{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Equation_sheet_midterm

Equation_sheet_midterm - Equation Sheet for Midterm Exam ME...

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

View Full Document Right Arrow Icon
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
Image of page 1

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

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern