Class 04 Handout

7/25/2019 Class 04 Handout

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Fluid Mechanics AS102

Class Note No: 04

Monday, August 6, 2007


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Review: Curvilinear C. S. & Tensor Analysis

introduce curvilinear coordinate systems,

like cylindrical c.s.,

spherical c.s. purpose:

to meet the need of writing equations of motion incurvilinear c.s.


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example: cylindrical c.s.

x1 =rcos, x2 =rsin, x3 =z,r=

(x1)2 + (x2)2, =tan

1 x2x1

, z=x3

O

x1

x2

x3

r

P

r

z

i1i2

i3

gr

ggz r u1 =u1(x1, x2)

u2 =u2(x1, x2)zu3 =u3(x3)

Figure:cylindrical coordinates


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tangent to the curve of uk:

gk := r

uk =

(xiii)

uk =

xiuk

ii, k=1, 2, 3 (2)

called thecovariant/ natural base vector foruk, k=1, 2, 3

(g1, g2, g3)provides a base for the coordinates(u1,u2,u3)

ii=uk

xigk (3)

gkgl= xiuk

xiul

(4)


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normal to the surface of uk =const:

gk := uk =uk

xiii (5)

contravariant/ dural base vector for uk, k=1, 2, 3

(g1, g2, g3)provides another base for the coordinates(u1,u2,u3)

gkgl =lk=kl (6)

gkgl =uk

xi

ul

xi(7)


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themetric tensor gij:

ds2 :=dxdx=gkldukdul,

gkl :=gkgl= xiuk

xiul

=glk (8)

covariant, symmetric

theconjugate metric tensor gkl:

gkl :=gkgl = uk

xi

ul

xi=glk

gklglm=km (9)

contravariant, symmetric

the inverse ofgij.

memorize these two definitions


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gk=gklgl, gk =gkl gl (10)

# gij canlowercontravariantindices to covariantindices

# gij canraisecovariantindices to contravariantindices

# they cancreatenew type tensors from the given (morelater)


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in a general curvilinear coordinate system,

several sets of base vectors:

# covariant/natural base (from the tangents)

(g1, g2, g3)

# contravariant/dural base (from the normals)

(g1, g2, g3)

(the bases may not be the same;e.g. g1 not parallel tog

1 ...)


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in a rectangular coordinate system,

#

(g1, g2, g3) (i1, i2, i3), (g1, g2, g3)(i1, i2, i3)

gij ij, g

ij

ij

=ij


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Curvilinear Coordinate Systems & Tensor Analysis

Todays topic:

# tensor transformations


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Specific examples of finding gi, gi, gij, g

ij:

# cylindrical coordinate systems

# spherical coordinate systems

(last homework assignment)


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cylindrical coordinate system:

O

x1

x2

x3

r

P

u1u2

u3

i1

i2

i3g1

g2g3

Figure:cylindrical coordinates


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x1 =u1cos(u2), x2 =u

1sin(u2), x3 =u

3

xiuj

=

cos(u2) u1sin(u2) 0sin(u2) u1cos(u2) 00 0 1

,g1 =cos(u

2)i1+sin(u2)i2,

g2 =u1cos(u2)i1+u1sin(u2)i2,

g3 =i3


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[gkl] =1 0 0

0 (u1

)2

00 0 1

=

1 0 00 r

2

00 0 1

# gkl=0, k =l, calledorthogonal

# memorizethe relation for gij

C ili C di S T A l i


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u1 =

(x1)2 + (x2)2, u2 =tan1

x2x1

, u3 =x3

uk

xl

=

x1/u1 x2/u1 0x2/(u1)2 x1/(u1)2 0

0 0 1

g1 =g1, g2 =

1(u1)2

g2, g3 =g3

only onebase for the cylin. c.s.

