{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Equation_sheet_final

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

This preview shows pages 1–2. Sign up to view the full content.

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 = + + + +

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 3

Equation_sheet_final - Equation Sheet for ME 521 prepared...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online