Talk:Reynolds transport theorem

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1 Reynold's Transport Theorem
2 Correctness?
3 Mass Flow
4 Fluid Mechanics Text
5 Formulation Error?

[edit] Reynold's Transport Theorem

I have no previous knowledge of the Reynold's Transport Theorem, but as a Physcist I make a big plea to whoever knows more on this topic to convert the intergral theorems into differential theorems. Integral theorems are much harder to apply and use than differential forms of the same equations. You can also "see" the physics more easily in differential form, than in integral form. At least, differential forms are taught these days, instead of integral forms.

The differential forms of these equations are known as the Navier-Stokes equations which you may have heard of. I have to disagree with your statements that the differential forms are more useful or more often used, though that may be true in your field. I am a mechanical engineer, and the integral forms of these are both very powerful, and a lot clearer to me about what they describe. Regardless, they have already been converted for you by some smart people a long time ago. An important note: I added this page and it was written from the perspective of a mechanical engineer. If there is something that could be done to make it more useful to other specialties, I welcome the additions. -EndingPop 17:27, 16 January 2006 (UTC)

On 2/8/2006 someone added "also magnetic" to the end of one of the sentences. That sentence ended in the words "electromagnetic fields". Even though magnetic fields were already covered, if you think something is missing but don't want to add it in final form yourself make sure to put it on here on the Talk page. -EndingPop 02:03, 9 February 2006 (UTC)

I might add that the purpose of the Reynolds Transport Theorem, for the most part, was to allow differential forms of transport equations to be developed. One can't accomplish this feat without doing something with that nasty substantial derivative outside the integral (which, of course, has limits of integration that depend on time, and so can't just be "taken inside" the integral) The Transport Theorem, by the way, is identical to the Leibniz integral rule for 1-dimensional domains.--71.98.78.28 04:39, 11 June 2007 (UTC)

If someone has the time, please consider simplifying the equations with $\overrightarrow{V} = \overrightarrow{V_b} + \overrightarrow{V_r}$ . hoo0 05:06, 20 February 2006 (UTC)

First, thanks for the additions to the page. I really like the introduction; it gives it a lot more depth. Now, on the point of combining the equations. I added them this way because it makes more sense from a teaching point of view. The difference between Vb and Vr needs to be further explained, but I think that for a person looking for the equations would find this more useful than a condensed version. The expanded versions simply showcases more about the physical phenomena. If you still disagree, I won't revert it if you decide to make the change. - EndingPop 04:26, 21 February 2006 (UTC)

Does anyone know of a way to have a formula with a strike through it to symbolize the volume? You can see on this page it is just put there in normal text mode, and it looks terrible. - EndingPop 15:08, 21 August 2006 (UTC)

As far as I can tell you can't do strikethroughs in TeX without an extension package, which I don't know how to use on Wikipedia (if you even can). I'd suggest changing this to some other standard symbol for differential volume--

d V

d Ω

d 3 x

or some such. Starryharlequin 03:27, 5 October 2006 (UTC)

V is already used as velocity, and the strike through is what I have seen in academia as the standard. Is there another symbol that is used in thermofluids? -EndingPop 11:50, 5 October 2006 (UTC)

In my thermofluids class, as well as our book, script v is used for velocity and capital V for volume. This seems a pretty good compromise to me. Something alone the lines of:

$\frac{DN_{sys}}{Dt} = \int_{c.v.}^{} \frac{\partial}{\partial t} (\rho \eta) dV + \int_{c.s.}^{} \rho \eta \vec{v_b}\cdot \widehat{n} dA+\int_{c.s.}^{} \rho \eta \vec{v_r}\cdot \widehat{n} dA$

I used \vec with lowercase v, because I don't really know tex enough to figure out a script vector. !jim 18:45, 6 October 2006 (UTC)

I have always seen the script v as specific volume in thermofluids, but since that isn't used here I'll try to make that change. - EndingPop 12:57, 9 October 2006 (UTC)

[edit] Correctness?

Is the equation shown correct? Shouldn't it be:

$\frac{DN_{sys}}{Dt} = \int_{c.v.}^{} \frac{\partial}{\partial t} (\rho \eta) dV -\int_{c.s.}^{} \rho \eta \vec{v_r}\cdot \widehat{n} dA$

Can anyone check the reference? —The preceding unsigned comment was added by 193.1.100.105 (talk • contribs) 09:14, January 3, 2007 UTC.

