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Lecture 2 Mathematical preliminaries and tensor analysis
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Transcript of Lecture 2 Mathematical preliminaries and tensor analysis
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
2
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
5
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
7
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
9
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
10
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
11
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
12
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.
13
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:
14
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
15
(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!
17
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
18
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
21