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# Midterm - z-direction through the annulus Calculate the...

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Midterm Progress Assessment ECH 3203, Fall 2010 Wednesday, October 20 Directions: Two hours. Open book. No notes. No calculator needed. Maximize your score by showing all work. Good luck to you!! Problem #1: A Flow Meter Calculation (30 points) An inviscid fuid oF constant density ρ fows steadily through the contraction shown in the ±gure. Derive an expression For the fow rate Q in terms oF ρ , D 1 , D 2 , and h . Problem #2: (30 points) The velocity potential φ = - k ( x 2 - y 2 ) , where k is a constant, may be used to represent fow against an in±nite plane boundary as shown. ²or fow in the vicinity oF a stagnation point, it is Frequently assumed that the pressure gradient alonog the surFace is oF the Form ∂P ∂x = Ax, where A is a constant. Use the given velocity potential to show this is true.

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Problem #3: Zero fow oF a fuid in an annulus (40 points) A solid, cylindrical piston with radius R p is centered in a tube of radius R t . The gap between the piston and the tube is Flled with a Newtonian ±uid of known viscosity μ and constant density ρ . A pressure drop, of P H - P L over the length L of the piston, drives a ±ow in the negative
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Unformatted text preview: z-direction 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 Navier-Stokes equations, cylindrical coordinates r-momentum: ρ ± ∂u r ∂t + u r ∂u r ∂r + u θ r ∂u r ∂θ-u 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 2-2 r 2 ∂u θ ∂θ + ∂ 2 u r ∂z 2 ´-∂P ∂r + ρg r θ-momentum: 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 z-momentum: 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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Midterm - z-direction through the annulus Calculate the...

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