I'm the original author of the article, so maybe my assurance doesn't count, but I assure you the formula is correct. The term with "vb" in it deals with the more general case of deformable control volumes. - EndingPop 14:54, 3 January 2007 (UTC)

You might encounter your version of this equation in a basic thermo class. That equation (yours) assumes that the control volume is fixed in whatever reference frame you're using, so the vb is zero, and the third term goes away. The minus sign is just related to a different sign convention, I believe. For what it's worth, my Thermo I professor also did away with the third term in the more general equation. !jim 04:21, 4 March 2007 (UTC)

Why are the $\int_{c.s.}^{} \rho \eta \vec{v_r}\cdot \widehat{n} dA$ and the $\int_{c.s.}^{} \rho \eta \vec{v_b}\cdot \widehat{n} dA$ two integral instead of just one $\int_{c.s.}^{} \rho \eta \vec{v}\cdot \widehat{n} dA$ where $\vec{v} = \vec{v_r} + \vec{v_b}$ i.e. the fluid flow at the surface? —The preceding unsigned comment was added by 129.94.6.28 (talk) 03:03, 22 March 2007 (UTC).

This was actually discussed above. It was felt that leaving the two velocities seperate was more clear or provided more information to someone not necessarily familiar with this concept. See hoo0's comment above. !jim 06:03, 22 March 2007 (UTC)

[edit] Mass Flow

Shouldn't the case where $η$ = 1 be included? This will reduce to the mass flow conservation equations. I believe that they are important enough to warrant its own heading along with the momentum and energy conservation. 203.59.183.57 07:43, 18 April 2007 (UTC) Tim

Somehow I missed a deletion of that section several edits back. I am currently working on integrating past constructive edits into the last version that had the mass formulation in it. Thanks for the head's up! - EndingPop 14:22, 18 April 2007 (UTC)

There, I've updated the page with the mass formulation back in there. The initial removal was my fault, I reverted some vandalism and missed the edit before by the same user that removed the mass formulation section. All subsequent constructive edits should have been included, but I'd appreciate someone checking my work. - EndingPop 14:34, 18 April 2007 (UTC)

I believe the case where $\frac{DN_{sys}}{Dt}=0$ for the mass flow equation only holds true in the steady flow conditions, and not in the general case. Consider a water tank with a hole, filling with water. The rate of change of the mass of the water tank is equal to the differences between the mass flow in and out. Also I think that the page should be edited so that it starts with the most general equation, then shows the equations with various simplifying forms. i.e. steady flow, fixed volume, incompressible flow, adiabatic flow, etc.

Mass of a system is always constant and therefore cannot change in time, meaning $\frac{DM_{sys}}{Dt}=0$ is always true. - Hacktivist 15:58, 4 November 2007 (UTC)

In this formulation, "system" is defined as an identifiable and fixed quantity of matter. We aren't accounting for nuclear reactions, so the time-rate of change of mass in the system is necessarily zero. Don't confuse the system with the control volume (Lagrangian vs. Eulerian viewpoints) - EndingPop 15:42, 5 November 2007 (UTC)

[edit] Fluid Mechanics Text

In my fluid mechanics textbook, the Momentum Equation is slightly different, some of it is just placement, but I was wondering what assumption my text might be making that it neglects the 3rd integral? I'm not seeing how there equivalent.

$\begin{align} \sum_{} \vec{F}&=\int_{c.v.}^{}\frac{\partial}{\partial t}(\rho \vec\upsilon) dV + \int_{c.s.}^{} \rho \vec\upsilon(\vec\upsilon_b \cdot \widehat{n})dA + \int_{c.s.}^{} \rho \vec\upsilon(\vec\upsilon_r \cdot \widehat{n})dA \\ \sum_{} \vec{F}&=\frac{\partial}{\partial t}\int_{cv}^{} \vec\upsilon \rho d\vec{\mathbb{V}} + \int_{cs}^{} \vec\upsilon \rho \vec{V} \cdot d\vec{A} \end{align}$

$\mathbb{V}$ = Volume

$\vec\upsilon$ = Velocity relative to an inertial reference frame

$\vec{V}$ = Fluid velocity relative to the control surface

Zath42 (talk) 19:34, 2 March 2008 (UTC)

Your text assumes a nondeformable control volume. If you allow it to move, you get the extra integral. Use Vr = V - Vb to convert between the two. - EndingPop (talk) 13:55, 3 March 2008 (UTC)

Thanks, that clears things up. Zath42 (talk) 18:39, 3 March 2008 (UTC)

[edit] Formulation Error?

I believe there is still a problem though. In going from the general form to the specialized form for mass, you are taking the time derivative outside the integral over the c.v. - this is only true for a non-deformable c.v. For a deformable cv, from Liebnitz theorem,

del/delt(integral_over_cv(rho*eta*dV))=integral_over_cv(del/delt(rho*eta*dV))+integral_over_cs(rho*eta*v_b.normal*dS)

so, only the relative velocity integral should show up - this issue is covered in Potter and Foss' book that is cited as a reference —Preceding unsigned comment added by 128.138.64.39 (talk) 21:38, 23 May 2008 (UTC)

I'm not following you. Could you point out the alleged discrepancy/error? What, specifically, is wrong, and why? Also, a page number from Potter would be good too, since you said it was dealt with in there. - EndingPop (talk) 20:31, 27 May 2008 (UTC)