surface where pressure is applied (i.e. velocity of the surface with respect to the ground). If you expand $H$ in terms of $P$ in your equation and rewrite, you'll find that this is exactly the case. I see nothing wrong with the equation you posted.
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