Fluid mechanics and circular flow

In summary, the conversation discusses a problem with the circular flow of fluid where the use of a circular infinitesimal volume results in a different answer due to the greater area on the side with higher pressure. The speaker asks for clarification and the other person explains the concept using cylindrical coordinates and the difference in forces between the inner and outer areas of the infinitesimal volume. It is also mentioned that in uniform circular motion, there must be an unbalanced force pointing towards the center for the motion to be possible.
  • #1
tomz
35
0
Dear all

I am having a problem on circular flow of fluid. On all books I have read they say
[itex]\frac{dp}{dr}[/itex]=[itex]\rho[/itex]v[itex]^{2}[/itex][itex]/r[/itex]

Which make sense by using infinitesimal square volume and take the force exert.

But if I use a circular infinitesimal volume (which is usually the case for circular things), i get a different answer because the side with greater pressure have greater area.

Why is this ?


Thank you
 
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  • #2
Please describe the situation you are dealing with.
What do you mean by "circular flow of fluid"?
To me that means the pipe goes in a circle, or that there is a vortex or whirlpool.
But I don't see how you can have an infinitesimal volume with different areas on different sides.
 
  • #3
Simon Bridge said:
Please describe the situation you are dealing with.
What do you mean by "circular flow of fluid"?
To me that means the pipe goes in a circle, or that there is a vortex or whirlpool.
But I don't see how you can have an infinitesimal volume with different areas on different sides.

Thanks for the reply. Sorry I did not make it clear.

I mean for a vortex, or any flow that is not along a straight line.

Because from what I deal with viscid fluid/statics/pressure vessel.. when there is rotation or cylindrical coordinate, we tend to use a curved shape (sector) infinitesimal area with area rdrdθ.
But for this shape, the force from outside is (r+dr)dθ(p+dp/dr*dr), force from inside is rdθ*p. Giving a difference of (dp/dr*r+p)*dθ*dr rather than just dp/dr*r*dθ*dr as for a square infinitesimal area.

Please tell me if I still make it unclear. Thank you
 
  • #4
In cylindrical coordinates, the force on the inner area of the infinitesimal volume ##dV## is ##F_1=p(\rho)\rho d\theta dz## while the outer area is ##F_2=(\rho+d\rho)p(\rho+d\rho) d\theta dz## and your assertion is that this means that ##F_1<F_2##. Where ##p(\rho)## is the pressure at radius ##\rho##.

Let's say there is no circular motion - the water is still.
Cylindrical coordinates have been chosen because of the cylindrical container or something.
Then ##F_2>F_1## would make the water tend to hump up in the middle wouldn't it?

Is that the jist of things?
 
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  • #5
Simon Bridge said:
In cylindrical coordinates, the force on the inner area of the infinitesimal volume ##dV## is ##F_1=p(\rho)\rho d\theta dz## while the outer area is ##F_2=(\rho+d\rho)p(\rho+d\rho) d\theta dz## and your assertion is that this means that ##F_1<F_2##. Where ##p(\rho)## is the pressure at radius ##\rho##.

Let's say there is no circular motion - the water is still.
Cylindrical coordinates have been chosen because of the cylindrical container or something.
Then ##F_2>F_1## would make the water tend to hump up in the middle wouldn't it?

Is that the jist of things?

Am I making the mistake of ignoring the tangential component of forces...

Thank you so much!
 
  • #6
In my example, there are no tangential forces.

You also get a similar issue vertically - the pressure of the water is higher as you go deeper, so the force up from the bottom of an infinitesimal volume is greater than the force down from the top...

These are things you've dealt with before.

If the water is in uniform circular motion, then there must be an unbalanced force on each volume element that points to the center or the motion is not possible right?
 

1. What is fluid mechanics?

Fluid mechanics is the branch of physics that studies the behavior of fluids (liquids and gases) at rest and in motion. It involves the study of how fluids flow and interact with surrounding objects and forces.

2. What is the difference between laminar and turbulent flow?

Laminar flow is characterized by smooth, orderly movement of fluids in parallel layers, while turbulent flow is chaotic and unpredictable, with fluid particles moving in all directions and mixing with each other.

3. How does Bernoulli's principle relate to fluid mechanics?

Bernoulli's principle states that as the speed of a fluid increases, its pressure decreases. This principle is often used to explain how airplanes are able to fly, as the faster-moving air above the wing creates a lower pressure, resulting in lift.

4. What is circular flow in fluid mechanics?

Circular flow refers to the continuous circulation of fluids in a closed loop, where the fluid is constantly being pushed by a source (such as a pump) and pulled by a sink (such as a drain). This concept is commonly used in hydraulic systems.

5. How is the continuity equation used in fluid mechanics?

The continuity equation is a fundamental equation in fluid mechanics that relates the flow rate of a fluid at one point to the flow rate at another point in a closed system. It is based on the principle of conservation of mass and is often used to analyze the flow of fluids in pipes and channels.

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