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phys documents (dragged) 38

phys documents (dragged) 38 - 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 v t + ( v · ) v = g - grad p + η 2 v C T t + C ( v · ) T = κ 2 T + 2 η D : D with C the thermal heat capacity. The force F on an object within a flow, 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 = - [ pn + v ( v · n )] d 2 A 9.3 Bernoulli’s equations Starting with the momentum equation one can find for a non-viscous medium for stationary flows, with ( v · grad) v = 1 2 grad( v 2 ) + (rot v ) × v and the potential equation g = - grad( gh ) that: 1 2 v 2 + gh + dp = constant along a streamline For compressible flows holds: 1 2 v 2 + gh + p/ = constant along a line of flow. If also holds rot v = 0 and the entropy is equal on each streamline holds 1 2 v 2 + gh + dp/ = constant everywhere. For incompressible flows this becomes: 1 2 v 2 + gh + p/ = constant everywhere. For ideal gases with constant
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