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 zdirection [positive zis “up”]. All forms of the Bernoulli equation shown below are derived from the Navier-Stokes equation, along with some useful vector identities: ()12ijjjjijkkjiuuuuuxxεω∂∂=−∂∂, and 2jikijkjjuujxxxωε⎛⎞∂∂∂∂⎜⎟∂∂∂⎝⎠. 1. Incompressible Flow: Start with the incompressible N-S equation: 2iiijijpugtxxxρρρμ∂∂+=−++jux∂∂∂a. Incompressible, Unsteady, and Irrotational (can be viscous): 212pqgzFtφ∂+++=∂twhere 2quu== magnitude of the velocity vector squared, is the velocity potentialdefined by u= ∇GGor ,iui=(is definable onlyif the flow is irrotational), and Fis a function of time, but not of space.
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