1

Continuum Mechanics The magnitude of the vector a is written a .

Scalar or dot product of two vectors

a •b = a b cos! where

! is the angle between

a and

b. It follows

from this definition that a •b = b •a and a + b( )• c = a • c + b • c .

Tensor product of two vectors

ab( )• c = a b • c( ) and c • ab( ) = c •a( )b . Sometimes written

a!b( )• c = a b • c( ) and c • a!b( ) = c •a( )b . It follows from this

definition that ab ! ba unless

a and

b are parallel. However

a + b( )c = ac + bc and a b + c( ) = ab + ac .

Vector or cross product of two vectors

c = a ! b = b • " •a in which ! is the permutation pseudotensor (see

below).

c is perpendicular to both

a and

b and has magnitude equal

to

a b sin! where

! is the angle from

a to

b looking along

c . It

follows from this definition that a ! b = "b ! a and

a + b( )! c = a ! c + b ! c .

Area as a vector

The area of each face of the tetrahedron is represented as an

outwards pointing vector perpendicular to the face and of

2

magnitude equal to the area of the face. Note that in looking along

the area vector 12v ! u we go clockwise from v to u .

The sum of the four areas, treating them as vectors is

12v ! u + 1

2u !w + 1

2w ! v + 1

2u "w( )! v "w( )

= 12v ! u + 1

2u !w + 1

2w ! v + 1

2u ! v " 1

2u !w " 1

2w ! v + 1

2w !w

= 0

since u ! v = "v ! u and w !w = 0 .

This result can be extended to any closed polyhedron by adding

tetrahedra – the vectors of the joing faces just cancel out. If we have

an infinitely large number of infinitely small tetrahedra, we have a

volume bounded by a smooth surface:

dA!V" = 0

in which the vector dA is an element of surface area and !V is the

entire surface enclosing the volume V .

General curvilinear coordinates

3

Basic relationships

r = X x1, x2, x3( )i + Y x1, x2, x3( )j + Z x1, x2, x3( )kg1 = r,1 = !r

!x1= !X

!x1i + !Y

!x1j + !Z

!x1k

g2 = r,2 = !r!x2

= !X

!x2i + !Y

!x2j + !Z

!x2k

g3 = r,3 = !r!x3

= !X

!x3i + !Y

!x3j + !Z

!x3k

gi •g j = ! ji =1 if i = j

= 0 if i ! j

"#$

%$ in which ! j

i is the Kronecker delta.

gi •g j = gij and gi •g j = gij .

g i are the covariant base vectors.

g i are the contravariant base vectors.

The curvilinear coordinates,

x i always have superscripts.

The unit tensor, I = gigi , and a = a • I = I•a for any vector, a .

An element of displacement, !r = ! xigi and ! s = !r gij! xi! x j .

Hence gij are known as the components of the metric tensor.

gikgjk = ! ji

! ijk = ! ji

k = gk • gi, j = "gk , j •g i

g = !123 = !231 = !312 = "!321 = "!213 = "!132 and !ijk = 0 unless

i ! j ! k ! i .

1g

= !123 = !231 = !312 = "!321 = "!213 = "!132 and

! ijk = 0 unless

i ! j ! k ! i .

g = g11g22g33 + g12g23g31 + g13g21g32 ! g13g22g31 ! g11g23g32 ! g12g21g33

g i ! g j = " ijkg k and

gi ! g j = "ijkgk .

v = vigi = v ig i where

vi = gijvj and

v i = g ijv j . The summation

i=1

3

! is

implied.

vi are the covariant components of

v.

4

v i are the contravariant components of

v.

grad! = !! = gi "!"xi

= gi!,i = gi!i!

!v = gi "v"xi

= giv,i = gi v jg

j( ),i= gi v j,ig

j + vjgj,i( ) = gi v j,ig j + vj g j,i •gk( )gk( )

= gi v j,igj # vj$ki

j gk( ) = gi v j,i # vk$ ijk( )g j

= !iv jgig j = v j,i + v

k$kij( )gig j = !iv

jgig j

divv = ! • v = gi • v,i = !ivjgi •g j = !iv

i

curlv = !" v = gi " v,i = !ivjgi " g j = !iv jg

i " g j = !iv j!ijkgk

Two dimensions

!12 = "!21 = g and !11 = !22 = 0 . !ij =1!ij

and

g = g11g22 ! g12g21 .

