Chapter 9: Transport phenomena41From this one can derive theNavier-Stokesequations for an incompressible, viscous and heat-conductingmedium:div±v=0²∂±v∂t+²(±v·∇)±v=²±g-gradp+η∇2±v²C∂T∂t+²C(±v·∇)T=κ∇2T+2ηD:DwithCthe thermal heat capacity. The force±Fon an object within a ±ow, when viscous effects are limited tothe boundary layer, can be obtained using the momentum law. If a surfaceAsurrounds the object outside theboundary layer holds:±F=-± ±±[p±n+v(±v·±n)]d2A9.3Bernoulli’s equationsStarting with the momentum equation one can ²nd for a non-viscous medium for stationary ±ows, with(±v·grad)±v=12grad(v2)+(rot±v)×±vand the potential equation±g=-grad(gh)that:12v2++±dp²=constant along a streamlineFor compressible ±ows holds:12v2++p/²=constant along a line of ±ow. If also holds rot±vandthe entropy is equal on each streamline holds12v2++²dp/²=constant everywhere. For incompressible
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