phys documents (dragged) 36

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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. ...
View Full Document

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.

Ask a homework question - tutors are online