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Unformatted text preview: zdirection through the annulus. Calculate the force F Z applied to the piston necessary to generate no net ±ow (volumetric ±owrate of zero). Equation oF continuity, cylindrical coordinates ∂ρ ∂t + 1 r ∂ ∂r ( ρru r ) + 1 r ∂ ∂θ ( ρu θ ) + ∂ ∂z ( ρu z ) = 0 NavierStokes equations, cylindrical coordinates rmomentum: ρ ± ∂u r ∂t + u r ∂u r ∂r + u θ r ∂u r ∂θu 2 θ r + u z ∂u r ∂z ² = μ ³ ∂ ∂r ± 1 r ∂ ∂r ( ru r ) ² + 1 r 2 ∂ 2 u r ∂θ 22 r 2 ∂u θ ∂θ + ∂ 2 u r ∂z 2 ´∂P ∂r + ρg r θmomentum: ρ ± ∂u θ ∂t + u r ∂u θ ∂r + u θ r ∂u θ ∂θ + u z ∂u θ ∂z + u r u θ r ² = μ ³ ∂ ∂r ± 1 r ∂ ∂r ( ru θ ) ² + 1 r 2 ∂ 2 u θ ∂θ 2 + 2 r 2 ∂u r ∂θ + ∂ 2 u θ ∂z 2 ´1 r ∂P ∂θ + ρg θ zmomentum: ρ ± ∂u z ∂t + u r ∂u z ∂r + u θ r ∂u z ∂θ + u z ∂u z ∂z ² = μ ³ 1 r ∂ ∂r ± r ∂u z ∂r ² + 1 r 2 ∂ 2 u z ∂θ 2 + ∂ 2 u z ∂z 2 ´∂P ∂z + ρg z...
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 Fall '10
 BUTLER
 Fluid Dynamics, uz, uθ ur uθ

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