# Fluid Dynamics AP Physics by fqy94797

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```									Fluid Dynamics

AP Physics
Streamlines, Laminar flow and Turbulence
Continuity Equation

• Volume flow rate: volume of a fluid passing a
ΔVolume     Δl
given point per second or             =A    = Av
Δt      Δt
• What Goes In Must Come Out!
• Therefore: A1v1 = A2v2 or Av = constant
Applies to incompressible fluids – the only kind we will study.
Bernoulli’s Principle

Where the velocity of a fluid is high, the pressure
is low; Where the velocity of a fluid is low, the
pressure is high.

Assumptions:
• Fluid is Incompressible i.e. constant density
• No friction
• Laminar flow (no turbulence)
Derivation of Bernoulli’s
Equation
Δl2
1
E total =     mv2 + mgh
2
Δl1
æF ö
W = Fd = ç ÷(Ad ) = P V
è ÷
çA ø
÷

Consider the change in total energy of the fluid as it moves from the inlet to the
outlet.
WNC = Wdone on fluid - Wdone by fluid = Δ Etotal
Δ Etotal = ΔKE +ΔPE = (1/2mv22 + mgh2) – (1/2mv12 + mgh1)
Wdone on fluid - Wdone by fluid = (1/2mv22 + mgh1) – (1/2mv12 + mgh2)
P2V2 - P1V1 = (1/2mv22 + mgh1) – (1/2mv12 + mgh2)           Note V1=V2 and now divide both sides by V
P2 – P1 = (1/2ρ v12 + ρ gh1) – (1/2ρ v12 + ρ gh1)

∴        P1 + 1/2ρ v12 + ρ gh1 = P2 + 1/2ρ v22 + ρ gh2
Understanding Bernoulli’s Equation

• Is it the faster velocity that causes lower pressure? Or
does lower pressure cause faster velocity?
• According to Newton’s Second Law, acceleration is caused
by a net force.
• So when the fluid accelerates in the direction of the fluid’s
velocity, there must be a net force in that direction – in
this case, a difference in pressure x area.
• Therefore, lower pressure generates faster velocity, not
the other way around.
• If we multiply Bernoulli’s Equation times V (undo the last
step in the derivation) each term has units of energy! – as
it should – Bernoulli’s Equation is just an expression of
Conservation of Energy.

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