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Unformatted text preview: At every point P(r, θ ,z) at every instant t, the velocity components u r , u θ and u z must satisfy this eq. The two u r can be combined into one term to give an equivalent alternate form, 1 r ∂ ∂r ( ru r ) + 1 r ∂u θ ∂θ + ∂u z ∂z = 0 . Question 3: Problem 4.20 Given 2D velocity field ~ U = u ˆ i + v ˆ j where u = K (1 e αy ) , and v = V at y = 0 (all x) Conservation of mass requires ∂u ∂x + ∂v ∂y = 0 since ∂u ∂x = 0, then we must have ∂v ∂y = 0. Thus, v = F ( x ) at most. At y = 0, we know that v = V for all x, therefore F ( x ) = V and v = V here. The solution is u = K (1 e αy ) v = V K must have units of velocity (m/s). ( αy ) must be dimensionless, thus α must have units of ( 1 m ). Question 4: Problem 4.25 Conservation of mass in cylindrical coordinates is 1 r ∂ ∂r ( rv r ) + 1 r ∂v θ ∂θ = 0 Given v r = Kcosθ 1 b r 2 ¶ then rV r = Krcosθ Kbcosθ r ∂ ∂r ( rv r ) = Kcosθ + Kbcosθ r 2 Given v θ = Ksinθ 1 + b r 2 ¶ =...
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This note was uploaded on 09/14/2011 for the course ME 362 taught by Professor Ceciledevaud during the Winter '11 term at Waterloo.
 Winter '11
 CecileDevaud

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