Bernoulli Equation Fluid Dynamics

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

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

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

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?

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