Demostración de La Ecuación de Continuidad en Coordenadas Cilíndricasy Esferica

Demostración de la ecuación de continuidad en coordenadas cilíndricas.

A=E−S+G

G=0 ( Nohay reacción) E−S= A

( ψA ) E−( ψA )S =

dϕ dV dt

Δ V =r Δθ . Δr . Δz Δ Ar =r Δ θ . Δz Δ Aθ =Δr . Δz Δ A z=r Δ θ . Δ r θ= [ 0,2 π ] ∂ρ ∂ V =Δ A r [( ρ v r )r −( ρ v r )r+ Δr ] + Δ A θ [ ( ρ v θ )θ −(ρ v θ)θ+ Δ θ ] + Δ A z [ (ρ v z )z −( ρ v z )z + Δz ] ∂t ∂ ρ r Δ θ Δz [ (ρ v r )r−( ρ v r )r + Δr ] Δ r Δz [( ρ v θ)θ −( ρ v θ )θ+ Δ θ ] r Δθ Δr [ (ρ v z) z−(ρ v z) z+ Δz ] = + + ∂t r Δθ ΔrΔz r Δθ ΔrΔz r Δ θ ΔrΔz ∂ ρ [ (r ρ v r )r−(r ρ v r )r+ Δr ] [ ( ρ v θ )θ−( ρ v θ)θ +Δ θ ] [( ρ v z ) z−(ρ v z ) z+ Δz ] = + + ∂t r . Δr r Δθ Δz Δ r → 0. Δ θ→ 0, Δ z → 0

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∂ ( r ρ V r ) ∂ ( ρV θ ) ∂ ( ρ V z ) ∂ρ =− + + ∂t r .∂r r∂θ ∂z

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∂ρ 1 ∂ ( ρV r ) 1 ∂ ( ρV θ ) ∂ ( ρ V z ) =− . + . + ∂t r ∂r r ∂θ ∂z

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ρ ∂V r ρ ∂ V θ ρ ∂V z V ∂ρ V ∂ρ V ∂ρ ∂ρ =− + + + r + θ + z ∂t r∂r r∂θ ∂z r∂r r∂θ ∂z

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∂ρ 1 Vr∂ ρ 1 Vθ∂ ρ Vz∂ ρ 1 ∂Vr 1 ∂Vθ ∂Vz + . + . + =−ρ . + . + ∂t r ∂r r ∂θ ∂z r ∂r r ∂ θ ∂z

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Demostración de la ecuación de continuidad en coordenadas esféricas.

A=E−S+G G=0 ( Nohay reacción)

E−S= A

( ψA ) E−( ψA )S =

dϕ dV dt

Δ V =r Δθ . Δr . rsenϕΔ θ

Δ Ar =r Δθ . rsenϕΔ θ Δ Aθ =r Δ θ . rsenϕ . Δ θ Δ A ϕ= Δr .rsenϕ . Δθ

θ= [ 0,2 π ] ϕ= [ 0,2 π ] ∂ρ ∂ V =Δ A r [( ρ v r )r −( ρ v r )r+ Δr ] + Δ A θ [ ( ρ v θ )θ −(ρ v θ)θ+ Δ θ ] + Δ A ϕ [ (ρ v ϕ) ϕ−(ρ v ϕ )ϕ + Δϕ ] ∂t

∂ ρ r Δ θ . rsenϕΔ θ [ (ρ v r )r −( ρ v r )r+ Δr ] r Δ θ . rsenϕΔ θ [(ρ vθ )θ−(ρ v θ )θ+ Δθ ] Δr . rsenϕ . Δ θ [ ( ρ v ϕ )ϕ −( ρ v ϕ )ϕ + Δ = + + ∂t r Δθ . Δr .rsenϕΔ θ r Δ θ . Δr . rsenϕΔ θ r Δθ . Δr . rsenϕΔ θ ∂ ρ [ ( ρ v r )r−( ρ v r )r + Δr ] [(ρ v θ )θ−(ρ v θ )θ+ Δθ ] [( ρ v ϕ )ϕ −( ρ v ϕ )ϕ + Δϕ ] = + + ∂t Δr Δr Δϕ Δ r → 0. Δ θ→ 0, Δ ϕ → 0

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[(

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∂ ( ρ V r ) ∂ ( ρV θ ) ∂ ( ρ V z ) ∂ρ =− + + ∂t ∂r r ∂θ ∂z

ρ ∂V r ρ ∂ V θ ρ ∂V z V ∂ρ V ∂ρ V ∂ρ ∂ρ =− + + + r + θ + z ∂t ∂r r∂θ ∂z ∂r r∂θ ∂z

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∂V r ∂ V θ ∂ V z ∂ρ Vr∂ ρ Vθ∂ ρ Vz∂ ρ + + + =−ρ + + ∂t ∂r r∂θ ∂z ∂r r∂θ ∂z

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