Chapter 9: Transport phenomena41From this one can derive theNavier-Stokesequations for an incompressible, viscous and heat-conductingmedium:divv=0∂v∂t+(v·∇)v=g-gradp+η∇2vC∂T∂t+C(v·∇)T=κ∇2T+ 2ηD:DwithCthe thermal heat capacity. The forceFon an object within a flow, 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=-[pn+v(v·n)]d2A9.3Bernoulli’s equationsStarting with the momentum equation one can find for a non-viscous medium for stationary flows, with(v·grad)v=12grad(v2) + (rotv)×vand the potential equationg=-grad(gh)that:12v2+gh+dp=constant along a streamlineFor compressible flows holds:12v2+gh+p/=constant along a line of flow. If also holds rotv= 0andthe entropy is equal on each streamline holds12v2+gh+dp/=constant everywhere. For incompressibleflows this becomes:12v2+gh+p/=constant everywhere. For ideal gases with constant
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