quiz2 - V Equation of continuity cylindrical coordinates...

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Quiz #2 ECH 3203, Fall 2010 Friday, September 24 Directions: Thirty minutes. Closed book. No notes. No calculator neededed. Maximize your score by showing all work. Good luck to you!! Lubrication of a piston A solid, cylindrical piston with radius R p is centered in a tube of radius R t . Applying a force on the piston in the axial ( z ) direction causes the piston to move with constant velocity V . The gap between the piston and the tube is Flled with a Newtonian ±uid of known viscosity μ and constant density ρ . ²ind the force necessary to move the piston with constant velocity V . a) Clearly write the governing di³erential equation for this problem along with the boundary conditions. b) Calculate the velocity u z as a function of radial position for R p < r < R t . c) Calculate the shear stress τ rz as a function of r . d) Now use the shear stress calculation to Fnd the force (per unit length of piston) necessary to move the piston with the speciFed velocity
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Unformatted text preview: V . 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 p ∂u r ∂t + u r ∂u r ∂r + u θ r ∂u r ∂θ-u 2 θ r + u z ∂u r ∂z P = μ b ∂ ∂r p 1 r ∂ ∂r ( ru r ) P + 1 r 2 ∂ 2 u r ∂θ 2-2 r 2 ∂u θ ∂θ + ∂ 2 u r ∂z 2 B-∂P ∂r + ρg r θ-momentum: ρ p ∂u θ ∂t + u r ∂u θ ∂r + u θ r ∂u θ ∂θ + u z ∂u θ ∂z + u r u θ r P = μ b ∂ ∂r p 1 r ∂ ∂r ( ru θ ) P + 1 r 2 ∂ 2 u θ ∂θ 2 + 2 r 2 ∂u r ∂θ + ∂ 2 u θ ∂z 2 B-1 r ∂P ∂θ + ρg θ z-momentum: ρ p ∂u z ∂t + u r ∂u z ∂r + u θ r ∂u z ∂θ + u z ∂u z ∂z P = μ b 1 r ∂ ∂r p r ∂u z ∂r P + 1 r 2 ∂ 2 u z ∂θ 2 + ∂ 2 u z ∂z 2 B-∂P ∂z + ρg z...
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quiz2 - V Equation of continuity cylindrical coordinates...

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