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phys documents (dragged) 36 - Chapter 9 Transport phenomena...

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Chapter 9 Transport phenomena 9.1 Mathematical introduction An important relation is: if X is a quantity of a volume element which travels from position r to r + dr in a time dt , the total differential dX is then given by: dX = X x dx + X y dy + X z dz + X t dt dX dt = X x v x + X y v y + X z v z + X t This results in general to: dX dt = X t + ( v · ) X . From this follows that also holds: d dt Xd 3 V = t Xd 3 V + X ( v · n ) d 2 A where the volume V is surrounded by surface A . Some properties of the operator are: div( φ v ) = φ div v + grad φ · v rot( φ v ) = φ rot v + (grad φ ) × v rot grad φ = 0 div( u × v ) = v · (rot u ) - u · (rot v ) rot rot v = grad div v - ∇ 2 v div rotv = 0 div grad φ = 2 φ 2 v ( 2 v 1 , 2 v 2 , 2 v 3 ) Here, v is an arbitrary vector field and φ an arbitrary scalar field. Some important integral theorems are: Gauss: ( v
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