Fluid Mechanics, Depth, Pressure

In summary, the conversation discusses the derivation for fluid pressure at certain depths, where the weight of the fluid is not neglected. The speaker is reconstructing the derivation and points out where they were confused. They question the additional dp in the downward forces and why pressure varies in altitude. Through the conversation, it is clarified that pressure varies due to the weight of the fluid and that the slab of height dy is in equilibrium. The expert also explains that the net force on parallel to gravity can be seen as Adp=-W.
  • #1
mathsciguy
134
1
I'm having a little problem with my book as I was reading about fluid mechanics. The book seems to have skipped a bit of some crucial part (at least for me) during the derivation for fluid pressure at certain depths (where the weight of the fluid is not neglected).

Here, I'll try to reconstruct the derivation and try to point out the parts where it confused me.

First suppose we have a fluid with definite volume where its density is the same throughout, hence it's uniform. Now, if we take an element fluid with thickness dy then and its top and bottom surfaces are the same, say A. Its volume is dV=A*dy, it's mass dm=(rho)*dV=(rho)*A*dy, and its weight w is dmg=(rho)*g*A*dy.

Where rho is the density of the fluid.

When the book gave an analysis of the forces on the y-component of that certain element fluid, the upward force is given by F(upward) = pA. I understand that part since there is pressure (p) pressing the fluid at its bottom area. Now, when the book gave the downward forces, it's given by F(downward) = (p+dp)*A and this confused me, where did the additional dp come from? There is the p that presses the upper area of the fluid but what about dp? Also the other downward force is the weight, but the fluid is in equilibrium, so:

(Sum)Fy = pA-(p+dp)*A-W=0 *The fact that there is force p*A upward and downward plus the weight is also non intuitive for me.*

So yeah, the biggest question for me is dp in the derivation, what is that?
 
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  • #2
hi mathsciguy! :smile:

(have a rho: ρ :wink:)
mathsciguy said:
… When the book gave an analysis of the forces on the y-component of that certain element fluid, the upward force is given by F(upward) = pA. I understand that part since there is pressure (p) pressing the fluid at its bottom area. Now, when the book gave the downward forces, it's given by F(downward) = (p+dp)*A and this confused me, where did the additional dp come from? There is the p that presses the upper area of the fluid but what about dp? Also the other downward force is the weight, but the fluid is in equilibrium, so:

(Sum)Fy = pA-(p+dp)*A-W=0 *The fact that there is force p*A upward and downward plus the weight is also non intuitive for me.*

So yeah, the biggest question for me is dp in the derivation, what is that?

it's more intuitive if we write p(y) instead of just p, where y is height

then the force upward at height y is Ap(y),

and the force downward at height y+dy is Ap(y+dy), = A{p(y) + dp} (where dp = (dp/dy) dy)

so the total force downward (excluding gravity) is A{p(y) + dp} - Ap(y) = Adp​

(and dp will be negative because y is height, not depth :wink:)
 
  • #3
Thanks, I thought no one would reply since my post was kinda messy.

So the pressure really is different at differing altitudes, but I thought that was caused by considering the force done by the gravity I.e. the weight. Since there is weight then it gives a downward pressure at the top surface (of course I know that if I think of it like that then the element fluid will not be in equilibrium, but I think that's another question).

I've thought of that, since from what I know, if we consider the case where there is negligible force done by the gravity then the pressure is the same throughout the volume of the fluid, it's the same even if we consider the pressure done by fluid onto a part of itself.

Edit: I think, my confusion is caused by not being sure of why pressure vary in altitude. Would it vary in altitude if we neglect the weight of the fluid?
 
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  • #4
mathsciguy said:
So the pressure really is different at differing altitudes, but I thought that was caused by considering the force done by the gravity I.e. the weight. Since there is weight then it gives a downward pressure at the top surface

yes, that's correct …

pressure = force per area, and the only force downward is the weight of the fluid above :smile:
(of course I know that if I think of it like that then the element fluid will not be in equilibrium, but I think that's another question).

not following you … the slab of height dy is in equilibrium :confused:
I've thought of that, since from what I know, if we consider the case where there is negligible force done by the gravity then the pressure is the same throughout the volume of the fluid, it's the same even if we consider the pressure done by fluid onto a part of itself.

Edit: I think, my confusion is caused by not being sure of why pressure vary in altitude. Would it vary in altitude if we neglect the weight of the fluid?

no, eg air is a fluid, and its density is negligible (for most bodies, not for balloons etc), which is why we usually regard atmospheric pressure as the same throughout the lab, regardless of height :wink:
 
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  • #5
tiny-tim said:
yes, that's correct …

pressure = force per area, and the only force downward is the weight of the fluid above :smile:not following you …*the slab of height dy is in equilibrium :confused:

Sorry, what actually goes through my mind is this:
Not considering dy, Fnet = Ap(y)-w-Ap(y) = ma *Well, because I thought at both top and bottom A there is that same pressure p, we just got an additional pressure done by w.

I kind of get it now though, if we solve for the net force on parallel to gravity we could see that Adp=-W, and this makes sense to me now.

Many thanks sir tiny-tim.
 
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  • #6
tiny-tim said:
yes, that's correct …
not following you …*the slab of height dy is in equilibrium :confused:

Sorry, what actually goes through my mind is this:
Not considering dy, Fnet = Ap(y)-w-Ap(y) = ma *Well, because I thought at both top and bottom A there is that same pressure p, we just got an additional pressure done by w.

I kind of get it now though, if we solve for the net force on the y-axis we could see that Adp=-W, and this makes sense to me now. It's kind of not very intuitive to see that Adp is an upward force. (I might be wrong all together on this, I hope not)

Many thanks sir tiny-tim.
 
  • #7
mathsciguy said:
It's kind of not very intuitive to see that Adp is an upward force.

it is! … there's more pressure underneath than on top, and that's the bouyant force that helps that slab of fluid to float! :biggrin:
 
  • #8
tiny-tim said:
it is! … there's more pressure underneath than on top, and that's the bouyant force that helps that slab of fluid to float! :biggrin:

Thanks, I haven't read about buoyant force in depth yet, it's a good thing I've got some idea.
 
  • #9
mathsciguy said:
Thanks, I haven't read about buoyant force in depth …

well, that's where to find it! :biggrin:
 

What is fluid mechanics?

Fluid mechanics is the study of how fluids, such as liquids and gases, behave when subjected to different forces and environments. It is a branch of physics and is used to understand and predict the behavior of fluids in various scenarios.

How is depth related to fluid mechanics?

Depth is an important factor in fluid mechanics as it affects the pressure and flow of fluids. As the depth increases, so does the pressure exerted by the fluid, which can impact its behavior and movement in a system.

What is pressure in fluid mechanics?

Pressure in fluid mechanics refers to the force per unit area that a fluid exerts on its surroundings. It is affected by factors such as depth, density, and velocity of the fluid, and is an important concept in understanding how fluids move and interact with their environment.

How does Bernoulli's principle relate to fluid mechanics?

Bernoulli's principle is a fundamental principle in fluid mechanics that states that as the speed of a fluid increases, its pressure decreases. This principle is important in understanding the behavior of fluids in motion and is used in various applications, such as airplane wings and water pumps.

What are some real-world applications of fluid mechanics?

Fluid mechanics has many practical applications in our daily lives, including in industries such as aviation, hydraulics, and marine engineering. It is also used in designing and optimizing various systems, such as pipelines, pumps, and turbines. Additionally, understanding fluid mechanics is crucial in fields like meteorology, oceanography, and environmental science.

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