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Equation_sheet_midterm

# Equation_sheet_midterm - Equation Sheet for Midterm Exam ME...

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Equation Sheet for Midterm Exam, ME 521, prepared by Professor J. M. Cimbala Tensor transformation rules for rotation : cos ij ij C α , where α ij = angle between the old ( i ) and new ( j ) axes. Then, for tensor A , m im i A A C ′ = mn im jn ij A C C A = mnp im jn kp ijk A C C C A = , etc. Epsilon-delta relation : ijk kmn im jn in jm ε ε δ δ δ δ = ( ) Cross product : ijk i j k a b a b ε × = G G Gauss, Stokes, and Leibniz theorems : G : i i A F d Fd x = v V V A S : ( ) A C u dA u ds ∇× = G G G G v L : ( ) ( ) ( ) ( , ) ( , ) ( , ) A t t A t d F x t F x t d d F x t u dA dt t = + G G G G G v V V V V Material derivative : j j DF F F u Dt t x = + (following a fluid particle) where F is some variable ( ) Strain rate tensor : 1 , , 2 ij i j j i e u u = + for any fluid. Principal strain rates (eigenvalues) found from det 0 ij ij e λδ = Reynolds transport theorem : j j (t) CV CS D F Fd d Fu dA Dt t = + v V V V where F can be any quantity per unit volume. Conservation of mass : 0 j j CV CS ρ d ρ u dA t = + v V ( ) 0 i i ρ ρ u t x + = Conservation of momentum : For any fluid: ( ) i i j j i ij j CV CS CV CS ρ u d ρ u u dA ρ g d τ dA t + = + v v V V ( ) ( ) ij i i j i j j τ ρ u ρ u u ρ g t x x + = + ij i i j τ Du ρ ρ g Dt x = + Constitutive equation (relation between stress and strain), with σ ij

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