Dimentional Analysis

Dimentional Analysis - Equations of Change for Isothermal...

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Dimensional analysis of the equations of change 2 xxxx x y zx x uuuu p uuu u g tx y ρ μρ ⎛⎞ ∂∂∂∂ +++ = + + ⎜⎟ ∂∂ ⎝⎠ Non-dimensionalizing Navier-Stokes equation Consider the x-component of the Navier-Stokes equation 00 0 0 0 2 0 // / ; y ; z ; / ; / ; xx l y l z l PP tu t l u u u P u === == = ± ±± ± ± ± Introducing the following dimensionless variables using characteristic length ( l 0 ), characteristic velocity ( u 0 ), and characteristic pressure ( P 0 ) Equations of Change for Isothermal Systems
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Dimensional analysis of the equations of change Non-dimensionalizing Navier-Stokes equation Inserting the dimensionless variables into the x-component Navier-Stokes equation gives 222 00 0 0 0 0 2 0 2 2 2 2 / xxxx xyz xx x x x x uu u u u u u u uuu lu t l l y l z p ug ll uuuu p u lg ul u ty z μ ρ μρ ⎛⎞ ∂∂∂∂ +++ ⎜⎟ ⎝⎠ =− + + +++= + + ± ± ±± ±±±± ±±± ± ± ± ± ± ± cf c f ± dg d g ± d g ± ± d g eh e h Equations of Change for Isothermal Systems
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Dimensional analysis of the equations of change Non-dimensionalizing Navier-Stokes equation (cont.) Or 0 2 00 0 2 11 Re Fr xxxx y x xz x xx uuuu p uuu u ty z lg ul u μ ρ ∂∂∂∂
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Dimentional Analysis - Equations of Change for Isothermal...

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