Various_Bernoulli_forms - Various Forms of the Bernoulli...

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Various Forms of the Bernoulli Equation Author: John M. Cimbala, Penn State University Latest revision: 25 September 2007 Note : The fluid is assumed to be Newtonian, and gravity is assumed to act in the negative z direction [positive z is “up”] . All forms of the Bernoulli equation shown below are derived from the Navier-Stokes equation, along with some useful vector identities : () 1 2 i j j j j ijk k ji u uu u u xx εω =− ∂∂ , and 2 j ik ijk j j u u j x x x ω ε ⎛⎞ ⎜⎟ ⎝⎠ . 1. Incompressible Flow : Start with the incompressible N-S equation : 2 ii i j ij p ug tx x x ρρ ρ μ += + + j u x a. Incompressible, Unsteady, and Irrotational (can be viscous) : 2 1 2 p qg z F t φ ++ + = t where 2 qu u = = magnitude of the velocity vector squared, is the velocity potential defined by u = ∇ G G or , i u i = ( is definable only if the flow is irrotational ), and F is a function of time, but not of space.
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