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16100lectre27_cg

# 16100lectre27_cg - Poiseuille Flow Through a Duct in 2-D y...

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Poiseuille Flow Through a Duct in 2-D Assumptions: Velocity is independent of 0 , = = x v x u x Incompressible flow Constant viscosity, µ Steady Pressure gradient along length of pipe is non-zero, i.e. 0 x p Boundary conditions: No slip: () 0 walls are not moving 0 uy h vy h = = To be clear, we now will take the compressible, unsteady form of the N-S equations and carefully derive the solution: Conservation of mass: 0 ) ( = + V t K ρ But 0 = t because flow is steady and incompressible. Also, since = constant, then V V K K = ) ( 0 = + = y v x u V K Finally, 0 = x u because of assumption #1 long pipe. 0 = y v h y + = h y = y x

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Poiseuille Flow Through a Duct in 2-D 16.100 2002 2 Now, integrate this: C v = = constant Apply boundary conditions: ()0 ( )0 vh v y ± =⇒ = We expect this but it is good to see the math confirm it. Now, let’s look at momentum y .
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16100lectre27_cg - Poiseuille Flow Through a Duct in 2-D y...

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