Stress

The force crossing an element of area !A is given by !F = !A •! .

The element of area is represented by the vector !A whose

direction is normal to the area and whose magnitude is equal to the

area. This is the definition of the stress ! which is a second order

tensor. Vectors are first order tensors and scalars are zeroth order.

Conservation of angular momentum (see below) tells us that the

stress tensor is symmetric, ! = !T

in which the ‘T’ means

transpose.

In curvilinear coordinates:

!F jg j = !Aigi • ! ajgag j( ) = !Ai!a

i" ajg j = !Ai"ijg j

!F j = !Ai"ij

.

In Cartesian coordinates:

5

!Fxi +!Fy j+!Fzk = !Axi +!Ay j+!Azk!" #$ •! xii +! xyij+! xzik +! yx ji +! y jj+! yz jk +! zxki +! zykj+! zkk

!

"

%%%

#

$

&&&

= !Ax! x +!Ay! yx +!Az! zx( )i + !Ax! xy +!Ay! y +!Az! zy( ) j+ !Ax! xz +!Ay! yz +!Az! z( )k

Applications

Conservation of mass

The rate of mass flowing through a stationary closed boundary is

equal to !v • dA"V! where

! is density and

v is fluid velocity. This

must be equal to minus the rate of change of the mass within the

boundary, ! ddt

! dVV"

#

$%&

'(= ! "!

"tdV

V" . Thus using the divergence

theorem, ! !"!tdV

V" = # • "v( )dV

V" . This must apply for any

boundary and therefore

!"!t

+ # • "v( ) = 0 .

In curvilinear coordinates !"!t

+!i "vi( ) = !"!t

+!i"vi + "!iv

i = 0

and in Cartesian coordinates

!"!t

+ !"!x

vx + !"!y

v y + !"!z

vz + " !vx

!x+!v y

!y+ !vz

!z

#

$ %

&

' ( = 0 .

Conservation of momentum

The net force applied by stress to the material across a stationary

closed boundary is dA •!!V" . If the body force per unit mass is p

then the total force applied to the material is dA •!!V" + pdV

V" . If

the only body force is own weight then p = !!gk .

Conservation of momentum applied to the material within the

boundary means that the total force must be equal to

ddt

!vdVV!

"

#$%

&'+ !vv • dA

!V! = !"

!tv + ! "v

!t"#$

%&' dV

V! + "vv • dA

!V! .

Therefore using the divergence theorem

! •" + p( )dVV# = !"

!tv + ! "v

"t$%&

'() dV

V# + ! • !vv( )dV

V# .

6

This must apply for any boundary and therefore

! •" + p = !"!tv + ! "v

"t+! • !vv( )

= !"!t

+ ! "v"t

+! • !v( )v + !v •!v .

Now applying conservation of mass, ! •" + p = ! "v"t

+ v •!v#$%

&'( in

which a = !v!t

+ v •!v is the acceleration vector.

In curvilinear coordinates !i!ij + p j = "a j = " #v j

#t+ vi!iv

j"#$

%&'

and

in Cartesian coordinates

!" x

! x+!# yx

! y+!# zx!z

+ px = $ax = $ !vx!t

+ vx!vx! x

+ vy!vx! y

+ vz!vx!z

!"#

$%&

!# xy

! x+!" y

! y+!# zy!z

+ py = $ay = $!vy!t

+ vx!vy! x

+ vy!vy! y

+ vz!vy!z

!"#

$%&

!# xz

! x+!# yz

! y+!" z

!z+ pz = $az = $ !vz

!t+ vx

!vz! x

+ vy!vz! y

+ vz!vz!z

!"#

$%&.

Conservation of angular momentum

The net moment about the point O caused by stress and body force

is r ! dA •"( )!V# + r ! pdV

V# = $ dA • " ! r( )

!V# $ p ! rdV

V# in which

we have used the scalar triple product and where

r is the ‘lever

arm’. Conservation of angular momentum applied to the material

within the boundary means that this must be equal to

ddt

!r ! vdVV"

#

$%&

'(+ dA • v! r ! v( )

!V"

= !"!tr ! v + !r ! !v

!t#$%

&'( dV

V" + dA • v! r ! v( )

!V"

.

