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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