Isothermal Equations of Change

Isothermal Equations of Change - Equations of Change for...

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1 Equations of Change for Isothermal Systems Mass conversation: the equation of continuity Mass balance over a fixed control volume () ( ) ( ) For incompressible fluids: . .0 xx x yy y y zz z z x xyz yz u u t xz u u t x u yu u t uu tx y z ρ ρρ ΔΔΔ Δ ⎡⎤ =ΔΔ Δ ⎣⎦ +ΔΔ Δ ⎢⎥ Δ ⎛⎞ ∂∂ ⎜⎟ =− + + =− ∇ ⎝⎠ ∇= u u y z x Mass influx (inward arrows) Mass outflux (outward arrows) x y z The equation of motion Derivation of the equation of motion using momentum principle The balance for x-momentum over a fixed control volume with , , x xx xx yx yx x y zx zx x z ix i x i yx x z x x xi x u t xy xyz g p i x y u g ty z ρδ τ ∂Φ Φ Φ Δ ++ + ΔΔΔ =ΔΔ Φ −Φ +Δ Δ Φ Δ + Δ Δ Δ Φ= + + = Quiz Derive y- and z- components Equations of Change for Isothermal Systems
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2 () ,, For Newtonian flow 2 . 3 u = == ⎡⎤ =− + + + ⎢⎥ ∂∂ ⎣⎦ + ⎛⎞ + + ⎜⎟ ⎝⎠ ∑∑ j ij i j i j j ixyz ij j i xyz j i ij ij u uu p g ti p g ii j u u xx ρ ρδ τ ρρ τμ κ μ δ The equation of motion in a general form Equations of Change for Isothermal Systems Simplification of the equation of motion 1. Constant density ρ and viscosity μ : Navier-Stokes equation For incompressible fluids: .0 = u 2 2 2 2 0 j ij j i j u u u u j i μμ = ∇= =− ∇ .u ±²³ ² ´± ²³ ²´ () ( ) j j i i j ji i j j i i u u u u i u u u u i =+ ⎤⎡ = =
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This note was uploaded on 01/27/2011 for the course PGE 323 taught by Professor Johns during the Spring '07 term at University of Texas at Austin.

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Isothermal Equations of Change - Equations of Change for...

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