Therefore using the divergence theorem

! " • # $ r( )dVV% ! p $ rdV

V%

= !"!tr $ v + !r $ "v

"t&'(

)*+ dV

V% + " • !v r $ v( )( )dV

V%

.

This must apply for any boundary and therefore

0 = ! • " # r( )+ p # r + !"!tr # v + !r # !v

!t+! • !v r # v( )( ) .

7

Using conservation of mass and conservation of momentum

0 = ! • " # r( )+ p # r $! • !v( )r # v+r # ! •" + p $ !v •!v( )+! • !v r # v( )( )= ! • " # r( )$! •" # r $ !r # v •!v( )+ !v! • r # v( )

.

The term

!!r " v •#v( )+ !v# • r " v( )= !!r •gk"kjmv

i#ivjgm + !vi r •gk"kjmv

jgm( ),i= !!r •gk"kjmv

i#ivjgm + !vigi •g

k"kjmvjgm + !vir • gk"kjmv

jgm( ),i= !!r •gk"kjmv

i#ivjgm + !vi" ijmv

jgm + !vir •gk"kjm#ivjgm

= 0

so that

0 = ! • " # r( )$! •" # r

= gi • " # r( ),i $ gi •",i # r= gi • " # gi( )=! ij" jikg

k

Therefore the stress tensor is symmetric, even in dynamics. Also,

provided that we take the stress tensor as symmetric, there is no

more information to be obtained by taking moments (although it’s

often very useful).

Deformation and strain The position vector of a material point in the initial or reference

configuration is

P X K( ) defined in terms of the curvilinear coordinate

system

X K which are known as the Lagrangian coordinates.

The position vector of the same material point in the deformed condition is

p xk( ) defined in terms of the curvilinear coordinate system

xk (which are

known as the Eulerian coordinates) and time

t .

This leads to the definition of the following vectors:

GK = !P!X K ,

g k = !p!xk ,

CK = !p!X K = !p

!xk

!xk

!X K = g k x k,K and

ck = !P!xk = !P

!X K

!X K

!xk = GK X K,k and hence to the definition of the

symmetric second order tensors with components

GKL = GK • GL ,

gkl = g k • g l ,

CKL = CK • CL = gklxk

,K x l,L and

ckl = ck • c l = GKL X K,k X L

, l .

8

Note that

X K,k x k

,L = !LK and

xk,K X K

, l = !lk so that if

X K,k and

xk,k are

written as matrices, one is the inverse of the other.

xk,k is the deformation

gradient and

X K,k is the inverse deformation gradient.

CKL are the components of the Green deformation tensor,

ckl , are the

components of the Cauchy deformation tensor,

EKL = ELK = 12

CKL !GKL( ) are the components of the Lagrangian strain

tensor and

ekl = elk = 12

gkl ! ckl( ) are the components of the Eulerian

strain tensor. There are also the components of the Piola deformation

tensor

C!1KL

= g kl X K,k X L

, l and the Finger deformation tensor

c!1kl

= GKL xk,K x l

,L .

dS is the distance between two adjacent material points in the reference

configuration and

ds is the distance between the same two material

points in the deformed condition. It follows from the above definitions that

dS2 = GKLdX K dX L = ckldxk dx l and

ds2 = gkldxk dx l = CKLdX K dX L .

There are a number of deformation measures: Cauchy

ds ! dSdS

, Swainger

ds ! dSdS

, Henky

logedsdS

!

" #

$

% & , Green

12

dsdS

!

" #

$

% &

2

'1(

) * *

+

, - - and Almansi

12

1! dSds

"

# $

%

& '

2(

) * *

+

, - - . Note that for small strain, when

dsdS

= 1+ ! as

! " 0 , all

the measures give the same result.

The Green deformation measure can be expressed as

12

dsdS

!"#

$%&2

'1(

)*

+

,- =

gkl ' ckl( )dxkdxl2cmndx

mdxn= ekldx

kdxl

2gmndxmdxn

=CKL 'GKL( )dXKdXL

2GMNdXMdXN = EKLdX

KdXL

2GMNdXMdXN

.

