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Bernoulli Equation Fluid Dynamics

Updated on April 4, 2014

How to Understand and Derive the Bernoulli Equation

Fluid dynamics is a very complex subject, but can be simplified by making some assumptions. The Bernoulli Equation describes the behaviour of incompressible, non-viscous fluids and is a useful way of approximately modeling the behaviour of fluids.

Here is a simple explanation of how the Bernoulli equation is derived, an introduction to fluid dynamics and some applications of it's use: How does an aerofoil work? (i.e. how does a aeroplane fly?) and why does a baseball, cricket ball or tennis ball etc. swing: follow a different trajectory to a non spinning ball (The Magnus Effect)?

Fluid Dynamics

Streamline Flow

Fluid Dynamics: Assumptions

To analyse fluid dynamics there are several assumptions that can be made to greatly simplify the maths:

The fluid is incompressible (i.e. constant density)

The fluid is non-viscous

Angular momentum is ignored (i.e. the fluid is "irrotational")

The flow is laminar or streamline and steady (smooth flow without turbulence)

These are all fairly unlikely in real life, but this is a good approximation

The Equation of Continuity

Conservation of Mass

Bernoulli Equation
Bernoulli Equation

Applying the law of Conservation of Mass (i.e. what enters a pipe at one end will come out of the other end - assuming no leaks) to the example shown above:

In time Δt a mass m1 enters the tube of area A1

m1 = ρ1 A1 v1 Δt

where ρ1 = density

a mass m2 leaves A2 at the same time

m2 = ρ2 A1 v2 Δt

m1 = m2 (conservation of mass)

=> ρ1 A1 v1 = ρ2 A2 v2

ρ1 = ρ2 (incompressible fluids)

A1 v1 = A2 v2

so as the area gets larger the speed of flow gets smaller and vice versa

The Bernoulli Equation

Bernoulli Equation - Change of speed of a fluid

Bernoulli Equation
Bernoulli Equation

Applying the law of conservation of energy to the example above (pipe narrowing from Area A1 at pressure P1 to A2 at P2):

Work done = Work done at A + Work done at B

i.e. force x distance in time Δt

W = (P1 A1)(v1 Δt) - (P2 A2)(v2 Δt)

A1 v1 = A2 v2 =>

W = (P1 - P2)(A1v Δt)

Gain in kinetic energy = m(v2^2 - v1^2) / 2 = (A1v1 Δt ρ) (v2^2 - v1^2) = W

(P1 -P2)(A1v1 Δt) = 1/2 (A1 v1 Δt ρ)(v2^2 - v1^2)

(P1 - P2) = 1/2ρ(v2^2 - v1^2)

P1 + 1/2 ρ v1^2 = P2 + 1/2 ρ v2^2

Bernoulli Equation - Change of height of fluid

Bernoulli Equation
Bernoulli Equation

Work Done = gain of potential energy

(P1 - P2)(Av Δt) = (Av Δt ρ) g (h2 - h1)

(P1 - P2) = ρ g (h2 -h1)

P1 + h1 ρ g = P2 + h2 ρ g

Changing speed and height gives us the Bernoulli Equation:

P + 1/2 ρ v^2 + h ρ g = constant

1/2 ρ v^2 = dynamic pressure

P + h ρ g = static pressure

How Does and Aircraft Wing Work? Why Does a Spinning Ball Swing? - Applications of Bernoulli's Equation

How Does and Aircraft Wing Work?
How Does and Aircraft Wing Work?

How Does and Aircraft Wing Work?

An aerofoil creates lift as it moves through the air and this lift can be explained by the Bernoulli Equation:

The shape is such that the air travels further over the top of the wing and therefore faster

v1 > v2

P + 1/2 ρ v^2 + h ρ g = constant

so P1 < P2

Why Does a Spinning Ball Swing?

A ball with "top-spin" (e.g. a tennis ball, baseball or cricket ball) will not follow the expected parabolic trajectory, because the pressure on either side of the ball is different in the same way as the aerofoil example above. This is because the ball is spinning and not because of it's shape.

On one side of the ball travelling at a velocity v the surface is travelling at v+v' and the other side v-v' where v' is speed of the surface of the ball. i.e. the air moves over the surface at different speeds and from the Bernoulli equation there will be a difference in pressure. This is called The Magnus Effect

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    • profile image

      anonymous 4 years ago

      Well, this is a rocket that you shot over my head for sure....FB liked to share with those who want a clear explanation of the Bernoulli Equation....I think A-Redneck may have an idea there! :)

    • Redneck Lady Luck profile image

      Lorelei Cohen 4 years ago from Canada

      You lost me at equation but that's because I am just beginning my first coffee. Wow is all that I have to say on the Bernoulli equation of fluid dynamics. Have you thought of applying for a spot on the Big Bang Theory?

    • Andy-Po profile image
      Author

      Andy 5 years ago from London, England

      @gypsyman27 lm: Thanks very much. Yes, perhaps this is not a typical Squidoo lens, but it is a subject that I find fascinating.

    • gypsyman27 lm profile image

      gypsyman27 lm 5 years ago

      I'm surprised to see a lens involving the bernoulli equation and fluid dynamics. You are a surprising man. I am an engineer, so I am thoroughly familiar with these equations and their use. Really good job of explaining for the layman. Stay well and happy. See you around the galaxy...

    • Philippians468 profile image

      Philippians468 6 years ago

      i learnt a thing or two from your lens! great job! cheers

    • MargoPArrowsmith profile image

      MargoPArrowsmith 6 years ago

      You explained it well. Whether I got it is something else.

    • TonyPayne profile image

      Tony Payne 7 years ago from Southampton, UK

      Too complicated for me, but a great lens Andy.