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12.2 Bernoulli’s equation  (Page 4/7)

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Wings and sails

The airplane wing is a beautiful example of Bernoulli's principle in action. [link] (a) shows the characteristic shape of a wing. The wing is tilted upward at a small angle and the upper surface is longer, causing air to flow faster over it. The pressure on top of the wing is therefore reduced, creating a net upward force or lift. (Wings can also gain lift by pushing air downward, utilizing the conservation of momentum principle. The deflected air molecules result in an upward force on the wing — Newton's third law.) Sails also have the characteristic shape of a wing. (See [link] (b).) The pressure on the front side of the sail, P front size 12{P rSub { size 8{"front"} } } {} , is lower than the pressure on the back of the sail, P back size 12{P rSub { size 8{"back"} } } {} . This results in a forward force and even allows you to sail into the wind.

Making connections: take-home investigation with two strips of paper

For a good illustration of Bernoulli's principle, make two strips of paper, each about 15 cm long and 4 cm wide. Hold the small end of one strip up to your lips and let it drape over your finger. Blow across the paper. What happens? Now hold two strips of paper up to your lips, separated by your fingers. Blow between the strips. What happens?

Velocity measurement

[link] shows two devices that measure fluid velocity based on Bernoulli's principle. The manometer in [link] (a) is connected to two tubes that are small enough not to appreciably disturb the flow. The tube facing the oncoming fluid creates a dead spot having zero velocity ( v 1 = 0 size 12{v rSub { size 8{1} } =0} {} ) in front of it, while fluid passing the other tube has velocity v 2 size 12{v rSub { size 8{2} } } {} . This means that Bernoulli's principle as stated in P 1 + 1 2 ρv 1 2 = P 2 + 1 2 ρv 2 2 size 12{P rSub { size 8{1} } + { {1} over {2} } ρv rSub { size 8{1} } "" lSup { size 8{2} } =P rSub { size 8{2} } + { {1} over {2} } ρv rSub { size 8{2} } "" lSup { size 8{2} } } {} becomes

P 1 = P 2 + 1 2 ρv 2 2 . size 12{P rSub { size 8{1} } =P rSub { size 8{2} } + { {1} over {2} } ρv rSub { size 8{2} } "" lSup { size 8{2} } "." } {}
Part a of the figure shows a picture of a wing. It is in the form of an aerofoil. One side of the wing is broader and the other end tapers. The direction of air is shown as lines along the length of the wing. The direction of air below the wing is shown as flowing along the length initially and at the tapered end of the wing it rises up. The pressure exerted by the air is given by P b is upward. The direction of air on the top or front part of the wing is shown as flowing along the length of the wing. The pressure exerted by the air is given by P f and it acts downward. Part b of the figure shows a boat with a sail. The direction of sail is almost across the boat. The direction of air in the sail is shown by lines on front and back sides of the sail. The air currents on the front exert a pressure P front toward the sail and air currents on the back sides of sail exerts a pressure P back again toward the sail.
(a) The Bernoulli principle helps explain lift generated by a wing. (b) Sails use the same technique to generate part of their thrust.

Thus pressure P 2 size 12{P rSub { size 8{2} } } {} over the second opening is reduced by 1 2 ρv 2 2 size 12{ { { size 8{1} } over { size 8{2} } } ρv rSub { size 8{2} } rSup { size 8{2} } } {} , and so the fluid in the manometer rises by h on the side connected to the second opening, where

h ∝ 1 2 ρv 2 2 . size 12{h prop { {1} over {2} } ρv rSub { size 8{2} } rSup { size 8{2} } "."} {}

(Recall that the symbol ∝ size 12{ prop } {} means “proportional to.”) Solving for v 2 size 12{v rSub { size 8{2} } } {} , we see that

v 2 ∝ h . size 12{v rSub { size 8{2} } prop sqrt {h} "."} {}

[link] (b) shows a version of this device that is in common use for measuring various fluid velocities; such devices are frequently used as air speed indicators in aircraft.

Part a shows a U-shaped manometer tube connected to ends of two tubes which are placed close together. Tube one is open on the end and shows a velocity v one equals zero at the end. Tube two has an opening on the side and shows a velocity v two across the opening. The level of fluid in the U-shaped tube is more on the right side than on the left. The difference in height is shown by h. Part b of the figure shows a velocity measuring device a pitot tube. Two coaxial tubes, one broader outside and other narrow inside are connected to a U-shaped tube. The U-shaped tube is also narrow at one end and broader at the other. The narrow end of the U-shaped tube is connected to the narrow inner tube and the broader end of the U-shaped tube is connected to the broader outer tube. The tube one has an opening at one of its edges and the velocity of the fluid at the end is v one equals zero. Tube two has an opening on the side and shows a velocity v two across the opening. The level of fluid in the U-shaped tube is more on the right side than on the left. The difference in height is shown by h.
Measurement of fluid speed based on Bernoulli's principle. (a) A manometer is connected to two tubes that are close together and small enough not to disturb the flow. Tube 1 is open at the end facing the flow. A dead spot having zero speed is created there. Tube 2 has an opening on the side, and so the fluid has a speed v across the opening; thus, pressure there drops. The difference in pressure at the manometer is 1 2 ρv 2 2 size 12{ { { size 8{1} } over { size 8{2} } } ρv rSub { size 8{2} } rSup { size 8{2} } } {} , and so h is proportional to 1 2 ρv 2 2 size 12{ { { size 8{1} } over { size 8{2} } } ρv rSub { size 8{2} } rSup { size 8{2} } } {} . (b) This type of velocity measuring device is a Prandtl tube, also known as a pitot tube.

Summary

  • Bernoulli's equation states that the sum on each side of the following equation is constant, or the same at any two points in an incompressible frictionless fluid:
    P 1 + 1 2 ρv 1 2 + ρ gh 1 = P 2 + 1 2 ρv 2 2 + ρ gh 2 . size 12{P rSub { size 8{1} } + { { size 8{1} } over { size 8{2} } } ρv rSub { size 8{1} } rSup { size 8{2} } +ρ ital "gh" rSub { size 8{1} } =P rSub { size 8{2} } + { { size 8{1} } over { size 8{2} } } ρv rSub { size 8{2} } rSup { size 8{2} } +ρ ital "gh" rSub { size 8{2} } } {}
  • Bernoulli's principle is Bernoulli's equation applied to situations in which depth is constant. The terms involving depth (or height h ) subtract out, yielding
    P 1 + 1 2 ρv 1 2 = P 2 + 1 2 ρv 2 2 . size 12{P rSub { size 8{1} } + { { size 8{1} } over { size 8{2} } } ρv rSub { size 8{1} } rSup { size 8{2} } =P rSub { size 8{2} } + { { size 8{1} } over { size 8{2} } } ρv rSub { size 8{2} } rSup { size 8{2} } } {}
  • Bernoulli's principle has many applications, including entrainment, wings and sails, and velocity measurement.

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Source:  OpenStax, College physics for ap® courses. OpenStax CNX. Nov 04, 2016 Download for free at https://legacy.cnx.org/content/col11844/1.14
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