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I just don't see how to approach this question without Bernoulli's equation, I can see that the Lagrangian derivative of the density is zero but that doesn't specifically show that the density is constant along streamlines.

Thanks.

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- Thread starter jmz34
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In summary, you can use the divergence theorem to show that the mass within a tube is constant along its streamlines.

- #1

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I just don't see how to approach this question without Bernoulli's equation, I can see that the Lagrangian derivative of the density is zero but that doesn't specifically show that the density is constant along streamlines.

Thanks.

- #2

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try proving it the same way you would for Bernoulli's equation …

consider a section of a tube bounded by streamlines

- #3

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By considering a tubular volume bounded by streamlines, I've considered the rate of change of mass inside this volume and equated it to the momentum flux through the surface.

This has led to the integral expression for the continuity equation- expanding out div.(pu) (where p=density and

INT(dp/dt)dV=INT(

I don't really see why

Thanks.

- #4

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(have a rho: ρ and a grad: ∇ )

i'm not sure what you've done

since no fluid can go through the sides of the tube, all you need do is consider the rate of gain or loss of fluid at the two ends of the tube

- #5

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tiny-tim said:

(have a rho: ρ and a grad: ∇ )

i'm not sure what you've done

since no fluid can go through the sides of the tube, all you need do is consider the rate of gain or loss of fluid at the two ends of the tube

Surely this is not a simple matter of saying:

(d/dt)INTρdV = INT(over end A) ρu(r).ds - INT(over end B) ρu(r+dr).ds

Where A and B are the ends of the tube.

I'm under the impression that it cannot be assumed that the densities at A, within the volume V and at B are the same.

This is probably a simple problem so sorry for not seeing it already.

- #6

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(d/dt) ∫ ρdV = ∫ (over end A) ρ

(and then replace each ∫ by the area )

- #7

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tiny-tim said:assumeρ depends on distance, and write it …

(d/dt) ∫ ρdV = ∫ (over end A) ρ_{A}u(r).ds - ∫ (over end B) ρ_{B}u(r+dr).ds …

(and then replace each ∫ by the area )

I'm still very confused.

Do we have to assume that the areas A and B are the same? Surely we can imagine a scenario where the streamlines diverge and so A and B are different.

Thanks very much for your help.

- #8

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you should get Aρ

(oh, I've just noticed, it needn't have been u(r) and u(r+dr) … the tube can be as long as we like )

- #9

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tiny-tim said:

you should get Aρ_{A}u_{A}= Bρ_{B}u_{B}

(oh, I've just noticed, it needn't have been u(r) and u(r+dr) … the tube can be as long as we like )

So you're assuming that the mass of fluid within the tube is constant? I don't see how it's ok to say this, I mean if the density can vary- the mass within the tube can also vary?

But if I can understand that then the expression you've given makes sense since the velocity times area along the tube (the flux) must be constant.

- #10

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jmz34 said:So you're assuming that the mass of fluid within the tube is constant?

no, I'm saying that mass out minus mass in = that integral, and if you put du/dx = 0, you should get that integral to be zero

(for example, using the divergence theorem)

- #11

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Ok thanks a lot for all your help :)

This refers to the property of an incompressible fluid where the density remains constant along a streamline, or a line that is tangent to the velocity vector at a given point. In other words, the density of the fluid does not change as it moves along a streamline.

The constant density along a streamline is a result of the incompressibility of the fluid. Incompressible fluids have a constant density, which means that the volume of the fluid does not change even when there is a change in pressure. This allows the density to remain constant along a streamline.

The density of an incompressible fluid can be calculated using the equation: density = mass / volume. Since the volume of an incompressible fluid remains constant, the density can be determined by measuring the mass of the fluid.

Some common examples of incompressible fluids include water, oil, and air at low pressures. These fluids have a constant density and are often used in hydraulic systems, as their volume remains the same even when under pressure.

The density of an incompressible fluid can affect its flow in several ways. A higher density fluid will typically have a slower flow rate compared to a lower density fluid. Additionally, changes in density can also affect the pressure and velocity of the fluid, which can impact the overall flow behavior.

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