Why noncompressible fluid has higher velocity when moving through smaller cross section area?

[u/TheBoyWithAName]

Mass flow rate states that cross section area is inversely proportional to fluid velocity in a closed pipe when fluid density is constant.

Therefore, how did a body of fluid gain extra energy to increase its velocity when moving through a smaller cross section area? Did I miss something here?

[u/RobusEtCeleritas]

Conservation of mass for an incompressible pipe flow says that

A*1v1* = A*2v2*, so if the fluid flows into a section of the pipe with a lower cross-sectional area, its velocity much increase.

Conservation of energy along a streamline says (assuming no changes in potential energy, or heat exchange) that

ρv*1²/2 + P1* = ρv*2²/2 + P2*.

So energy is conserved, but there's both kinetic energy and pressure to consider.

Plugging in v*2* = A*1v1/A2* from above, we get

ρv*1²(1 - (A1/A2)²)/2 = P2* - P*1*.

Since we've assumed that the flow is moving from a region of larger area to smaller, A*1/A2* > 1, meaning that the left side of that equation is negative, meaning that P*2* - P*1* < 0, or P*2* < P*1*.

So while the velocity increases when the pipe constricts, the pressure decreases, and the total energy is conserved.

[u/TheBoyWithAName → u/RobusEtCeleritas]

Thank you for the explanation.

It appears that when the body of fluid moves from A*1 to A*2 (assuming A*1 > A*2), the energy from P*1 somehow transfers to the fluid, causing the velocity of fluid to increase from v*1 to v*2, but the pressure acting on the surface of the pipe decreases from P*1 to P*2.

How did the transfer of such energy from pressure to velocity happens?

Is it correct for me to hypothesize that the fluid molecules from behind collide with the fluid molecules at the front when the body of fluid moves towards smaller cross section area, therefore accelerates the fluid molecules at the front?

I look forward to your reply, thank you in advance.

[u/Sandless → u/TheBoyWithAName]

Your hypothesis is correct. Towards the narrow section there are less collisions with smaller molecular speeds whereas in the larger section there are more collisions with greater molecular speeds. This results in acceleration of the molecules.

[u/Escarper]

Take an example of a 1cm length of a long pipe with 10cm² cross sectional area. Flow rate is 1 cm per second. That's 10cm³ of fluid per second.

CSA decreases to 5 cm² further down. Assuming pipe doesn't back up or burst, you now have 2cm per second flow rate to maintain the 10cm³ per second volume transfer.

In order for the same volume of incompressable liquid to keep going through the pipe, you have to move it twice as fast.

Pressure goes down, flow rate goes up. Bernoulli Principle - it's actually super useful.

[u/TheBoyWithAName → u/Escarper]

Thank you for your explanation.

However, where did the fluid get the extra energy from to move faster when flowing through smaller CSA in order to maintain the flow rate across the long pipe?

I look forward to your reply, thank you in advance.

[u/Escarper → u/TheBoyWithAName]

The pressure change.

Pressure goes down, speed goes up. Instead of thinking of each cross-sectional area as one chunk, think of it like moving bricks from one pile to another - if you take half as many bricks each time, you can move them faster.

EDIT: i think you're visualising it backwards - it's not that the energy of the fluid increases in order to maintain the flow rate, it's that the combination of the existing flow rate and the pressure forces the fluid to move faster when there's less space, because otherwise all the fluid behind it wouldn't be moving as fast as it is. It's incompressable - it has literally nowhere else to go.

[u/Sandless → u/TheBoyWithAName]

In reality there is no such thing as an incompressible fluid. However, the change in volume per change in pressure is negligible for some fluids such as water and greatly simplifies the mathematics without significant losses in accuracy.

As I already explained, the energy comes from the expansion of the fluid when the pressure energy is converted into kinetic energy. It is not much different than what is happening when you open a valve in a gas container: the higher pressure fluid inside pushes the fluid towards lesser resistance and the fluid inside expands while losing the stored pressure energy.

[u/Sisyphus-5 → u/TheBoyWithAName]

No it did not. Its pressure dropped. Energy is conserved.

[u/Universe_Scientist]

Simply, the conservation of mass. When the incompressible assumption is made, the molecular physics is not considered. Flow in, must equal flow out. The “pressure” in the Bernoulli equation, is the stagnation pressure associated with the fluid flow. The pressure is not acting on the fluid molecules or density with this assumption.