f13_fall

# f13_fall - Fluids – Lecture 13 Notes 1 Bernoulli Equation...

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Unformatted text preview: Fluids – Lecture 13 Notes 1. Bernoulli Equation 2. Uses of Bernoulli Equation Reading: Anderson 3.2, 3.3 Bernoulli Equation Derivation – 1-D case The 1-D momentum equation, which is Newton’s Second Law applied to ﬂuid ﬂow, is written as follows. ∂u ∂u ∂p ρ + ρu = − + ρg x ∂t ∂x ∂x We now make the following assumptions about the ﬂow. • Steady ﬂow: ∂/∂t = 0 • Negligible gravity: ρg x ≃ 0 • Negligible viscous forces: ( F x ) viscous ≃ 0 • Low-speed ﬂow: ρ is constant + ( F x ) viscous These reduce the momentum equation to the following simpler form, which can be immedi- ately integrated. du dp ρu + = 0 dx dx 1 d ( u 2 ) dp ρ + = 0 2 dx dx 1 ρ u 2 + p = constant ≡ p o 2 p The final result is the one-dimensional Bernoulli Equation , which uniquely relates velocity and pressure if the simplifying assumptions listed above are valid. The constant of integration o is called the stagnation pressure , or equivalently the total pressure , and is typically set by known upstream conditions. Derivation – 2-D case The 2-D momentum equations are ∂u ∂u ∂u ∂p ρ + ρu + ρv = − + ρg x + ( F x ) viscous ∂t ∂x ∂y ∂x ∂v ∂v ∂v ∂p ρ + ρu + ρv = − + ρg y + ( F y ) viscous ∂t ∂x ∂y ∂y Making the same assumptions as before, these simplify to the following....
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f13_fall - Fluids – Lecture 13 Notes 1 Bernoulli Equation...

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