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Unformatted text preview: 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 X dX X X X X X dx + dy + dz + dt = vx + vy + vz + x y z t dt x y z t X dX = + (v dt t d dt )X . t Xd3 V + X(v n )d2 A operator are: rot grad = 0 div rotv = 0 This results in general to: From this follows that also holds: Xd3 V = where the volume V is surrounded by surface A. Some properties of the div(v ) = divv + grad v div(u v ) = v (rotu ) - u (rotv ) div grad = 2 rot(v ) = rotv + (grad) v rot rotv = grad divv - 2 v 2 v ( 2 v1 , 2 v2 , 2 v3 ) Here, v is an arbitrary vector field and an arbitrary scalar field. Some important integral theorems are: Gauss: Stokes for a scalar field: Stokes for a vector field: This results in: Ostrogradsky: (v n )d2 A = ( et )ds = (v et )ds = (divv )d3 V (n grad)d2 A (rotv n )d2 A (rotv n )d2 A = 0 (n v )d2 A = (n )d2 A = (rotv )d3 A (grad)d3 V ds. Here, the orientable surface d2 A is limited by the Jordan curve 9.2 Conservation laws
On a volume work two types of forces: 1. The force f0 on each volume element. For gravity holds: f0 = 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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This note was uploaded on 11/16/2010 for the course ENGR 201 taught by Professor Elder during the Spring '10 term at Blinn College.
- Spring '10