Fundamentals of Fluid Mechanics B Questions and Answers

The midterm is open book so you can consult your assignments, class notes, books if you want. However, if I were you, I would stick to consulting the tables: you won't waste time and score higher this way.	03.25.20

Question by AME536B Student I was attempting to figure out the mean free path derivation question you gave us previously on how to get the \[\sqrt2\] term. What I came up with using your tip of \begin{equation} q_{rms}=c*q_{avg} \end{equation} is, \begin{equation}\overline{V_{rel}^{2}}=\overline{V_1^{2}}+\overline{V_2^{2}}\end{equation} Is this process correct?

I don't understand where your equation (2) is coming from. You need to clarify this.

Question by AME536B Student On the tables, the equation given for the strain rate $S_{\theta r}$ in spherical coordinates has a term of $\frac{v_{\theta}}{r}$; is that a typo? Shouldn't the shear strain equation for spherical coordinates be given by: $$ S_{r\theta} = \frac{1}{2} \left( \frac{1}{r}\frac{\partial v_r}{\partial \theta} + \frac{\partial v_{\theta}}{\partial r} \right) $$	04.06.20

The equation in the tables is correct: there is no typo.

Question by AME536A Student For question 1. b. on Assignment 6, do you want us to derive the expression you gave for the streamfunction? Or just start from the given equation?

If it's given in the question don't derive it.

Question by AME536B Student For problem 3b. Do we need to derive the Oseen's stream function or can we use the stream function provided in class and in the tables?	04.08.20

Use the one in the tables, don't derive it.

Question by AME536B Student For problem 3 on HW 6, can we assume that the droplet spends a much more significant time at terminal velocity than it does getting to terminal velocity, and thus say that: time=\frac{Distance}{Terminal Velocity}

You could do this as a first approximation but you should ultimately integrate fully the equations starting from rest till terminal velocity. And typeset properly your comments/questions using LATEX please.

Question by AME536B Student Can you give us the results for problem #3, so we can use them as a reference?	04.13.20

You can obtain a reference on your own using a simplified solution assuming no acceleration. When setting the height to a large value, both solutions (simplified and complete) will give the same answer for the time.

Question by AME536B Student I have a question on problem #3 on HW #6. We can solve the Stoke's equations analytically, and using those solutions we can determine a Reynold's number to verify if the Stoke's solution is valid or not. Assuming that the Reynold's number becomes very small, can we say that the Oseen drag equation converges to the Stoke's drag equation? Obviously this wouldn't work when the Re approaches 1, but would it be considered an okay assumption for a very small Re?

No, you should solve for the Oseen equation fully. This will give a different solution than the Stokes solution and you need to determine this difference. Also, you should write you code or analytical expression in general form so it can be applied to any droplet size, not just those listed in the question.

Question by AME536B Student Would it be too much to ask for the homework to be posted Thursday after class or Friday morning please? Most classes give you at least 1 week to work in homework problems. Thanks!	04.20.20

I think this assignment is significantly easier than the previous one, and can be completed in 4 days. The deadline is not changed. Besides, the 3 questions were already mentioned to you in class more than 1 week ahead of the deadline. You had plenty of time to work on them.

This thread is for questions related to solving the assignment not to try to delay the due date.

Question by AME536A Student In homework 8.2, by total drag do you mean the skin friction drag or the skin friction drag plus the form drag (due to a pressure gradient)?	04.27.20

If you can calculate form drag plus skin friction drag accurately then do so. Otherwise provide best estimate.

Question by AME536A Student In homework 8 question 1, can we assume an infinitely long plate in z, so that the derivatives in z vanish?

You can not make extra assumptions on top of those involved in deriving the boundary layer equations.

Question by AME536B Student Concerning question 4, the velocity at the wake was obtained by assuming that $x$ goes to infinity. Why we can't say that $\frac{u_w}{u_{inf}}=0.99999$ at the edge of the wake in order to find its height, as this strategy causes problems only close to the cylinder.	04.30.20

It's because far away from the cylinder, $u_w$ will be almost exactly equal to $u_\infty$ everywhere within the profile. As $x \rightarrow \infty$, $u_w \rightarrow u_\infty$. Thus, the point $u_w=0.999999 u_\infty$ won't lie anywhere on the profile! The way you determine the height must work equally well close to the cylinder and when $x$ is very large. Think about this more.

Question by AME536B Student For Q#2,I am able to reach this point: $$ \frac{\partial}{\partial t}\int_{C\forall}\rho V_{i}d\forall + \oint_{CS}\left(V_{C\forall}\cdot\hat{n}\right)\rho V_{i}dS = -\oint_{CS}\left(P\cdot\hat{n}_i\right)dS + \oint_{CS}\tau_{ij}\hat{n}_{j} dS + \int_{C\forall}\vec{f}_{i}d\forall $$ But I am not sure about the 2nd term on the LHS of the equation, would you provide a hint for this please? $$ \oint_{CS}\rho V_{i}\left(V_{C\forall}\cdot\hat{n}\right)dS \stackrel{?}{=} \oint_{CS}\rho V_{i} \left(V - V_{C\forall}\right)\cdot\hat{n}dS $$ Note: I started by contracting the momentum eq in conservative form to its vector form $$ \frac{D}{Dt}\left(\rho\vec{V}\right) = -\nabla\cdot \vec{P} + \nabla\cdot\vec{\tau} + \vec{f}_b $$ I applied the Reynolds transport on the LHS and the Divergence theorem to the pressure gradient term and the shear gradient

05.02.20

I do not agree with your last equation. How can you have a pressure vector? Pressure is a scalar. Also, you shouldn't transform the momentum equation convection terms as a substantial derivative. I don't see how this can help you. Simply substitute the time derivative of the momentum equation within the first term on the RHS of the Reynolds transport theorem listed on the last page of the tables. Then, there'll be several integrals you need to rewrite using Gauss's theorem. The ones involving the velocity are easy. The ones involving pressure and the shear stress are more challenging.

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