See also[edit]

• Terminology in fluid dynamics

• Navier–Stokes equations – for the flow of a viscous fluid

• Euler equations – for the flow of an inviscid fluid

• Hydraulics – applied fluid mechanics for liquids

• Venturi effect

• Inviscid flow

References[edit]

1. Jump up^ Clancy, L.J., Aerodynamics, Chapter 3.

2. ^ Jump up to:^a b Batchelor, G.K. (1967), Section 3.5, pp. 156–64.

3. Jump up^ "Hydrodynamica". Britannica Online Encyclopedia. Retrieved 2008-10-30.

4. Jump up^ Streeter, V.L., Fluid Mechanics, Example 3.5, McGraw–Hill Inc. (1966), New York.

5. Jump up^ "If the particle is in a region of varying pressure (a non-vanishing pressure gradient in the x-direction) and if the particle has a finite size l, then the front of the particle will be ‘seeing’ a different pressure from the rear. More precisely, if the pressure drops in the x-direction (dp/dx < 0) the pressure at the rear is higher than at the front and the particle experiences a (positive) net force. According to Newton’s second law, this force causes an acceleration and the particle’s velocity increases as it moves along the streamline... Bernoulli’s equation describes this mathematically (see the complete derivation in the appendix)."Babinsky, Holger (November 2003), "How do wings work?", Physics Education

6. Jump up^ "Acceleration of air is caused by pressure gradients. Air is accelerated in direction of the velocity if the pressure goes down. Thus the decrease of pressure is the cause of a higher velocity." Weltner, Klaus; Ingelman-Sundberg, Martin, Misinterpretations of Bernoulli's Law

7. Jump up^ " The idea is that as the parcel moves along, following a streamline, as it moves into an area of higher pressure there will be higher pressure ahead (higher than the pressure behind) and this will exert a force on the parcel, slowing it down. Conversely if the parcel is moving into a region of lower pressure, there will be an higher pressure behind it (higher than the pressure ahead), speeding it up. As always, any unbalanced force will cause a change in momentum (and velocity), as required by Newton’s laws of motion." See How It Flies John S. Denkerhttp://www.av8n.com/how/htm/airfoils.html

8. ^ Jump up to:^a b Batchelor, G.K. (1967), §5.1, p. 265.

9. Jump up^ Mulley, Raymond (2004). Flow of Industrial Fluids: Theory and Equations. CRC Press. ISBN 0-8493-2767-9., 410 pages. See pp. 43–44.

10. Jump up^ Chanson, Hubert (2004). Hydraulics of Open Channel Flow: An Introduction. Butterworth-Heinemann. ISBN 0-7506-5978-5., 650 pages. See p. 22.

11. Jump up^ Oertel, Herbert; Prandtl, Ludwig; Böhle, M.; Mayes, Katherine (2004). Prandtl's Essentials of Fluid Mechanics. Springer. pp. 70–71. ISBN 0-387-40437-6.

12. Jump up^ "Bernoulli's Equation". NASA Glenn Research Center. Retrieved 2009-03-04.

13. ^ Jump up to:^a b Clancy, L.J., Aerodynamics, Section 3.5.

14. Jump up^ Clancy, L.J. Aerodynamics, Equation 3.12

15. ^ Jump up to:^a b Batchelor, G.K. (1967), p. 383

16. Jump up^ White, Frank M. Fluid Mechanics, 6e. McGraw-Hill International Edition. p. 602.

17. Jump up^ Clarke C. and Carswell B., Astrophysical Fluid Dynamics

18. Jump up^ Clancy, L.J., Aerodynamics, Section 3.11

19. Jump up^ Landau & Lifshitz (1987, §5)

20. Jump up^ Van Wylen, G.J., and Sonntag, R.E., (1965),Fundamentals of Classical Thermodynamics, Section 5.9, John Wiley and Sons Inc., New York

21. ^ Jump up to:^a b c Feynman, R.P.; Leighton, R.B.; Sands, M. (1963).The Feynman Lectures on Physics. ISBN 0-201-02116-1., Vol. 2, §40–3, pp. 40–6 – 40–9.

