Class 05 Handout

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Fluid Mechanics AS102

Class Note No: 05

Tuesday, August 7, 2007


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Review: C. C. S. & Tensor Analysis - transformations

xi=xi(uj), xi=x

i(uj);

ui =ui(uj), ui =ui(uj) (1)

x1

x2

x3

r

P

u1u2

u3g1

g2g3

u1u2

u3

g1

g

2

g

3

Figure:representing two curvilinear coordinate systems


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gk = um

ukgm, g

k =uk

um gm,

gkl= um

uku

n

ul gmn,

gkl =uk

umul

un gmn (2)

# 2nd order covariant tensor & 2nd order contravarianttensor

# understand the transformation rules


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0th order tensors (scalars):

= (xi) = (ui) = (ui) (3)


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1st order tensors (vectors):

a = aiii=ai

(u)gi=ai(u)gi

= ak(u)gk =a

i(u)gi (4)

# Hereurepresentsum andu representsum

# thepairingin thesummationfor the representations in a

cuvilinear c. s.:e.g. indexk, one up & one down

ai(u) = ui

um

am(u) =ui

xm

am

ai(u) =

um

uiam(u) =

xm

uiam

ai(u) =gijaj(u)

ai(u) =gijaj(u) (5)


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1st order tensors (vectors):

# hints to derive the above transformations:

e.g. dot product both sides ofak(u)gk=ai(u)gi with

gm

e.g. dot product both sides ofak(u)gk=ai(u)gi withgm

# covariant tensor & contravariant tensor

# understand the transformation rules:positions of & correspondences between the indexes

# HOMEWORK Assignment derive the relations in (5)


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other 2nd order tensors:

A=Aijiiij=Aij(u)gigj=Aij(u)g

igj =Aijgig

j

=Aij(u)gi g

j =A

ij(u) gigj =Ai

jg

igj (6)

# Hereurepresentsum andu representsum

# thepairingin thesummationfor the representations in acuvilinear c. s.:

e.g. indexi, one up & one down

Aij(u) =ui

umuj

unAmn(u) =

ui

xm

uj

xnAmn

Aij(u) =

um

u

i

un

u

j

Amn(u) = xm

u

i

xn

u

j

Amn

Aij(u

) =ui

umun

ujAm

n(u) =ui

xm

xn

ujAmn

Aij(u) =gimgjnAmn(u), Aij(u) =gimgjnAmn(u)

Aij(u) =g

im

Amj(u) =gjnAin

(u) ... (7)


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other 2nd order tensors:

# hints to derive the above transformations:

e.g. dot product both sides ofAij(u)gig

j =aij(u)gigj

withgm andgn consecutively, each operating on its

immediate neighboring base vectore.g. dot product both sides ofAij(u)gigj=Aij(u)g

igj withgm andgn consecutively, each operating on its immediateneighboring base vector

# covariant tensor & contravariant tensor

# understand the transformation rules:positions of & correspondences between the indexes

# HOMEWORK Assignment derive the relations in (7)


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2nd order tensors:

# working example 1

Aij(u) = ui

umu

j

unAmn(u) ux= u

i

xmu

j

xnAmn (8)

from

A= Aij(u)gi g

j =Aij(u)gigj (9)

(use the hints mentioned above).


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2nd order tensors:

# working example 2

Aij(u) = ui

umu

n

ujAmn(u) u

x= ui

xmxnuj

Amn (10)

from

A= Aij(u

)gi gj =Ai

j(u)gigj (11)



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2nd order tensors:

# working example 3

Aij(u) =gimgjnAmn(u) (12)

from

A= Aij(u)gigj=Aij(u)gigj (13)



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Curvilinear Coordinate Systems & Tensor Analysis

Todays topic:

# tensor differentiations

# example: a= aiii=ai(u)gi=ai(u)g

i

in a rectangular c.s.

xja=

xj(aiii) =

aixj

ii =ai,j ii (14)

in a curvilinear c.s.

uja =

uj

a

i(u)gi=

ai(u)

uj gi

Q?

uja = ai;j(u) gi ? a

i;j(u) = ?

form similar to the case of rectangular c.s. ai,j in(14)

(YES !! below ...)


