convective_boundary_value_problems

Haghighi by substituting for using the relationship e

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Unformatted text preview: es of the convective terms is: ⎛T ∂φ ∂φ ⎞ T ∫A ⎜ [N] ρ φu ∂x + [N] ρ φ v ∂y ⎟ dA ⎟ ⎜ ⎠ ⎝ New Chapter Page 3 K. Haghighi Substituting {} (e ) = − [N]T ⎛ D ∂φ cos θ + D ∂φ sin θ ⎞ dΓ ⎟ R y ∫Γ ⎜ x ∂x ⎜ ⎟ ∂y ⎝ ⎠ ⎛ ⎛ ∂[N]T ∂φ ∂[N]T ∂φ ⎞ ⎞ ⎟ dA ⎟ + ⎜ ∫ ⎜ Dx + Dy ⎜ A⎜ ∂ x ∂x ∂y ∂y ⎟ ⎟ ⎝ ⎠⎠ ⎝ ⎛⎛T ∂φ ⎞ ⎞ ∂φ T + ⎜ ∫ ⎜ [N] ρ φu + [N] ρ φ v ⎟ dA ⎟ ⎜ A⎜ ∂x ∂y ⎟ ⎟ ⎠⎠ ⎝⎝ + ∫ G[N]T φdA − ∫ Q[N]T dA A A New Chapter Page 4 K. Haghighi {} By substituting for φ using the relationship φ (e ) = [N] Φ (e ) and rearranging: {} (e ) = − [N]T ⎛ D ∂φ cos θ + D ∂φ sin θ ⎞ dΓ ⎟ R y ∫Γ ⎜ x ∂x ⎟ ⎜ ∂y ⎠ ⎝ {} ⎛ ⎛ ∂[N]T ∂[N] ∂[N]T ∂[N] ⎞ ⎞ (e ) ⎟ dA ⎟ Φ + ⎜ ∫ ⎜ Dx + Dy ⎜ A⎜ ∂x ∂ x ∂y ∂y ⎟ ⎟ ⎝ ⎠⎠ ⎝ {} ⎛⎛T ∂[N] ⎞ ⎞ (e ) ∂[N] T ⎟ dA ⎟ Φ − ⎜ ∫ ⎜ [N] ρ φu + [N] ρ φ v ⎜ A⎜ ∂x ∂y ⎟ ⎟ ⎠⎠ ⎝⎝ + (∫A G[N]T [N]dA ){Φ (e )}− ∫A Q[N]T dA New Chapter Page 5 K. Haghighi which has the general form {R (e )}= {I(e)}+ [k (e ) ]{Φ (e )}− {f (e)} where {} (e ) = − [N]...
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This document was uploaded on 01/29/2014.

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