1 cee 451g environmental fluid mechanics lecture 1: scalars, vectors and tensors a scalar has...

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CEE 451G ENVIRONMENTAL FLUID MECHANICS

LECTURE 1: SCALARS, VECTORS AND TENSORS

A scalar has magnitude but no direction.An example is pressure p.The coordinates x, y and z of Cartesian space are scalars.

A vector has both magnitude and directionLet denote unit vectors in the x, y and z direction. The hat denotes a magnitude of unity

The position vector (the arrow denotes a vector that is not a unit vector) is given as

k,j,i

x

kzjyixx

x

y

z

i j

kx

2


The velocity vector is given as

The acceleration vector is given as

The units that we will use in class are length L, time T, mass M and temperature °. The units of a parameter are denoted in brackets. Thus

u

kdt

dzj

dt

dyi

dt

dx

dt

xdu

a

kdt

zdj

dt

ydi

dt

xd

dt

xdk

dt

dwj

dt

dvi

dt

du

dt

uda

2

2

2

2

2

2

2

2

?]a[

LT]u[

L]x[1

�

Newton’s second law is a vectorial statement: where denotes the force vector and m denotes the mass (which is a scalar)

amF

F

2LT

3


The components of the force vector can be written as follows:

The dimensions of the force vector are the dimension of mass times the dimension acceleration

Pressure p, which is a scalar, has dimensions of force per unit area. The dimensions of pressure are thus

kFjFiFF zyx

2122 TML)L/(MLT]p[

The acceleration of gravity g is a scalar with the dimensions of (of course) acceleration:

2x MLT]F[]F[

2LT]g[

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A scalar can be a function of a vector, a vector of a scalar, etc. For example, in fluid flows pressure and velocity are both functions of position and time:

)t,x(uu,)t,x(pp

A scalar is a zero-order tensor. A vector is a first-order tensor. A matrix is a second order tensor. For example, consider the stress tensor .

zzzyzx

yzyyyx

xzxyxx

The stress tensor has 9 components. What do they mean? Use the following mnemonic device: first face, second stress

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Consider the volume element below.

x

y

z

Each of the six faces has a direction.For example, this faceand this faceare normal to the y direction

A force acting on any face can act in the x, y and z directions.

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x

y

z

yy yz


Consider the face below.

The face is in the direction y.

yx

The force per unit face area acting in the x direction on that face is the stress yx (first face, second stress).

The forces per unit face area acting in the y and z directions on that face are the stresses yy and yz.

Here yy is a normal stress (acts normal, or perpendicular to the face) and yx and yz are shear stresses (act parallel to the face)

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x

y

z

yy yz


Some conventions are in order

Normal stresses are defined to be positive outward, so the orientation is reversed on the face located y from the origin

yx

Shear stresses similarly reverse sign on the opposite face face are the stresses yy and yz.

yy

yz

yx

Thus a positive normal stress puts a body in tension, and a negative normal stress puts the body in compression. Shear stresses always put the body in shear.`

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Another way to write a vector is in Cartesian form:

)z,y,x(kzjyixx

The coordinates x, y and z can also be written as x1, x2, x3. Thus the vector can be written as

)x,x,x(x 321

or as

3..1i,)x(x i

or in index notation, simply as

ixx

where i is understood to be a dummy variable running from 1 to 3.

Thus xi, xj and xp all refer to the same vector (x1, x2 and x3) , as the index (subscript) always runs from 1 to 3.


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Scalar multiplication: let be a scalar and = Ai be a vector. Then

A

)A,A,A(AA 32ii

is a vector.

Dot or scalar product of two vectors results in a scalar:

scalarBABABABA 332211

In index notation, the dot product takes the form

3

1rrr

3

1kkk

3

1iii BABABABA

Einstein summation convention: if the same index occurs twice, always sum over that index. So we abbreviate to

rrkkii BABABABA

There is no free index in the above expressions. Instead the indices are paired (e.g. two i’s), implying summation. The result of the dot product is thus a scalar.

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Magnitude of a vector:

ii

2

AAAAA

A tensor can be constructed by multiplying two vectors (not scalar product):

333231

332221

131211

jiji

BABABA

BABABA

BABABA

3..1j,3..1i,)BA(BA

Two free indices (i, j) means the result is a second-order tensor

Now consider the expression

jji BAA

This is a first-order tensor, or vector because there is only one free index, i (the j’s are paired, implying summation).

)A,A,A)(BABABA(BAA 321232211jji That is, scalar times vector = vector.

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Kronecker delta ij

100

010

001

jiif0

jiif1ij

Since there are two free indices, the result is a second-order tensor, or matrix. The Kronecker delta corresponds to the identity matrix.

Third-order Levi-Civita tensor.

0

k,j,iif1

k,j,iif1

ijl

cycle clockwise: 1,2,3, 2,3,1 or 3,1,2

cycle counterclockwise: 1,3,2, 3,2,2 or 2,1,3

otherwise

Vectorial cross product:

kjijk BABxA

One free index, so the result must be a vector.

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Vectorial cross product: Let be given as

BxAC

Then

C

2

2

1

1

321

321

2

2

1

1

321

321

321

321

321

321

B

A

j

B

A

i

BBB

AAA

kji

B

A

j

B

A

i

BBB

AAA

kji

BBB

AAA

kji

BBB

AAA

kji

detC

kBABAjBABAiBABA 122131132332

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Vectorial cross product in tensor notation:

kjijki BAC

Thus for example

111112313232123kjjk11 BABABABAC= 1 = -1 = 0

a lot of other terms that all = 0

2332 BABA

i.e. the same result as the other slide. The same results are also obtained for C2 and C3.

The nabla vector operator :

321 xk

xj

xi

or in index notation

ix

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The gradient converts a scalar to a vector. For example, where p is pressure,

kx

pj

x

pi

x

pp)p(grad

321

or in index notation

ix

p)p(grad

The single free index i (free in that it is not paired with another i) in the above expression means that grad(p) is a vector.

The divergence converts a vector into a scalar. For example, where is the velocity vector,

k

k

i

i

3

3

2

2

1

1

x

u

x

u

x

u

x

u

x

u)u(div

u

Note that there is no free index (two i’s or two k’s), so the result is a scalar.

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The curl converts a vector to a vector. For example, where is the velocity vector,

u

kx

u

x

uj

x

u

x

ui

x

u

x

u

uuuxxx

kji

ux)u(curl

2

1

1

2

1

3

3

1

3

2

2

3

321

321

or in index notation,

j

kijk x

u)u(curl

One free index i (the j’s and the k’s are paired) means that the result is a vector

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A useful manipulation in tensor notation can be used to change an index in an expression:

ijij uu

This manipulation works because the Kronecker delta ij = 0 except when i = j, in which case it equals 1.


1 cee 451g environmental fluid mechanics lecture 1: scalars, vectors and tensors a scalar has...

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