22. Jump up^ Tipler, Paul (1991). Physics for Scientists and Engineers: Mechanics (3rd extended ed.). W. H. Freeman. ISBN 0-87901-432-6., p. 138.

23. Jump up^ Feynman, R.P.; Leighton, R.B.; Sands, M. (1963). The Feynman Lectures on Physics. ISBN 0-201-02116-1., Vol. 1, §14–3, p. 14–4.

24. Jump up^ Physics Today, May 1010, "The Nearly Perfect Fermi Gas", by John E. Thomas, p 34.

25. Jump up^ Resnick, R. and Halliday, D. (1960), Physics, Section 18–5, John Wiley & Sons, Inc., New York ("[streamlines] are closer together above the wing than they are below so that Bernoulli's principle predicts the observed upward dynamic lift.")

26. Jump up^ Eastlake, Charles N. (March 2002). "An Aerodynamicist’s View of Lift, Bernoulli, and Newton".The Physics Teacher 40. "The resultant force is determined by integrating the surface-pressure distribution over the surface area of the airfoil."

27. Jump up^ Clancy, L.J., Aerodynamics, Section 3.8

28. Jump up^ Mechanical Engineering Reference Manual Ninth Edition

29. Jump up^ Castro-Orgaz, O. & Chanson, H. (2009). "Bernoulli Theorem, Minimum Specific Energy and Water Wave Celerity in Open Channel Flow". Journal of Irrigation and Drainage Engineering, ASCE, 135 (6): 773–778.doi:10.1061/(ASCE)IR.1943-4774.0000084.

30. Jump up^ Chanson, H. (2009). "Transcritical Flow due to Channel Contraction". Journal of Hydraulic Engineering, ASCE135 (12): 1113–1114.

31. Jump up^ Chanson, H. (2006). "Minimum Specific Energy and Critical Flow Conditions in Open Channels". Journal of Irrigation and Drainage Engineering, ASCE 132 (5): 498–502. doi:10.1061/(ASCE)0733-9437(2006)132:5(498).

32. Jump up^ Glenn Research Center (2006-03-15). "Incorrect Lift Theory". NASA. Retrieved 2010-08-12.

33. Jump up^ Chanson, H. (2009). Applied Hydrodynamics: An Introduction to Ideal and Real Fluid Flows. CRC Press, Taylor & Francis Group, Leiden, The Netherlands, 478 pages. ISBN 978-0-415-49271-3.

34. Jump up^ "Newton vs Bernoulli".

35. Jump up^ Ison, David. Bernoulli Or Newton: Who's Right About Lift? Retrieved on 2009-11-26

36. Jump up^ Phillips, O.M. (1977). The dynamics of the upper ocean(2nd ed.). Cambridge University Press. ISBN 0-521-29801-6. Section 2.4.

37. Jump up^ Batchelor, G.K. (1967). Sections 3.5 and 5.1

38. Jump up^ Lamb, H. (1994) §17–§29

39. Jump up^ Weltner, Klaus; Ingelman-Sundberg, Martin. "Physics of Flight – reviewed". "The conventional explanation of aerodynamical lift based on Bernoulli’s law and velocity differences mixes up cause and effect. The faster flow at the upper side of the wing is the consequence of low pressure and not its cause."

