Bernoulli's Principle and the Venturi Tube

posted on 2 Oct 2013 by guy
last changed 6 Jul 2016

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The Bernoulli principle states that a region of fast flowing fluid exerts lower pressure on its surroundings than a region of slow flowing fluid. The principle applies to the motion of air over an airplane wing, to air flow through a carburetor, to a flag flapping in the breeze, and to the low pressure systems in hurricanes. This lesson describes Bernoulli's principle from several different perspectives appropriate for a range of age groups and lists a number of activities and demonstrations that use Bernoulli's principle.

For more ideas on Bernoulli's principle, check out our summary curriculum on Bernoulli Effects.

subjects: Physics
keywords: Bernoulli, fluid flow, venturi

Fig. 1: Venturi tube showing that the narrow portion of the tube at the left has lower pressure than the wider portion of the tube at the right. The water in the U-bend at the bottom of the picture is pushed up towards the left by the pressure difference.

the Bernoulli equation

The Bernoulli principle states that a region of fast flowing fluid exerts lower pressure on its surroundings than a region of slow flowing fluid. It is named after Daniel Bernoulli, a Dutch-Swiss scientist who published the principle in his book Hydrodynamica in 1738. Bernoulli derived his principle from the conservation of energy, though it can also be derived directly from Newton's second law.1 To see Bernoulli's approach, write the energy density for a flowing fluid as:

$${energy\over volume} = P + {1\over2}\rho v^2 + \rho g h$$

where $P$ is the pressure at the location of interest, $\rho$ is the mass density, $v$ is the flow velocity, $g$ is the gravitational constant, and $h$ is the altitude. The second term ${1\over2}\rho v^2$ represents the kinetic energy of the fluid due to its average flow, and the third term $\rho g h$ represents it potential energy in the earth's gravity field. The first term $P$ represents the energy associated with the pressure of the fluid, and has the dimensions of force per area, or equivalently, energy per volume.

According to the conservation of energy, the energy density is constant, so that for two different regions in the flow:

$$P_1 + {1\over2}\rho v_1^2 + \rho g h_1 = P_2 + {1\over2}\rho v_2^2 + \rho g h_2$$

For two regions at the same height ($h_1 = h_2$), an increase in flow velocity in one region must necessarily correspond to a decrease in pressure in order to keep the equation balanced. Kinetic energy is increased at the expense of pressure energy, while the total energy remains constant.

Note for geeks: Bernoulli's derivation above makes two assumptions:

  1. Energy losses due to friction from viscous forces are negligible.
  2. The fluid is not compressed, i.e. $\rho$ does not change.  This condition is true for most liquids, and for gases under some circumstances (generally at low velocities). However, equations similar to Bernoulli's can be derived even for compressible fluids. See for examples.

Fig. 2: The flow of material through a Venturi tube. High pressure regions are dark blue; low pressure regions are white. As the fluid goes through the constriction, it speeds up, and the pressure drops.

the Venturi effect

The Venturi effect, published in 1797 by Giovanni Venturi, applies Bernoulli's principle to a fluid that flows through a tube with a constriction in it, such as in figure 2. The Venturi tube provides a handy method for mixing fluids or gases, and is popular in carburetors and atomizers, which use the low pressure region generated at the constriction to pull the liquid into the gas flow. It also offers a particularly clear example of the Bernoulli principle.

As the fluid flows through the constriction, the fluid molecules speed up, as indicated by the animation in figure 2. The molecules must speed up in the constricted region in order for the total flow rate to remain the same. However many molecules enter the tube in a given time must be the same as the number of molecules going through the constriction and coming out the other end. What goes in must come out. Since the cross section is smaller in the constriction, the molecules must move faster in order for enough molecules to get through in the specified time.

Since the molecules are flowing faster in the constriction, Bernoulli's principle indicates that the pressure in the constriction should be lower than it is outside. Indeed, in order for the molecules to speed up as they enter the constriction, and then slow down again as they leave, there must be a pressure difference at the entrance and exit of the constriction. High pressure before the constriction accelerates molecules into the low pressure region of the constriction, and high pressure after the constriction slows them down again as they exit.

the microscopic view

At the microscopic level, how do we understand that molecules speed up as they enter the constriction of a Venturi tube and produce lower pressure on the walls of the tube?

The short answer is that molecules in the constriction are more likely to be traveling in the direction of flow, rather than bouncing up and down, and therefore they hit the walls of the tube less frequently, and with less force.

To understand how this happens, we must first recognize that molecules are generally moving much faster than the typical flow speed of a fluid. Even in "still" air, air molecules are constantly bouncing around and colliding with each other at high speed. A stiff breeze may blow at a few meters per second, but the average speed of an air molecule at room temperature is around 500 m/s (=1100 mi/hr), and is known as the "thermal velocity". The picture we have of a gentle summer breeze is then a collection of molecules moving in all directions and colliding at high speed, but as a whole gently drifting in one direction on average.

