Bernoulli — Pipe Flow
Push fluid through a narrowing pipe — it speeds up and its pressure drops.
About this simulation
Fluid flows through a pipe whose inlet and throat radii you set; continuity forces the speed up at the throat while Bernoulli drops the pressure there, shown by manometer heights and faster-moving streamlines. This is the Venturi effect behind aspirators, carburettors and lift.
How to use it
- Set Inlet radius and Throat radius and watch the particles: their streaks get longer where the pipe is narrow, because the water moves faster there.
- Read the three manometer tubes labelled inlet, throat and outlet; the throat column is lower and turns red when its pressure falls below the inlet pressure.
- Compare v₂, printed in the throat, with v₁ at the inlet, and check that the speeds are in the inverse ratio of the areas.
- Press 'Uniform pipe' to make both radii equal: the speeds and pressures match everywhere along the pipe.
- Raise Inlet speed, or press 'Aspirator', and watch the gap between the inlet and throat manometer readings grow (120 kPa against 46 kPa for Aspirator).
What to notice
- The same volume of water must pass every section each second, so where the pipe narrows the water speeds up.
- In a horizontal pipe, faster flow means lower pressure: the pressure energy per unit volume is turned into kinetic energy per unit volume.
- The pressure drop grows with the square of the speed, so doubling the inlet speed makes the drop four times larger.
- After the throat the pipe widens again, the water slows, and in this ideal model the pressure returns to its inlet value.
The formulas
- A₁v₁ = A₂v₂
- Equation of continuity: cross-sectional area A times flow speed v is the same at every section of the pipe.
- P₁ + ½ρv₁² = P₂ + ½ρv₂²
- Bernoulli's equation for a horizontal pipe: pressure P plus kinetic energy per unit volume is constant; ρ is the water's density.
Questions
What is Bernoulli's principle?
For an ideal fluid in steady, streamlined flow, the sum of pressure, kinetic energy per unit volume and potential energy per unit volume is constant along a streamline: P + ½ρv² + ρgh = constant. It is the conservation of energy applied to a moving fluid. In a horizontal pipe it means the fluid's pressure is lowest where it moves fastest.
Why does fluid speed up in a narrow pipe?
A liquid is nearly incompressible, so the volume flowing past any section each second must be the same everywhere. That volume rate is area × speed, so if the area falls the speed must rise. Halving the radius quarters the area and makes the fluid four times faster, which you can check with the Throat radius slider.
What is a venturimeter?
A venturimeter measures the flow rate in a pipe. It has a narrow throat between two wider sections, and a manometer measures the pressure difference between the wide part and the throat. Using continuity and Bernoulli's equation, that pressure difference gives the flow speed. The simulation's pipe, with its inlet and throat manometers, has the same layout.
What are the assumptions of Bernoulli's equation?
The fluid is incompressible and non-viscous, the flow is steady and streamlined rather than turbulent, and the equation is applied along a single streamline. Real fluids lose some pressure to viscosity, so a real pipe's outlet pressure is a little lower than its inlet even when the two sections are the same size.
How does an atomiser or aspirator work?
Air or water is forced at high speed through a narrow nozzle, so the pressure there drops below the surrounding atmospheric pressure. Liquid in a tube that opens into the fast stream is pushed up into it by the higher outside pressure and broken into a spray. The 'Aspirator' preset shows how far the throat pressure falls when the constriction is narrow.
