Equations for the Conservation of Energy Author: John M. Cimbala, Penn State University Latest revision: 25 September 2007 Mechanical Energy EquationNon-conservative form: ()12ijiiiiijDu uu guDtxτρρ∂=+∂Conservative form: ()()1212ijiijiiiiijjτρu uuρu uρu gutxx∂∂∂+=+∂∂∂or ()ijjiiijjτEu Eρu gutxx∂∂∂+=+∂∂∂where 212mVis the kinetic energy (the conserved quantity), 12iiEu uρ≡is the kinetic energy per unit volume, and 12iiu uis the kinetic energy per unit mass. The terms on the right are sources (or sinks) of kinetic energy per unit volume. Alternate conservative form: ()()jjiiijijjuEu Eρu gτuptxxxjφ∂∂∂∂+=++−∂∂∂∂where iijjuσxφ∂≡∂= rate of viscous dissipation of kinetic energy per unit volume (increases internalenergy at expense of kineticenergy). φis always positivesince friction is an irreversibleprocess. For a Newtonian fluid,
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