40. Jump up^ "Bernoulli's law and experiments attributed to it are fascinating. Unfortunately some of these experiments are explained erroneously..." Misinterpretations of Bernoulli's Law Weltner, Klaus and Ingelman-Sundberg, Martin Department of Physics, University Frankfurt http://www-stud.rbi.informatik.uni-frankfurt.de/~plass/MIS/mis6.html

41. Jump up^ "This occurs because of Bernoulli’s principle — fast-moving air has lower pressure than non-moving air." Make Magazine http://makeprojects.com/Project/Origami-Flying-Disk/327/1

42. Jump up^ " Faster-moving fluid, lower pressure. ... When the demonstrator holds the paper in front of his mouth and blows across the top, he is creating an area of faster-moving air." University of Minnesota School of Physics and Astronomyhttp://www.physics.umn.edu/outreach/pforce/circus/Bernoulli.html

43. Jump up^ "Bernoulli's Principle states that faster moving air has lower pressure... You can demonstrate Bernoulli's Principle by blowing over a piece of paper held horizontally across your lips."http://www.tallshipschannelislands.com/PDFs/Educational_Packet.pdf

44. Jump up^ "If the lift in figure A were caused by "Bernoulli principle," then the paper in figure B should droop further when air is blown beneath it. However, as shown, it raises when the upward pressure gradient in downward-curving flow adds to atmospheric pressure at the paper lower surface." Gale M. Craig PHYSICAL PRINCIPLES OF WINGED FLIGHThttp://www.regenpress.com/aerodynamics.pdf

45. Jump up^ "In fact, the pressure in the air blown out of the lungs is equal to that of the surrounding air..." Babinskyhttp://iopscience.iop.org/0031-9120/38/6/001/pdf/pe3_6_001.pdf

46. Jump up^ "...air does not have a reduced lateral pressure (or static pressure...) simply because it is caused to move, the static pressure of free air does not decrease as the speed of the air increases, it misunderstanding Bernoulli's principle to suggest that this is what it tells us, and the behavior of the curved paper is explained by other reasoning than Bernoulli's principle." Peter Eastwell Bernoulli? Perhaps, but What About Viscosity? The Science Education Review, 6(1) 2007http://www.scienceeducationreview.com/open_access/eastwell-bernoulli.pdf

47. Jump up^ "Make a strip of writing paper about 5 cm X 25 cm. Hold it in front of your lips so that it hangs out and down making a convex upward surface. When you blow across the top of the paper, it rises. Many books attribute this to the lowering of the air pressure on top solely to the Bernoulli effect. Now use your fingers to form the paper into a curve that it is slightly concave upward along its whole length and again blow along the top of this strip. The paper now bends downward...an often-cited experiment, which is usually taken as demonstrating the common explanation of lift, does not do so..." Jef Raskin Coanda Effect: Understanding Why Wings Workhttp://karmak.org/archive/2003/02/coanda_effect.html

48. Jump up^ "Blowing over a piece of paper does not demonstrate Bernoulli’s equation. While it is true that a curved paper lifts when flow is applied on one side, this is not because air is moving at different speeds on the two sides... It is false to make a connection between the flow on the two sides of the paper using Bernoulli’s equation." Holger Babinsky How Do Wings Work Physics Education 38(6)http://iopscience.iop.org/0031-9120/38/6/001/pdf/pe3_6_001.pdf

49. Jump up^ "An explanation based on Bernoulli’s principle is not applicable to this situation, because this principle has nothing to say about the interaction of air masses having different speeds... Also, while Bernoulli’s principle allows us to compare fluid speeds and pressures along a single streamline and... along two different streamlines that originate under identical fluid conditions, using Bernoulli’s principle to compare the air above and below the curved paper in Figure 1 is nonsensical; in this case, there aren’t any streamlines at all below the paper!" Peter EastwellBernoulli? Perhaps, but What About Viscosity? The Science Education Review 6(1) 2007http://www.scienceeducationreview.com/open_access/eastwell-bernoulli.pdf

50. Jump up^ "The well-known demonstration of the phenomenon of lift by means of lifting a page cantilevered in one’s hand by blowing horizontally along it is probably more a demonstration of the forces inherent in the Coanda effect than a demonstration of Bernoulli’s law; for, here, an air jet issues from the mouth and attaches to a curved (and, in this case pliable) surface. The upper edge is a complicated vortex-laden mixing layer and the distant flow is quiescent, so that Bernoulli’s law is hardly applicable." David Auerbach Why Aircreft Fly European Journal of Physics Vol 21 p 295 http://iopscience.iop.org/0143-0807/21/4/302/pdf/0143-0807_21_4_302.pdf

