Bernoulli's Theorem: Non-Viscous & < Critical Fluid Velocity

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    Bernoulli's Theorem
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Discussion Overview

The discussion revolves around the assumptions of Bernoulli's Theorem, specifically focusing on the conditions of non-viscous fluids and the concept of critical fluid velocity. Participants explore the implications of these assumptions in the context of fluid dynamics, including laminar versus turbulent flow and the characteristics of idealized fluids.

Discussion Character

  • Exploratory
  • Debate/contested
  • Technical explanation

Main Points Raised

  • Some participants question how the assumptions of non-viscous fluids and critical velocity can coexist, particularly noting that if viscosity is zero, the critical velocity would also be zero, implying no flow.
  • One participant suggests that critical velocity might refer to the speed of sound, while others clarify that it pertains to the transition from laminar to turbulent flow.
  • There is a discussion about the lack of a universal equation for determining laminar flow, with some asserting that there is no specific velocity where laminar flow definitively ends.
  • Another participant introduces the concept of a critical Reynolds number, suggesting that the assumptions imply the flow can be treated as inviscid while remaining non-turbulent.
  • Several participants highlight the assumptions of Bernoulli's Theorem, including inviscid flow, steady state, and incompressibility, noting that these are idealizations rather than strict realities.
  • Concerns are raised about the implications of inviscid flows, particularly regarding the interaction between streamlines and the nature of energy transfer in incompressible fluids.
  • One participant argues against the claim that pressure times volume does not represent potential energy in incompressible fluids, providing an example involving a nozzle to illustrate energy transformation in fluid flow.

Areas of Agreement / Disagreement

Participants express differing views on the interpretation of critical velocity and the implications of Bernoulli's assumptions. There is no consensus on the nature of critical velocity or the validity of certain claims regarding energy transfer in incompressible fluids.

Contextual Notes

Some statements made by participants rely on specific definitions of terms like "critical velocity" and "laminar flow," which may vary in different contexts. The discussion also highlights the limitations of applying idealized models to real fluid behavior.

kamaljeet_pec
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Two of the assumptions in Brenoulli's Theorem say

1. The fluid should be non-viscous
2. The velocity of fluid should be less than its critical velocity

Now how can these two co-exist. We know that

Vc = (N.n)/(D.p)

Vc= Critical Velocity
N= Raynolds No.
n= viscosity coef.
D= Dia of pipe
p= dencitu of fluid

if n=0 the Vc=0 i.e. fluid is static.

If there is no flow then what are we studying here.
Please explain anybody.
 
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Isn't what they mean by the "critical velocity" in this case actually the speed of sound?
 
No. Its the velocity till which the flow of a perticular fluid is laminar.
 
kamaljeet_pec said:
No. Its the velocity till which the flow of a perticular fluid is laminar.

There is no general equation for which a fluid is laminar in a given situation. There is no magic velocity where laminar flow stops.
 
kamaljeet_pec said:
Two of the assumptions in Brenoulli's Theorem say

1. The fluid should be non-viscous
2. The velocity of fluid should be less than its critical velocity

Now how can these two co-exist. We know that

Vc = (N.n)/(D.p)

Vc= Critical Velocity
N= Raynolds No.
n= viscosity coef.
D= Dia of pipe
p= dencitu of fluid

I take it your Reynolds No. is actually supposed to be a "critical Reynolds No.," which is around 2000 for pipe flow.

If so, the assumptions are really saying:

1. the fluid flow can be approximated as an inviscid flow (viscous dissipation is not big), yet

2. the flow is not turbulent

Here I assume the non-turbulent part is included because you are going to use (or derive) the form of Bernoulli for an irrotational flow (Also, dissipation is going to become important at some point if you have turbulence).

I know they're funny if you think about the Reynolds Number carefully, but well there you have it. Most physics are done in a "magic region" of parameter space.
 
Last edited:
The four assumptions for bernoulli's are as follows:
1) inviscid/frictionless - in other words no forces on the fluid due to viscosity
2) applied along a streamline
3) steady state
4) incompressible

The other assumption you refer to has do do with turbulence. When we have turbulence assumptions 2 and 3 break down.

Remember these are assumptions, no fluid is truly without friction, but that doesn't mean inviscid isn't a good assumption in certain flows. i.e. away from walls and/or boundary layers.
 
Inviscid flows are indeterminant. Since there's no viscosity, there's no interaction between stream lines, so all of the net mass flow of an inviscid fluid could be constrained to a very narrow stream line in an otherwise non-moving inviscid fluid, such as the high speed flow exiting a narrow diameter section of pipe into a larger diameter section of pipe.

Incompressable fluids also presents problems. Pressure times volume is no longer a form of potential energy, because no energy is transferred when pressure is changed (you have force but zero distance). The speed of sound in an incompressable fluid is infinite.

Idealized Bernoulli relies on an idealized fluid with magical properties where the fluid is inviscid and non-inviscid at the same time depending on when it's convenient to fit the model.
 
Thanks Guys.. :-)
 
rcgldr said:
Incompressable fluids also presents problems. Pressure times volume is no longer a form of potential energy, because no energy is transferred when pressure is changed (you have force but zero distance).

Not true. The internal energy has a PV term even for incompressible fluids.

Just consider water going through a nozzle. It enters the nozzle with high pressure and low velocity and exits with lower pressure and higher velocity. Where do you think the kinetic energy came from? A nozzle swaps pressure energy for kinetic energy. Indeed, that is precisely the meaning of Bernoulli's equation: P/ρ is the pressure energy per unit mass, and V2/2 is the kinetic energy per unit mass. The mechanical energy of a fluid is constant along a streamline in steady flow, provided the flow is inviscid.

BBB
 

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