Equation_sheet_final

Equation_sheet_final - Equation Sheet for ME 521, prepared...

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Equation Sheet for ME 521, prepared by Professor J. M. Cimbala Tensor transformation rules for rotation : cos ij ij C α , where ij = angle between the old ( i ) and new ( j ) axes. Then, for tensor A , mi m i A AC ′ = mn im jn ij A CCA = mnp im jn kp ijk A CCCA = , etc. Epsilon-delta relation : i j kk m n i mj n i nj m ε εδ δ =− ( ) Cross product : ijk i j k ab a b ×= G G Gauss, Stokes, and Leibniz theorems : G : i i A F dF d x = ∫∫ v V V A S : ( ) ud A ud s ∇× = G GG G v L : () (,) A ttA t x t Fxtd d Fxtu dA dt t = +⋅ G G G v VV 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 eu u for any fluid. Principal strain rates (eigenvalues) found from det 0 ij ij e λδ = Reynolds transport theorem : j j (t) CV CS DF F dd F u Dt t v V d A where F can be any quantity per unit volume Conservation of mass : 0 j j CV CS ρ d ρ udA t v V 0 i i ρ ρ u tx + = Conservation of momentum : For any fluid: i i jj i i ρ ρ uu dA ρ gd τ dA t += + vv ij ii j i j j τ ρ u ρ uu ρ g x + ij i i j τ Du ρρ g Dt x Constitutive equation (relation between stress and strain), with σ ij defined as the deviatoric stress tensor : ij ij ij τ p δσ = −+ For Newtonian fluid: 2 ij ij ij mm ij τ p δ μ e λ e δ + + , and the famous Navier-Stokes equation results: j j i ji j ij j i i u p ρ u ρ g μλ x xx x ⎡⎤ ⎛⎞ + + + + ⎢⎥ ⎜⎟ ⎝⎠ ⎣⎦ j For incompressible flow: 2 ij ij ij τ p δ μ e + 2 i j u p ρ u ρ g μ x x + + j x Incompressible conservation equations of mass and momentum in Cartesian coordinates (x,y,z) : 0 uvw xyz ∂∂∂ ++ = 222 2 2 22 x yz =++ uuvw x ⋅∇= + + G G x uuu u p u u u g txyz x x y z μ +++ = + + ⎝⎠ ⎝⎠ y vvv v p v v v g y x y z = z w ww w p www g z x y z = + + + +
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Equation_sheet_final - Equation Sheet for ME 521, prepared...

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