NS - Cylindrical ρ ∂v r ∂t + v r ∂v r ∂r + v θ r...

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ENU 4134 N-S Equations, Fall 2009 – D. Schubring 1 ENU 4134 – Navier-Stokes Equations – Fall 2009 Navier-Stokes Equations, in component form for Cartesian and cylindrical geometries, assuming a constant viscosity (Newtonian fluid), incompressible behavior with a constant density, and no flow area changes. All of these assumptions will be valid for any work in this course which requires solution of the N-S equations. ~ f is a body force per unit volume. It is equal to ρ~g when the only body force is gravity. Cartesian ρ ± ∂v x ∂t + v x ∂v x ∂x + v y ∂v x ∂y + v z ∂v x ∂z ² = - ∂p ∂x + μ " 2 v x ∂x 2 + 2 v x ∂y 2 + 2 v x ∂z 2 # + f x (1) ρ ± ∂v y ∂t + v x ∂v y ∂x + v y ∂v y ∂y + v z ∂v y ∂z ² = - ∂p ∂y + μ " 2 v y ∂x 2 + 2 v y ∂y 2 + 2 v y ∂z 2 # + f y (2) ρ ± ∂v z ∂t + v x ∂v z ∂x + v y ∂v z ∂y + v z ∂v z ∂z ² = - ∂p ∂z + μ " 2 v z ∂x 2 + 2 v z ∂y 2 + 2 v z ∂z 2 # + f z (3)
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Unformatted text preview: Cylindrical ρ ∂v r ∂t + v r ∂v r ∂r + v θ r ∂v r ∂θ + v z ∂v r ∂z-v θ 2 r ! = f r-∂p ∂r + μ " ∂ ∂r ± 1 r ∂ ∂r ( rv r ) ² + 1 r 2 ∂ 2 v r ∂θ 2 + ∂ 2 v r ∂z 2-2 r 2 ∂v θ ∂θ # (4) ρ ± ∂v θ ∂t + v r ∂v θ ∂r + v θ r ∂v θ ∂θ + v z ∂v θ ∂z + v θ v r r ² = f θ-1 r ∂p ∂θ + μ " ∂ ∂r ± 1 r ∂ ∂r ( rv θ ) ² + 1 r 2 ∂ 2 v θ ∂θ 2 + ∂ 2 v θ ∂z 2 + 2 r 2 ∂v r ∂θ # (5) ρ ± ∂v z ∂t + v r ∂v z ∂r + v θ r ∂v z ∂θ + v z ∂v z ∂z ² = f z-∂p ∂z + μ " 1 r ∂ ∂r ± r ∂v z ∂r ² + 1 r 2 ∂ 2 v z ∂θ 2 + ∂ 2 v z ∂z 2 # (6)...
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This note was uploaded on 07/14/2011 for the course ENU 4133 taught by Professor Schubring during the Spring '11 term at University of Florida.

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