C4_Isothermal_Equations_of_Change_2

# C4_Isothermal_Equations_of_Change_2 - Equations of Change...

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Unformatted text preview: Equations of Change for Isothermal Systems Simplifications of the general equation of motion 1. Constant density ρ and viscosity μ : Navier-Stokes equation 3 1 j ij j i j i Du p g Dt x x δτ δ ρ ρ δ δ = = − − + ∑ 2 2 3 3 3 2 2 1 1 1 j ij j i j i i i i i j i u u u u x x x x δτ δ δ δ µ µ µ δ δ δ δ = = = ∇ ∇ = − − = − ∇ ∑ ∑ ∑ .u ¡¢£¢ ¤ ¡£¤ 2 2 or j j j j Du p D g u p Dt x Dt δ ρ ρ µ ρ µ ρ δ = − − + ∇ = −∇ + ∇ + u u g The general equation of motion 2 . 3 j i ij ij i j u u x x δ δ τ µ κ µ δ δ δ ⎛ ⎞ ⎡ ⎤ ⎡ ⎤ = − + + − ∇ ⎜ ⎟ ⎢ ⎥ ⎢ ⎥ ⎜ ⎟ ⎣ ⎦ ⎢ ⎥ ⎣ ⎦ ⎝ ⎠ u Equations of Change for Isothermal Systems Simplifications of the general equation of motion 1. Constant density ρ and viscosity μ : Navier-Stokes equation (con.) 2 D p Dt ρ µ ρ = −∇ + ∇ + u u g ( ) Introducing the force potential gh = −∇Π = −∇ g where is the elevation and the gravitational potential engergy per unit mass h gh ( ) 2 2 gives D p gh P Dt ρ ρ µ µ = −∇ + + ∇ = −∇ + ∇ u u u Note : for gas flow, absolute pressure could be related to gas density through the P-V-T relation for gas. For liquid flow, pressure gradient remains determinate by the equation of motion, and absolute pressure in the whole domain could only be determined if the pressure is known at some point in the domain. Equations of Change for Isothermal Systems The equation of motion 2. Small variation of density ρ and constant viscosity μ : Boussinesq (1903) approximation For small variation of ρ , the continuity equation has the incompres-sible form ( ) . ∇ = u ( ) ( ) 2 ' ' p D p p Dt ρ ρ ρ ρ µ = −∇ − + − + ∇ u g u ¡¢£¢ ¤ ¡ ¢£¢¤ For '/ 1 1 '/ 1 ρ ρ ρ ρ ⇒ + ¥ ¦ But the equation of motion is slightly different from Navier-Stokes equation. Indeed, consider a hypothetical static reference state in which the density is everywhere and the pressure is . Subtracting this equation from the Navier-Stokes equation, and dividing by gives p ρ ∇ = g ρ ρ N 2 ' 1 ' 1 ' p D p Dt ρ ρ µ ρ ρ ρ ρ =∇ ⎛ ⎞ ⇔ + = − ∇ + + ∇ ⎜ ⎟ ⎝ ⎠ u g u but the buoyancy term is imporant and cannot by nelgected ' / g ρ ρ Equations of Change for Isothermal Systems The equation of motion ( ) ( ) 2 ' ' p D p p Dt ρ ρ ρ ρ µ = −∇ − + − + ∇ u g u ¡¢£¢ ¤ ¡ ¢£¢¤ N 2 ' 1 ' 1 ' p D p Dt ρ ρ µ ρ ρ ρ ρ =∇ ⎛ ⎞ ⇔ + = − ∇ + + ∇ ⎜ ⎟ ⎝ ⎠ u g u fluid For '/ 1 1 '/ 1 ρ ρ ρ ρ ⇒ + ¥ ¦ but the buoyancy term is imporant and cannot by nelgected ' / g ρ ρ 2 '( ) D p T Dt ρ ρ µ = −∇ + + ∇ u g u Equations of Change for Isothermal Systems...
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C4_Isothermal_Equations_of_Change_2 - Equations of Change...

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