CEE 262A HYDRODYNAMICS

Lecture 2Mathematical preliminaries and tensor

analysis

1

Right-handed, Cartesian coordinate system

3xz

1xx

2xy 3a

1a

2a

~x

ka

ja

ia

ˆ)1,0,0(

ˆ)0,1,0(

ˆ)0,0,1(

3

2

1

Unit vectors

Position vector3

32

21

1

~xaxaxax

~x),,( 321 xxxWhere are components of

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Vector: Kundu- “…Any quantity whose components change like the components of a position vector under the rotation of the coord. system.”

1 2 3~( )x x x x

Scalar: Any quantity that does NOT change with rotation or translation of the coord. system

e.g. density () or temperature (T)

3

Tensor: Assigns a vector to each direction in space ( 2nd order)

e.g. 11 12 13

21 22 23

31 32 33

A A AA A A A

A A A

ijA

Rows Columns

(a) Isotropic – Components are unchanged by a rotation of frame of reference (i.e. independent of direction - e.g. “Kronecker Delta ij”)

(b) Symmetric : Aij = Aji (in general Aij = ATji)

(c) Anti-symmetric: Aij = -Aji

(d) Useful result: Aij = 1/2 (Aij+Aij)+1/2 (Aji-Aji) = 1/2 (Aij+Aji)+1/2 (Aij-Aji) = Symmetric+ Anti-symmetric

4

Einstein summation conventionA) If an index occurs twice in a term a summation over the

repeated index is implied

B) Higher-order tensors can be formed by multiplying lower order tensors:a) If Ui and Vj are 1st-order tensors then their product Ui Vj = Wij is a 2nd-order tensor. Also known as vector outer product ( ).b) If Aij and Bkl are 2nd-order tensors then their product Aij Bkl = Pijkl is a 4th-order tensor.

e.g. 11 22 33

31 2

1 2 3

/

ii

i i

U U U UUU UU X

X X X

* Result is a scalar quantity When a summation occurs over a repeated index contraction

VU

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C) Lower-order tensors are obtained from contractions

kjijkiklij

ABABBA

(b) Tensor multiplied by a vector

i

jij

uAuA~

CBAjiij

(a) Contraction of two 2nd-order tensors

CAB

:

(c) Double-contraction of two 2nd-order tensors

6

D) Kronecker delta ij

ijij UU

ij =1 if i=j=0 otherwise

100010001

Isotropic tensor of 2nd order

*

RHSUUU lll 332211 Expand:

If 31

ll

3

1

URHSURHS

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E) Alternating tensor ijk

= 1 ijk in cyclic order e.g. 123,231,312= -1 ijk in anticyclic order e.g. 321,132,213= 0 if any two indices are equal

ijk

(a) An index moved 2 places to right / left won’t change value

jkiijk or ijkkij

(b) An index moved 1 place to right / left will change sign

ikjijk

(c) Epsilon-Delta Relation

jlimjmilklmijk

FLOI = first(il)last (jm) - outer(im)inner(jl)8

~~~~//0 baba

~~~~0 baba

Basic vector operations

A) Dot Product (“Inner” Product)

* “Magnitude of one vector times component of other in direction of first vector”

UV

cosUV~~VU

332211~~~~VUVUVUVUbUVVU ii

Vector . Vector = Scalar

implies

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B) Outer product ( )

ijkkij cbaii )(

e.g. ijji cvui )(

~~VUC) Cross product

U

V～W

“…is the vector, , whose magnitude is , and whose direction is perpendicular to the plane formed by and such that

form a right-handed system”

sin UV~W

~~~,, WVU

~V

~U

VU

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Cross-product rules

3

~12212

~31131

~2332~~)()()( aVUVUaVUVUaVUVUVU

1 2 3

~ ~ ~

1 2 3~ ~

1 2 3

a a a

U V U U UV V V

(a)

(b)

(c)

ijk ikj jik

~ ~ ~ ~U V V U

Now since

~~~~//0 baba

~~~~0 baba

If:

~~~WWVUVU kjiijk

k

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1 2 3 i

~ ~ ~ ~1 2 3 i

a a a ax x x x

A) Gradient – “Grad”: increase tensor order

The “Del” ( ) operator

Vector

1C

3C2C

“…( ) is perpendicular to lines and gives magnitudeand direction of maximum spatial rate of change of ”

C

iix

j

i

UU

x

If we apply to a vector, we produce a second-order tensor

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B) Divergence – “div”: Reduce tensor order

3i 1 2~ ~

i 1 2 3

UU U UdivU Ux x x x

[ Scalar ]

j

ij

i xU

U

[ Vector ]e.g.

div is not defined

Our application will be to the divergence of a flux of various quantities.

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C) Curl

~ ~curl U U

kijk i~ ~i j

UUx

e.g. 0ijk

3 32 21 123 132

2 3 2 3

U UU Ux x x x

If j=1 or k=1i=1:

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Important div/grad/curl identities

(b)

(c)

(a)

23,321 orkorjiIf

023

2

13232

2

123

xxxx

+1 -1

If is a scalar 0)(

kjijk xx

jkkj xxxx

22

ikjijk Now But

Solenoidal

UUIf

~~

0

~~~~0 baba

~~~~//0 baba

alIrrotation

UUIf

~~

//0

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(d) 0)(~

U

02

ji

kijk

j

kijk

i xxU

xU

x

“Curl of a vector is non divergent”

(see above)

(e)~ ~ ~ ~ ~ ~

( ) ( 2) a a a a a a

~~baLet

l

mjklmijk

l

mklm

kk

kjijki

xaaaa

xaabAnd

baba

~~

~

~~

But: jlimjmilklmijk

16

l

mjjlim

l

mjjmil x

aaxaaaa

~~

li 1 if

mj 1 if

mi 1 if

lj 1 if

~~

~~

~

2

)()2/(

aaaa

aaaax

xaa

xaa

jjmmi

j

ij

i

mm

ja)(~

We will make good use of this result!

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A) Gauss’ Theorem

Integral theorems

~n

dVdA

AAreaSurfaceVVolume

(outward unit normal vector to surface element)

(infinitesimal surface area)

(infinitesimal volume)

“…Relates a volume integral to a surface integral A ” V

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V A ~i

dV n dAQ Qx

If is a scalar, vector, or any order tensor)(Q x

Specifically, if is a vector)(Q xQ

Q Q

ii iV A

i

dV n dAx

~

V AQdV Q ndAor

“Divergence Theorem”: Integral over volume of divergence of flux = integral over surface of the flux itself 19

V A ~q dV q n dA

Examples… (a)

V A ~T dV T n dA

(b)

(c) If a scalar is (advectively) transported

by the velocity ~U

)(~~ i

i

advadvUFUF

A advVi

iadv dAnFdVxF

~~

AV adv

dAnFdVF~~~

or

Divergence of flux within volume = Net flux across20

~n

ds

dA

C (bounding curve)

A (open surface)

B) Stokes' Theorem

“…Surface integral of the curl of a vector, ~U , equals

the line integral of ~U along the bounding curve”

~ ~ ~( ) A C

u ndA u ds

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