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phys documents (dragged) 45 - ( e t ) ds = ( n grad ) d 2 A...

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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 + d ± r 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 )r o t r o t ± v =gradd iv ± v -∇ 2 ± v div rot ± v=0 div grad φ = 2 φ 2 ± v ( 2 v 1 , 2 v 2 , 2 v 3 ) Here, ± v is an arbitrary vector ±eld and φ an arbitrary scalar ±eld. Some important integral theorems are: Gauss: ± ± ± ( ± v · ± n ) d 2 A = ±± (div ± v
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Unformatted text preview: ( e t ) ds = ( n grad ) d 2 A Stokes for a vector eld: ( v e t ) ds = (rot v n ) d 2 A This results in: (rot v n ) d 2 A = 0 Ostrogradsky: ( n v ) d 2 A = (rot v ) d 3 A ( n ) d 2 A = (grad ) d 3 V Here, the orientable surface d 2 A is limited by the Jordan curve ds . 9.2 Conservation laws On a volume work two types of forces: 1. The force f on each volume element. For gravity holds: f = g . 2. Surface forces working only on the margins: t . For these holds: t = n T , where T is the stress tensor ....
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