51. Jump up^ "Millions of children in science classes are being asked to blow over curved pieces of paper and observe that the paper "lifts"... They are then asked to believe that Bernoulli's theorem is responsible... Unfortunately, the "dynamic lift" involved...is not properly explained by Bernoulli's theorem." Norman F. Smith "Bernoulli and Newton in Fluid Mechanics" The Physics Teacher Nov 1972

52. Jump up^ "Bernoulli’s principle is very easy to understand provided the principle is correctly stated. However, we must be careful, because seemingly-small changes in the wording can lead to completely wrong conclusions." See How It Flies John S. Denkerhttp://www.av8n.com/how/htm/airfoils.html#sec-bernoulli

53. Jump up^ "A complete statement of Bernoulli's Theorem is as follows: "In a flow where no energy is being added or taken away, the sum of its various energies is a constant: consequently where the velocity increasees the pressure decreases and vice versa."" Norman F Smith Bernoulli, Newton and Dynamic Lift Part I School Science and Mathematics Vol 73 Issue 3http://onlinelibrary.wiley.com/doi/10.1111/j.1949-8594.1973.tb08998.x/pdf

54. Jump up^ "...if a streamline is curved, there must be a pressure gradient across the streamline, with the pressure increasing in the direction away from the centre of curvature." Babinsky http://iopscience.iop.org/0031-9120/38/6/001/pdf/pe3_6_001.pdf

55. Jump up^ "The curved paper turns the stream of air downward, and this action produces the lift reaction that lifts the paper." Norman F. Smith Bernoulli, Newton, and Dynamic Lift Part II School Science and Mathematics vol 73 Issue 4 pg 333http://onlinelibrary.wiley.com/doi/10.1111/j.1949-8594.1973.tb09040.x/pdf

56. Jump up^ "The curved surface of the tongue creates unequal air pressure and a lifting action. ... Lift is caused by air moving over a curved surface." AERONAUTICS An Educator’s Guide with Activities in Science, Mathematics, and Technology Education by NASA pg 26http://www.nasa.gov/pdf/58152main_Aeronautics.Educator.pdf

57. Jump up^ "Viscosity causes the breath to follow the curved surface, Newton's first law says there a force on the air and Newton’s third law says there is an equal and opposite force on the paper. Momentum transfer lifts the strip. The reduction in pressure acting on the top surface of the piece of paper causes the paper to rise." The Newtonian Description of Lift of a Wing-Revised David F. Anderson & Scott Eberhardt http://home.comcast.net/~clipper-108/Lift_AAPT.pdf

58. Jump up^ '"Demonstrations" of Bernoulli's principle are often given as demonstrations of the physics of lift. They are truly demonstrations of lift, but certainly not of Bernoulli's principle.' David F Anderson & Scott EberhardtUnderstanding Flight pg 229http://books.google.com/books?id=52Hfn7uEGSoC&pg=PA229

59. Jump up^ "As an example, take the misleading experiment most often used to "demonstrate" Bernoulli's principle. Hold a piece of paper so that it curves over your finger, then blow across the top. The paper will rise. However most people do not realize that the paper would not rise if it were flat, even though you are blowing air across the top of it at a furious rate. Bernoulli's principle does not apply directly in this case. This is because the air on the two sides of the paper did not start out from the same source. The air on the bottom is ambient air from the room, but the air on the top came from your mouth where you actually increased its speed without decreasing its pressure by forcing it out of your mouth. As a result the air on both sides of the flat paper actually has the same pressure, even though the air on the top is moving faster. The reason that a curved piece of paper does rise is that the air from your mouth speeds up even more as it follows the curve of the paper, which in turn lowers the pressure according to Bernoulli." From The Aeronautics File By Max Feilhttp://webcache.googleusercontent.com/search?q=cache:nutfrrTXLkMJ:www.mat.uc.pt/~pedro/ncientificos/artigos/aeronauticsfile1.ps+&cd=29&hl=en&ct=clnk&gl=us