As fluid molecules approach the constriction in a Venturi tube, the molecules that are most likely to enter the constriction are those whose thermal motion happens to be in the direction of flow, with only a small component perpendicular to the flow (towards the walls of the tube). Those molecules travel through the constriction faster by using some of the speed from their thermal velocity. They are also less likely to collide with the walls, and if they do collide, they will do so at a glancing angle, thereby exerting less pressure on the walls. Other molecules that are not headed along the flow initially are likely to encounter collisions with other molecules (or the walls of the tube outside the constriction) until their velocities are redirected in the right direction. In this way, the constriction acts as a velocity filter, selecting molecules whose thermal velocities are more likely along the direction of flow. That filter both increases the average flow speed of the molecules, and decreases the likelihood for collision with the walls.

As the molecules leave the constriction, they are once again free to scatter in all directions. Since the typical distance they travel between collisions is less than 100 nanometers2, collisions quickly randomize their directions again, and the average flow along the tube slows down.

Video by Manish Kumar showing the effect of blowing air over the top of a piece of paper.

Fig. 3: Air flow past an airplane wing, showing that air flow above the wing is faster than air flow below the wing.

applications and activities

The classic example of Bernoulli's principle involves airflow over airplane wings. Air flows faster over the top of an airfoil than under the bottom, thereby making Bernoulli's principle relevant (Figure 3). According to Bernoulli, the faster flow over the top corresponds to a lower pressure and provides lift to the wing. But see the discussion below under "controversies".

Bernoulli lift can be demonstrated in the classroom with just a piece of paper (see video above). Hold a piece of paper by the edge and blow over the top to see the paper rise into the air stream. This demo works best if your mouth is very close to the top of the paper, and the leading edge of the paper is held steady so as to prevent the air stream from traveling downward.

A flag flapping in the wind works much like an airplane foil. When the flag is curved one way, low pressure is created on the outside of the curve, causing the tail end of the flag to swing in that direction. Once the tail moves into the air stream, the bow of the flag reverses and low pressure develops on the other side of the flag, causing it to swing back. Over the length of a long flag, there can be many curves back and forth.

Hurricanes and tornadoes are vortexes of swirling winds, something like the swirling waters in a bathtub drain (except that air travels upwards in the hurricane and water travels downwards in the bathtub). In these storms, air near the center moves much faster than air far from the center due to the conservation of angular momentum. This fast moving air near the center exerts very little pressure on surrounding layers, following Bernoulli's principle. This feature is why hurricanes can sometimes lift the roofs off of houses, or blow the windows outward; the pressure outside the house in the high wind is lower than the pressure inside the house in relatively still air. Together with the force of thermal updrafts, the low pressure caused by Bernoulli effects helps a hurricane suck up water over the ocean. When the hurricane makes landfall, the water carried along with the hurricane can lead to storm surges.

One of the most popular demonstrations involving the Bernoulli principle uses a strong stream of air from a leaf blower or shop vacuum to levitate a beach ball in the air. A ball can be made to hang in the air far from the ground, gently rocking back and forth. It's a magical effect. Check out our lesson on Bernoulli's Beach Ball for details.

some controversies involving Bernoulli's principle and airplane wings

It's worth noting a couple of common arguments surrounding Bernoulli's principle, just so you can seem extra smart when you say you've already heard that old line.

  1. Some folks are fond of claiming that the lift in an airplane wing is not due to Bernoulli's principle at all, but rather due to a transfer of momentum following Newton's 3rd law when air molecules strike the underside of the wing. Most modern writings agree that both principles are relevant in describing the lift of an airplane wing.3,4,5
  2. A more legitimate argument involves a related principle called the equal transit time, which claims that the air flowing over an airplane wing must travel over the top of the wing in the same amount of time as the air under a wing. Although air does flow faster over the top of the wing, thereby making Bernoulli's principle relevant, the equal transit time theory does not give the correct velocity.6

further resources

A number of demonstrations of Bernoulli's principle are popular in the classroom. The School of Physics and Astronomy at the University of Minnesota has compiled a nice list here.

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Thank you, excellent lesson.

Thank you, excellent lesson. You mention deriving the Bernoulli effect using conservation of energy. The Venturi effect also falls easily out of conservation of energy, and might make an easy (easier) example. If U = KE + PE = total energy in and out - then since (barring friction and assuming incompressible gas flow) no work is being done on the gas ==> U in the restriction is also the same. The Venturi equation almost begs this interpretation by making the differences in pressure (PE) equal to difference in KE through velocity. I think adding this might help join the physics of Bernoulli and Venturi. The lesson is good as it stands - so its only a 'suggested' change.

In 20-20 hindsight - in your

In 20-20 hindsight - in your Bernoulli equation with h1 = h2, solve to P1-P2 = (rho/2) (V(2)^2 - V(1)^2) which is the Venturi equation obtained through conservation of Energy