phys documents (dragged) 47

# phys documents (dragged) 47 - Chapter 9 Transport phenomena...

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Chapter 9: Transport phenomena 41 From this one can derive the Navier-Stokes equations for an incompressible, viscous and heat-conducting medium: div ± v =0 ² ∂± v t + ² ( ± v · ) ± v = ²± g - grad p + η 2 ± v ² C T t + ² C ( ± v · ) T = κ 2 T +2 η D : D with C the thermal heat capacity. The force ± F on an object within a ±ow, when viscous effects are limited to the boundary layer, can be obtained using the momentum law. If a surface A surrounds the object outside the boundary layer holds: ± F = - ± ± ± [ p ± n + v ( ± v · ± n )] d 2 A 9.3 Bernoulli’s equations Starting with the momentum equation one can ²nd for a non-viscous medium for stationary ±ows, with ( ± v · grad) ± v = 1 2 grad( v 2 )+(rot ± v ) × ± v and the potential equation ± g = - grad( gh ) that: 1 2 v 2 + + ± dp ² = constant along a streamline For compressible ±ows holds: 1 2 v 2 + + p/ ² = constant along a line of ±ow. If also holds rot ± v and the entropy is equal on each streamline holds 1 2 v 2 + + ² dp/ ² = constant everywhere. For incompressible
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