Applying Reynold's Theorem to Infinitesimal Element: Fluid Dynamics

In summary, the transport equation states that if you are moving a finite volume with a fluid, you will get a term in the transport equation that is the convective derivative plus a term that is the area under the curve of the function that represents the volume changing shape.
  • #1
nonequilibrium
1,439
2
So Reynold's transport theorem states that [itex]\frac{\mathrm d}{\mathrm d t} \int_{V(t)} f \; \mathrm d V = \int_{V(t)} \partial_t f \; \mathrm d V + \int_{V(t)} \nabla \cdot \left( f \mathbf v \right) \; \mathrm d V[/itex].

Now I would expect (on basis of conceptual reasoning) that if I were to apply this to an infinitesimal element around [itex]\mathbf r(t)[/itex], I should get the well-known (the so-called convective derivative) result [itex]\frac{\mathrm d}{\mathrm d t} f(\mathbf r(t)) = \partial_t f + \left( \mathbf v \cdot \nabla \right) f[/itex]

However, it's straight-forward to see that one gets [itex]\frac{\mathrm d}{\mathrm d t} f(\mathbf r(t)) = \partial_t f + \nabla \cdot \left( f \mathbf v \right)[/itex]

I get a term [itex]f \; \nabla \cdot \mathbf v[/itex] too much. What gives?
 
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  • #3


You're correct, thank you. To keep it clean I fixed my mistake in the OP. However, it seems the problem remains...
 
  • #4


On further thought, both results might be correct and they might indicate a fundamental difference between a "point" and an "infinitesimal volume": the extra term indicates the volume change of the infinitesimal volume, which apparently is not negligible.
 
  • #5


The transport equation you wrote applies to any finite volume which moves with the fluid (and changes size and shape as it moves, in general).

If you want to work with a fixed volume in space, you get a different form of the transport equation. The difference between the two (i.e. the Lagrangian and Euleran formulations) is fundamental. Of course you can transform one equation into the other. In one form you the boundary moves. In the other, there is fluid flow through the fixed boundary.

But you can't chop a finite sized moving volume into a sum of "infinitesimal" parts that are fixed in space, and just hope things will work out OK!
 
  • #6


Oh but I wasn't keeping it fixed in space, I had simply not entertained the thought that an infinitesimal volume would also change in size, which it does, by a factor of [itex]\nabla \cdot \mathbf v[/itex]. Keeping this in mind, for the infinitesimal volume we get [itex]\frac{\mathrm d}{\mathrm d t}\left( f \mathrm d V \right) = \left( \frac{\mathrm d}{\mathrm d t}f \right) \mathrm d V + f \; \nabla \cdot \mathbf v \; \mathrm d V[/itex] where the second derivative w.r.t. to time is the convective derivative.

Alternatively, this leads to a different way to prove Reynold's transport theorem, i.e. one can show on grounds of physical reasoning that an infinitesimal volume changes in size with factor [itex]\nabla \cdot \mathbf v[/itex]. Reynold's transport theorem follows from this.
 
  • #7


mr. vodka said:
I had simply not entertained the thought that an infinitesimal volume would also change in size

That's one reason why I try to stick with finite volumes, whenever possible :smile:

One way to dig yourself out of this sort of hole is to back off and think about an analogous 1-D problem. You can convince yourself of the difference between say $$\frac{d}{dt}\int_a^b f(x,t)\,dx\quad\mathrm{and}\quad\frac{d}{dt}\int_{a(t)}^{b(t)} f(x,t)\,dx$$ by drawing pictures of the area under the curve at times ##t## and ##t + \delta t##. But drawing pictures of 3-D vector fields in arbrtrary regions is hard!
 

1. What is Reynold's Theorem and how does it apply to fluid dynamics?

Reynold's Theorem is a fundamental principle in fluid dynamics that describes the conservation of mass and momentum in a fluid flow. It states that the time rate of change of any extensive property of a fluid volume is equal to the sum of the net flux of that property across the boundaries of the volume and the local rate of change of the property within the volume. This theorem is used to analyze and solve problems in fluid mechanics.

2. How is Reynold's Theorem applied to infinitesimal elements in fluid dynamics?

In fluid dynamics, infinitesimal elements refer to very small volumes of fluid. Reynold's Theorem can be applied to infinitesimal elements by considering the properties of the element at a specific point in time and analyzing the fluxes of those properties across the boundaries of the element. This allows for a more detailed and accurate analysis of fluid flow dynamics.

3. What are the main assumptions made when applying Reynold's Theorem to infinitesimal elements?

There are several assumptions made when applying Reynold's Theorem to infinitesimal elements. These include assuming that the fluid is incompressible, continuous, and has a constant density. It is also assumed that the fluid flow is steady and the properties of the fluid are uniform within the infinitesimal element.

4. What are some practical applications of using Reynold's Theorem to infinitesimal elements in fluid dynamics?

Reynold's Theorem is commonly used in various applications in fluid dynamics, such as designing fluid flow systems, analyzing aerodynamics, and studying ocean currents. It is also used in industries such as aerospace, automotive, and marine engineering to optimize the performance of fluid flow systems and reduce drag.

5. Are there any limitations to using Reynold's Theorem to analyze fluid dynamics?

While Reynold's Theorem is a powerful tool in fluid dynamics, it does have some limitations. It assumes that the fluid is homogeneous and has a constant density, which may not always be the case in real-world scenarios. Additionally, the theorem is based on certain simplifications and assumptions, so the results may not always be entirely accurate. It is important to consider these limitations when applying Reynold's Theorem to fluid dynamics problems.

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