Lecture 2 on Vectors and Tensors 3-7-2008

Louisiana Tech UniversityRuston, LA 71272Learning ObjectivesYou should be able to!"e#ine and recogni$e scalars, vectors, and tensors%!"escribe the di##erence bet&een continuu' and statistical 'echanics, the advantages and disadvantages o# each, and their a((lications!Understand and (er#or' standard vector and tensor 'athe'atical o(erations!"e#ine the 'aterial derivative and use it to convert bet&een s(atial and 'aterial coordinates and to describe 'otion in engineering (roble's!"e#ine, calculate, and use (athlines, strea'lines, and strea)lines #or a given #lo&Louisiana Tech UniversityRuston, LA 71272*otivating +uestionYou &ish to generate a ,A" 'odel o# an arterial bi#urcation, &hich you re(resent as the union o# t&o cylinders%1% -o& do you describe the hori$ontal cylinder 'athe'atically.2% -o& do you describe the diagonal cylinder 'athe'atically.Louisiana Tech UniversityRuston, LA 71272*otivating +uestion1% -o& do you describe the hori$ontal cylinder 'athe'atically./t de(ends on the coordinate syste' you choose%( ) ( )( ) ( )2 2 2, , , , 0, , , , 0r z f r z r Rxyz f xyz x y R + Louisiana Tech UniversityRuston, LA 71272*otivating +uestion-o& do you describe the diagonal cylinder 'athe'atically.You have already done it #or the hori$ontal cylinder%/# you &ere to choose a coordinate syste' #or &hich the a0is is aligned &ith the $1coordinate, you just need to rotate( )( ) ( )2 2 21 1 1 1 21 1 1, , 01 0 0where , , , , 0 cos sin0 sin cosf x y z x y Rx y z xyz + 1 1 1 1 ]Louisiana Tech UniversityRuston, LA 71272*otivating +uestion( )( ) ( )2 2 21 1 1 1 21 1 1, , 0cos sin 0where , , , , sin cos 00 0 1f x y z x y Rx y z xyz + 1 1 1 1 ]This set o# e2uations loo)s co'(licated, but you &ill not need to &orry about the details%The so#t&are you use &ill ta)e care o# the tedious calculations%Rotation 'atri0 3tensor4vectorLouisiana Tech UniversityRuston, LA 71272*otivating +uestion( )( ) ( )2 2 21 1 1 1 21 1 1 0, , 0where , , , ,f x y z x y Rx y z x x yz + /# you &anted to shi#t the cylinder in the 01direction, you &ould do so'ething li)e this( ) ( ) ( )1 1 1 01 0 0, , , , 0 cos sin , 0, 00 sin cosx y z xyz x 1 1 1 1 ]And i# you &anted to rotate and then shi#tLouisiana Tech UniversityRuston, LA 71272*otivating +uestion/# you &anted a 'ore general rotation( )( ) ( )2 2 21 1 1 1 211 12 131 1 1 21 22 2331 32 33, , 0cos cos coswhere , , , , cos cos coscos cos cosf x y z x y Rx y z xyz + 1 1 1 1 ]5here ij is the angle through &hich the i1a0is in the original coordinate syste' 'ust rotate to align &ith the j1a0is in the ne& coordinate syste'%Louisiana Tech UniversityRuston, LA 71272A0is Rotation612Louisiana Tech UniversityRuston, LA 71272A0is Rotation126126131112Louisiana Tech UniversityRuston, LA 712727ur(ose o# the 8tress Tensor9or solids11 12 133 1 221 22 231 2 331 32 333 1 1 2 11 2 1 3 13 2 1 2 21 2 2 3 23 11 31 0 00 1 00 0 11 12 21 122 212u u ux x xu u u u ux x x x xu u u u uGx x x x xu ux x 11 _ 11 + + + 11 , 11 ] ] _ _ + + , , _ _ + + , , + 3 3 22 3 312u u ux x x 1 1 1 1 1 1 1_ _ 1+ 1 , , ]E One di'ensionalThree1"i'ensional, &here u is dis(lace'entLouisiana Tech UniversityRuston, LA 712727ur(ose o# the 8tress Tensor9or #luids11 12 1321 22 2331 32 333 1 1 2 11 2 1 3 13 2 1 2 21 2 2 3 23 3 1 21 3 2 31 0 00 1 00 0 11 12 21 122 21 12 2Pu u u u ux x x x xu u u u ux x x x xu u u ux x x x 11 11 + 11 11 ] ] _ _ + + , , _ _ + + , , _ _ + + , ,33ux 1 1 1 1 1 1 1 1 1 ]vy One di'ensionalThree1"i'ensional, &here u is velocityLouisiana Tech UniversityRuston, LA 71272Tensor :otation3 1 1 2 11 2 1 3 13 2 1 2 21 2 2 3 23 3 3 1 21 3 2 3 31 12 21 12 21 12 2u u u u ux x x x xu u u u ux x x x xu u u u ux x x x x 1 _ _ + + 1 , , 1 1 _ _ 1 + + 1 , , 1 _ _ 1+ + 1 , , ]/s called the rate o# strain tensor%/t can be &ritten 'ore si'(ly 3in tensor notation4 as12jiijj iuux x _ + ,and2ij ij ijP +Louisiana Tech UniversityRuston, LA 71272 A 8calar! -as 'agnitude only 3e%g% T;te'(erature4! Re(resented by a single nu'ber A 8calar 9ield! A scalar as #unction o# (osition 3e%g% T;T30,y,$44! Re(resented by a single nu'ber &hose value varies in s(ace% A