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CheatSheet1

# CheatSheet1 - ChBE 6200 Cheat Sheet Vector calculus...

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ChBE 6200 Cheat Sheet Vector calculus relationships: ~ ~ ~ ~ ~ ~ ~ ) ( C A B A C B A + = + ; ( 29 ~ ~ ~ ~ ~ ~ ~ C B C A C B A + = + ; ( 29 ( 29 ~ ~ ~ ~ ~ ~ C B A C B A = ; ( 29 ( 29 ~ ~ ~ ~ ~ ~ B A C B A C = i i b a b a = ~ ~ ; k j ijk b a b a e = × ~ ~ ; ij i T n T n = ~ ; j ij n T n T = ~ ; j i u x u = ~ ; j i x u u = ~ ; i i x u u = ~ Divergence theorem for a scalar Φ : Φ = Φ A V dA n dV ~ , vector ~ G : = A V dA n G dV G ~ ~ ~ and tensor T : = A V dA T n dV T ~ , where ~ n is the normal to the surface. Transport relationships: ( 29 Φ + Φ = Φ ~ v t Dt D , Φ scalar or vector, and ~ v is the fluid velocity. Reynolds Transport Theorem: ( 29 dA n v S dV t S dV Dt DS SdV Dt D Am Vm Vm Vm + = = ~ ~ Unidirectional Flow: 0 ~ ~ = v v ; Incompressible Flow: 0 ~ = v Stress vector- stress tensor relationships: ( 29 ( 29 ~ ~ ~ ~ n t n t - - = ; ( 29 = T n n t ~ ~ ~ ; ji ij T T = . ~ n is the normal to the surface where ( 29 ~ ~ n t is applied. Force applied on a surface A : ( 29 dA T n dA n t F A A = = ~ ~ ~ ~ Torque about an axis: ( 29 dA n t x L A × = ~ ~ ~ ~ , ~ x is the position vector originating from the axis. Non-dimensional Group Definitions: m r UL = Re ; mass D UL = Pe , where L, U

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