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ui =ui(x) , xi=xi(u) ,

r=xiii=xi(u) ii=: r(u) (15)

O

x1

x2

x3

r

Pu1

u2

u3

i1i2

i3

g1 g2

g3

Figure:curvilinear coordinate system


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scalars:

= (x) = (u) {= (u)} (16)

def

= ik

xk

c. r.= ik

um

um

xk

def=

umgm =;mg

m

{u2 u

=

umgm} (17)

um =

u

n

um

un (18)

1st order covariant tensor ?


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Curvilinear Coordinate Systems & Tensor Analysisvectors:

a= ak(u)gk=ak(u)gk

{=ak(u)gk =ak(u)gk} (19)

a def

= ik

xka

def= gl

ula= ? (20)

ula=

ul

ak(u)gk

=

ak(u)

ul gk+ a

k(u)gk

ul

ula=

ul

ak(u)g

k

=ak(u)

ul gk + ak(u)

gk

ul

gk

ul = ?

gk

ul = ? the key to the diff

C ili C di S T A l i


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one can check (defns + chain rules + tricks)

gk

ul =

m

k l

gm

gkul

=

kl n

gn (21)

mk l

:= 12

gmn(gnk

ul +

gnl

uk

gkl

un) (22)

the christoffel symbols of the second kind

C ili C di S & T A l i


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some hints to show (21):

#

gkul

= 2

xiuluk

ii= um

xi

2

xiuluk

gm

#

um

xi=gmn

xi

un

C ili C di t S t & T A l i


18/24



#

um

xi

2xi

uluk =gmn

xi

un2xi

uluk

=gmn ul

xiun

xiuk

2

xiulun

xiuk

=gmngnkul

un

xiul

xiuk

+

xiul

2xiunuk

=gmngnk

ul

glk

un +

ukxiul

xi

un

2xi

ukul

xi

un

=gmngnk

ul

glk

un +

gln

uk

xi

un2xi

ukul

...

C ili C di t S t & T A l i


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#

gk

ul =

ul(gkigi) = g

ki

ulgi+ gkigi

ul

gkj gjl=kl gkj

ungjl+ g

kjgjl

un =0

...

C r ilinear Coordinate S stems & Tensor Anal sis


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a=gl

ula= ?

case 1:

a = gl

ul(ak

gk) =glak

ul gk+ akgk

ul

= gl

am

ul gm+ a

k

m

k l

gm

=a

m

ul +m

k l

a

kg

l

gm=a

m

;lg

l

gm

am;l :=

am

ul +

m

k l

ak

2nd o.mixed t. Understand the rule (23)



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how to show thatam;lis a 2nd o. mixed t.?

am;lg

lgm= a=iia

xi

uu= am

;lglgm

am;l :=

am

ul +

m

k l

ak

a

m

;l=

um

up

uq

ul a

p

q (24)



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a=g

l

ula= ?

case 2:

a = gl

ul(akgk

) =glak

ul gk

+ akgk

ul

= gl

am

ul gmak

k

l n

gn

=am

ul k

m l

akg

l

gm=am;lgl

gm

am;l :=am

ul

k

m l

ak

2nd o.covariant t. understand the rule(25)



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how to show thatam;lis a 2nd o. covariant t.?

am;lglgm = a= ii

a

xi

uu= am;lg

lgm

am;l := amul

k

m l

ak

a

m;l=

up

um

uq

ul ap;q (26)



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Curvilinear Coordinate Systems & Tensor Analysisuseful formula:

diva := a= gk

uk[algl] =g

k

uk[algl]

= gk al;kgl=a

k;k (27)

2 := =gl

ul[gm

um] =gl

ul[,mg

m]

= gl(,m);lgm =glm(

2

ulum

k

m l

uk) (28)

curla := a=gk

uk(alg

l) =gk

uk(alg

l)

= gk(al;kgl) =al;kgkgl (29)

(b)a = (bmgm)gk

uka= bk

uk(algl)

= bk al;

k

gl (30)

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