60. Jump up^ "Some people blow over a sheet of paper to demonstrate that the accelerated air over the sheet results in a lower pressure. They are wrong with their explanation. The sheet of paper goes up because it deflects the air, by the Coanda effect, and that deflection is the cause of the force lifting the sheet. To prove they are wrong I use the following experiment: If the sheet of paper is pre bend the other way by first rolling it, and if you blow over it than, it goes down. This is because the air is deflected the other way. Airspeed is still higher above the sheet, so that is not causing the lower pressure." Pim Geurts. sailtheory.comhttp://www.sailtheory.com/experiments.html

61. Jump up^ "Finally, let’s go back to the initial example of a ball levitating in a jet of air. The naive explanation for the stability of the ball in the air stream, 'because pressure in the jet is lower than pressure in the surrounding atmosphere,' is clearly incorrect. The static pressure in the free air jet is the same as the pressure in the surrounding atmosphere..." Martin Kamela Thinking About BernoulliThe Physics Teacher Vol. 45, September 2007http://tpt.aapt.org/resource/1/phteah/v45/i6/p379_s1

62. Jump up^ "Aysmmetrical flow (not Bernoulli's theorem) also explains lift on the ping-pong ball or beach ball that floats so mysteriously in the tilted vacuum cleaner exhaust..." Norman F. Smith, Bernoulli and Newton in Fluid Mechanics" The Physics Teacher Nov 1972 p 455

63. Jump up^ "Bernoulli’s theorem is often obscured by demonstrations involving non-Bernoulli forces. For example, a ball may be supported on an upward jet of air or water, because any fluid (the air and water) has viscosity, which retards the slippage of one part of the fluid moving past another part of the fluid." The Bernoulli Conundrum Robert P. Bauman Professor of Physics Emeritus University of Alabama at Birminghamhttp://www.introphysics.info/Papers/BernoulliConundrumWS.pdf

64. Jump up^ "In a demonstration sometimes wrongly described as showing lift due to pressure reduction in moving air or pressure reduction due to flow path restriction, a ball or balloon is suspended by a jet of air." Gale M. Craig PHYSICAL PRINCIPLES OF WINGED FLIGHThttp://www.regenpress.com/aerodynamics.pdf

65. Jump up^ "A second example is the confinement of a ping-pong ball in the vertical exhaust from a hair dryer. We are told that this is a demonstration of Bernoulli's principle. But, we now know that the exhaust does not have a lower value of ps. Again, it is momentum transfer that keeps the ball in the airflow. When the ball gets near the edge of the exhaust there is an asymmetric flow around the ball, which pushes it away from the edge of the flow. The same is true when one blows between two ping-pong balls hanging on strings." Anderson & Eberhardt The Newtonian Description of Lift on a Wing http://lss.fnal.gov/archive/2001/pub/Pub-01-036-E.pdf

66. Jump up^ "This demonstration is often incorrectly explained using the Bernoulli principle. According to the INCORRECT explanation, the air flow is faster in the region between the sheets, thus creating a lower pressure compared with the quiet air on the outside of the sheets. UNIVERSITY OF MARYLAND PHYSICS LECTURE-DEMONSTRATION FACILITYhttp://www.physics.umd.edu/lecdem/services/demos/demosf5/f5-03.htm

67. Jump up^ "Although the Bernoulli effect is often used to explain this demonstration, and one manufacturer sells the material for this demonstration as "Bernoulli bags," it cannot be explained by the Bernoulli effect, but rather by the process of entrainment." UNIVERSITY OF MARYLAND PHYSICS LECTURE-DEMONSTRATION FACILITYhttp://www.physics.umd.edu/lecdem/outreach/QOTW/arch13/a256.htm