PGE 323 Test _1.key - Key PGE 323 Test#1 1 30 pts(a 20 pts...

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Key PGE 323 Test #1 February 14, 2006 1. 30 pts (a) 20 pts Starting with: ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 r z t t t r z r z r r z z r r z r r z r z u r r u r z u t r z u r r u r z u t θ +∆ θ θ +∆ +∆ θ+∆θ ∆θ∆ ∆ φρ - ∆θ∆ ∆ φρ = ∆θ∆ ρ + ∆θ∆ ρ + ∆ ∆ ρ - ∆θ∆ ρ + ∆θ∆ ρ + ∆ ∆ ρ Divide through by r r z ∆θ∆ ∆ and t ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 r r z z t t t r r r z z z r r z r z t u u r r t u u r r z t r r z t r r z t r z t u u r r z t +∆ +∆ +∆ θ θ θ θ+∆θ ∆θ∆ ∆ φρ - φρ ∆θ∆ ∆ ρ - ρ ∆θ∆ ∆ ρ - ρ = + ∆θ∆ ∆ ∆ ∆θ∆ ∆ ∆ ∆θ∆ ∆ ∆ ∆ ∆ ∆ ρ - ρ + ∆θ∆ ∆ ∆ Simplify and pull out a (-1) from each term on the right hand side ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 r r z z t t t r r r z z z r u r u u u 1 t r r z u u 1 r +∆ +∆ +∆ θ θ θ+∆θ θ � � φρ - φρ ρ - ρ ρ - ρ � � = - - ρ - ρ - ∆θ Take the limit of each term as the denominator 0 . ( 29 { } ( 29 ( 29 ( 29 r z r u u u 1 1 t r r z r θ ° ρ ° ρ ¨ φρ ρ = - + + X % - θ (b) 10 pts Darcy's Law for Cylindrical Coordinates (neglecting gravity): r r k P u r .
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