C ili C di S & T A l i


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gkl

=1 0 0

0 (u1

)2

00 0 1

=

1 0 00 r

2

00 0 1

# the inverse of gij

# memorizethis relation

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


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spherical coordinate system:

O

x1

x2

x3

P

u1u2

u

3i1

i2i3

g1

g2

g3

Figure:spherical coordinates

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


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x1 =u1sin(u2)cos(u3), x2=u

1sin(u2)sin(u3),

x3 =u1cos(u2)

xiuj

=

sin(u2)cos(u3) u1cos(u2)cos(u3) u1sin(u2)sin(u3)

sin(u2)sin(u3) u1cos(u2)sin(u3) u1sin(u2)cos(u3)cos(u2) u1sin(u2) 0

g1 =sin(u

2)cos(u3)i1+sin(u

3)i2

+cos(u2)i3,

g2 =u1cos(u2)

cos(u3)i1+sin(u

3)i2

u1sin(u2)i3,

g3 =u1sin(u2)sin(u3)i1+cos(u3)i2

C r ilinear Coordinate S stems & Tensor Anal sis


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[gkl] =1 0 00 (u1)2 0

0 0 (u1sin(u2))2

= 1 0 00 r2 00 0 (rsin)2

# gkl=0, k =l, orthogonal



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u1 =

(x1)2 + (x2)2 + (x3)2,

u2 =cos

1 x3

(x1)2 + (x2)2 + (x3)2

,

u3 =tan1x2

x1

.

uk

xl

= ?



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gkl= gij1

= 1 0 0

0 (u1)2 0

0 0 (u1sin(u2))

2

g1 =g11g1, g2 =g22g2, g

3 =g33g3

only onebase for the spher. c.s.

only onebase for a curvilinear c.s. with orthogonal gij



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if we have a cylindrical c.s. & a spherical c.s.

x1

x2

x3

rP

u1u2

u3g1

g2

g3

u1u2

u3

g1

g2

g3

Figure:cylindrical & spherical coordinate systems sketch



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# between(xi)& cylindrical c.s. ui,u1 =

(x1)2 + (x2)2, u

2 =tan1x2

x1, u3 =x3

# between(xi)& spherical c.s.ui

, (bit off from thesketch)

x1=u1sin(u2)cos(u3), x2 =u

1sin(u2)sin(u3),

x3=u1cos(u2)

# (spher.c.s.) & (cylind.c.s.) linked:

u1 =u1sin(u2), u2 =u3, u3 =u1cos(u2);

u1

=

(u1

)

2

+ (u2

)

2

, u2

=tan1 u

1

u3 , u3

=u2

.



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generalize to arbitrary(ui), (ui)

xi=xi(uj

), xi=x

i(uj

);ui =ui(uj), ui =ui(uj) (11)

(with the help of chain rule)

gk = um

uk gm, gk = u

k

um gm,

gkl= um

ukun

ulgmn,

gkl

=

uk

umul

un gmn

(12)

# 2nd order covariant tensor & 2nd order contravarianttensor

# understand the transformation rules



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0th order tensors (scalars):

= (xi) = (ui) = (ui) (13)



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1st order tensors (vectors):

a = aiii=ai(u)gi=ai(u)g

i

= ak(u)gk =a

i(u)gi (14)

Hereurepresentsum andu representsum.

(HW)

ai(u) = ui

umam(u) =

ui

xmam

ai(u) =

um

uiam(u) =

xmui

am

ai

(u) =gij

aj(u)ai(u) =gija

j(u) (15)

# covariant tensor & contravariant tensor

# understand the transformation rules



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other 2nd order tensors:

A = Aijiiij=Aij

(u) gigj=Aij(u) gi

gj

=Aijgig

j

= Aij(u) gi g

j =A

ij(u) gi gj =Ai

jg

i gj (16)

(HW)

Aij(u) = ui

umuj

unAmn(u) =

ui

xm

uj

xnAmn

Aij(u) =

um

uiun

ujAmn(u) =

xmui

xnuj

Amn

Aij(u

) = ui

umu

n

ujAmn(u) = u

i

xmxnuj

Amn

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

Aij(u) =g

imAmj(u) =gjnAin(u) ... (17)



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tomorrows topic:

tensor differentiations

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