Fluid Mechanics Cheat Sheet

Fluid Mechanics Cheat Sheet - Akram Ayache AUB MECH-314...

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Unformatted text preview: Akram Ayache AUB MECH-314: Introduction to Fluid Mechanics Final Cheat Sheet Material Derivative: D g2930 Π = ∂ g2930 Π + (V g4652g1318 ∙ ∇ g4652g1318 ) Π Conservation of Mass: 0 = ∂ g2930 g3512 ρ d g2288 g2887g2906 + g3518 ρ (V g4652g1318 g2916 / g2887g2903 ∙ d g2285 g1318 ) g2887g2903 0 = ∂ g2930 ρ + ∇ g4652g1318 ∙ g3435 ρ V g4652g1318 g3439 0 = ∂ g2930 ρ + ρ ∇ g4652g1318 ∙ V g4652g1318 + V g4652g1318 ∙ g3435 ∇ g4652g1318 ρ g3439 Using ∂ g2930 ρ = 0 and Ma < 0.3 we get incompressible continuity: ∇ g4652g1318 ∙ V g4652 = 0 Conservation of Linear Momentum: ∑F g4652g1318 = ∂ g2930 g3512 ρ V g4652g1318 d g2288 g2887g2906 + g3518 ρ V g4652g1318 (V g4652g1318 g2916 / g2887g2903 ∙ d g2285 g1318 ) g2887g2903 ρ g g4652g1318 − ∇ g4652g1318 P + ∇ g4652g1318 ∙ τ g1318 g1318 = ρ D g2930 V g4652g1318 ∑F g4652g1318 − g3505 a g4652 g2934g2935 / g2908g2909 dm = ∂ g2930 g3512 ρ V g4652g1318 d g2288 g2887g2906 + g3518 ρ V g4652g1318 (V g4652g1318 g2916 / g2887g2903 ∙ d g2285 g1318 ) g2887g2903 ∑F g4652g1318 − ma g4652 g2934g2935 / g2908g2909 = m g4662 g2925 u g2925 − m g4662 g2919 u g2919 ρ g3437 g g2934 g g2935 g g2936 g3441 − g4684 ∂ g2934 P ∂ g2935 P ∂ g2936 P g4685 + g4684 ∂ g2934 τ g2934g2934 + ∂ g2935 τ g2935g2934 + ∂ g2936 τ g2936g2934 ∂ g2934 τ g2934g2935 + ∂ g2935 τ g2935g2935 + ∂ g2936 τ g2936g2935 ∂ g2934 τ g2934g2936 + ∂ g2935 τ g2935g2936 + ∂ g2936 τ g2936g2936 g4685 = ρ g4684 ∂ g2930 V g2934 + V g2934 ∂ g2934 V g2934 + V g2935 ∂ g2935 V g2934 + V g2936 ∂ g2936 V g2934 ∂ g2930 V g2935 + V g2934 ∂ g2934 V g2935 + V g2935 ∂ g2935 V g2935 + V g2936 ∂ g2936 V g2935 ∂ g2930 V g2936 + V g2934 ∂ g2934 V g2936 + V g2935 ∂ g2935 V g2936 + V g2936 ∂ g2936 V g2936 g4685 For Newtonian Fluids τ g1318 g1318 = μ ∇ g4652 V g4652g1318 and τ g2919g2920 = τ g2920g2919 = μ (∂ g2919 V g2920 + ∂ g2920 V g2919 ) thus momentum balance yields the Navier-Stokes Equation: ρ g g4652g1318 − ∇ g4652g1318 P + μ ∇ g2870 V g4652g1318 = ρ D g2930 V g4652g1318 ρ g3437 g g2934 g g2935 g g2936 g3441 − g4684 ∂ g2934 P ∂ g2935 P ∂ g2936 P g4685 + μ g4684 ∂ g2934g2934 g2870 V g2934 + ∂ g2935g2935 g2870 V g2934 + ∂ g2936g2936 g2870 V g2934 ∂ g2934g2934 g2870 V g2935 + ∂ g2935g2935 g2870 V g2935 + ∂ g2936g2936 g2870 V g2935 ∂ g2934g2934 g2870...
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This note was uploaded on 09/06/2011 for the course ASE 13180 taught by Professor Goldstein during the Spring '10 term at University of Texas.

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Fluid Mechanics Cheat Sheet - Akram Ayache AUB MECH-314...

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