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f13_fall - Fluids Lecture 13 Notes 1. Bernoulli Equation 2....

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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 Newtons Second Law applied to uid ow, is written as follows. u u p + u = + g x t x x We now make the following assumptions about the ow. Steady ow: /t = 0 Negligible gravity: g x 0 Negligible viscous forces: ( F x ) viscous 0 Low-speed ow: is constant + ( F x ) viscous These reduce the momentum equation to the following simpler form, which can be immedi- ately integrated. du dp u + = 0 dx dx 1 d ( u 2 ) dp + = 0 2 dx dx 1 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 + u + v = + g x + ( F x ) viscous t x y x v v v p + u + v = + g 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 2....

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