Continuity principle in practice

[David43214]

TL;DR

Q=AV, what happens as V approaches infinity and A approaches 0?

If you imagine putting your thumb at the end of a garden hose and slowly restricting the area until the area is 0, according to the continuity principle, the flow rate stays constant because the velocity increases to make up for the smaller area.

However obviously this can't be completey accurate in real life.

Are there any specific values where this principle no longer applies in real life?

For example, if the area is 1m^2 and the velocity is 1m/s, Q=A×V=1m^3 per second.

If you then changed the area to 0.0000001m^2., theoretically the velocity would be 10,000,000 meters per second which I don't think would happen in real life.

 

[jedishrfu]

You’re right that the continuity equation Q = AV (which follows from the conservation of mass in fluid dynamics) is a useful approximation, but it has limitations in real life.

In real-world fluids, viscosity resists infinite acceleration. As the area A decreases, the velocity V increases; however, at a certain point, the flow becomes turbulent instead of laminar. Turbulence generates chaotic eddies and energy dissipation, which prevents the velocity from increasing indefinitely.

The continuity equation assumes incompressibility, which is a reasonable assumption for low-speed liquid flows. However, when velocity approaches the speed of sound in the fluid, compressibility effects become significant. In air, for example, once the flow reaches Mach 1, a choked flow condition occurs, meaning that no further increase in velocity is possible without a corresponding increase in upstream pressure.

 

[256bits]

TL;DR Summary: Q=AV, what happens as V approaches infinity and A approaches 0?

according to the continuity principle, the flow rate stays constant because the velocity increases to make up for the smaller area.

TL;DR Summary: Q=AV, what happens as V approaches infinity and A approaches 0?

For example, if the area is 1m^2 and the velocity is 1m/s, Q=A×V=1m^3 per second.

If you then changed the area to 0.0000001m^2., theoretically the velocity would be 10,000,000 meters per second

That is an incorrect statement(s).
The continuity principle applies to the flow of the fluid from source to sink for the particular system under analysis.
Change the system parameters, and a new Q=AV has to be calculated most likely different from the previous system.
An example of this is the flow of fluid of a tank from two orifices of different area.
The velocity from each orifice would be the same since the pressure head is the same for both. The areas are different, hence the flow Q from each is different.

 

[Rive]

Are there any specific values where this principle no longer applies in real life

Well, when your pump on the other end of that hose is out of beef an cannot provide the same amount of water any longer, then the example fails.

Apart from that triviality, if things pushed far enough then fluids are compressible, can cavitate, experience phase change, can change chemical composition... Just a few things off the bat. There may be further cases.

 

[russ_watters]

That is an incorrect statement(s).
The continuity principle applies to the flow of the fluid from source to sink for the particular system under analysis.
Change the system parameters, and a new Q=AV has to be calculated most likely different from the previous system.

I just want to reiterate this, as it is such a common error/misconception in fluids. We see it here a lot. Applied here, it means that continuity isn't as fragile as on might think. Start closing a valve on most systems and the cross sectional area at the valve opening goes down, velocity at the valve opening goes up, but flow rate also goes down. Violation of continuity? No, it's a new scenario in which continuity still holds.

Some of the more common limitations to continuity in real life would be compressible flow (if you apply a constant density assumption) and leaky systems.

 

[jrmichler]

If you imagine putting your thumb at the end of a garden hose and slowly restricting the area until the area is 0, according to the continuity principle, the flow rate stays constant because the velocity increases to make up for the smaller area.

Your use of the continuity principle assumes constant flow rate. If the flow stays constant, decreasing the the area will increase the velocity. The pressure upstream of the restriction will also increase. If the area gets very small, the pressure gets very high. This relationship is expressed in Bernoulli's Principle. Search that term, the Wikipedia link is a good one. In practice, the maximum pressure of a garden hose is set by the pump that put the water in the garden hose. That pressure is typically about 50 PSI, which sets the maximum possible velocity.

Bernoulli's Principle has its own limitations. It breaks down when the fluid is compressed, or the discharge velocity approaches the speed of sound in the fluid. Not to mention other effects, such as vena contracta.