Fixed Cartesian coordinate system

Let us consider a fixed Cartesian system in which

G11 = G22 = G33 = g11 = g22 = g33 = 1 and

G12 = G23 = G31 = g12 = g23 = g31 = 0 . Also

CKL = x1,K x1

,L + x2,K x2

,L + x3,K x3

,L so that the Green deformation

measure becomes

9

x1,K x1,L + x

2,K x

2,L + x

3,K x

3,L( )dXKdXL ! dX1( )2 + dX 2( )2 + dX 2( )2"

#$%

2 dX1( )2 + dX 2( )2 + dX 2( )2"#

$%

.

Convected coordinate system

In a convected coordinate system the coordinates are dragged along with

the material. Hence

X 1 = x1,

X 2 = x2 and

X 3 = x3 and we can

dispense with using both capital and small letters for indices. Also

Ckl = gkl and

ckl = Gkl .

Then dS2 =Gkldxkdxl and ds2 = gkldx

kdxl so that

12

dsdS

!"#

$%&2

'1(

)*

+

,- =

! kldxkdxl

Gmndxmdxn

in which the componets of strain are

! kl =12gkl !Gkl( ) .

Cylindrical polar coordinates

Basic relationships

r

!

z

x

y

r

r = x1,

! = x2 and

z = x3.

r = x1 cos x2i + sin x2j( ) + x3k

g1 = cos x2i + sin x2j,

g2 = x1 !sin x2i + cos x2j( ) and

g3 = k .

g1 = cos x2i + sin x2j,

g2 =!sin x2i + cos x2j( )

x1 and

g3 = k .

10

g11 = 1,

g22 = x1( )2,

g33 = 1, and

gij = 0 if

i ! j .

g11 = 1,

g22 = 1

x1( )2 ,

g33 = 1 and

g ij = 0 if

i ! j .

g = x1( )2

g1,1 = 0

g1,2 = g2,1 = ! sin x2i + cos x2 j

g2,2 = !x1 cos x2i + sin x2 j( )g3,i = gi,3 = 0

All ! ijk = 0 except !22

1 = "x1 and !122 = !21

2 =1x1

.

Static equilibrium

!i!ij =! ij

,i +!kj"ki

i +! ik"kij = 0 if no body forces.

! 11,1 +!

21,2 +!

31,3 +

! 11

x1! x1! 22 = 0

! 12,1 +!

22,2 +!

32,3 + 3

! 12

x1= 0

! 13,1 +!

23,2 +!

33,3 = 0

where

! r =g11g11

! 11 =! 11

!" =g22g22

! 22 = x1( )2! 22 = r2! 22

! z =g33g33

! 33 =! 33

! r" =g11g22

# 12 = x1# 12 = r# 12

!"z =g22g33

# 23 = x1# 23 = r# 23

! zr =g33g11

# 31 =# 31

so that

11

!"r!r

+ 1r!#$r!$

+ !# zr!z

+ " r % "$r

= 0

!#r$!r

+1r!"$!$

+!# z$!z

+2r#r$ = 0

!#rz!r

+!#$z!$

+!" z!z

= 0

x

y

! r!"

#r"

Note stresses !z , !"z and !zr not shown on diagram.

Small strain elasticity of initially unstressed

body

Statics

!i!ij = 0 if no body forces.

Small strains

! ij = 12 gik! jv

k + gjk!ivk( ) = 1

2 !iv j +! jvi( ) where vi are

displacements.

Stress-strain relationships

!U = 12

!U!" ij

+ !U!" ji

!

"#$

%&!! ij ="

ij !! ij

! ij = 12

"U"# ij

+ "U"# ji

!

"#$

%& where U is the strain energy per unit volume.

Linear elasticity

U = 12Cpqmn! pq! mn

C pqmn = Cqpmn = Cmnpq

12

The components of stress, ! ij , are defined by the relationship

!U =" ij!# ij in which !U is the increment in strain energy

corresponding to the strain increments !" ij

!U = 12Cijmn!" ij" mn +

12Cijmn" ij!" mn

= 12Cijmn!" ij" mn +

12Cmnij" mn!" ij

= 12Cijmn +Cmnij( )! mn"! ij

= Cijmn! mn( )!" ij

Alternatively we can write

! ij = 12

"U"# ij

+ "U"# ji

!

"#$

%&

= 1212Cijmn + 1

2Cmnij + 1

2C jimn + 1

2Cmnji!

"#$%& # mn

= Cijmn# mn

.

In matrix notation

U = 12

! 11 ! 22 ! 33 2! 12 2! 23 2! 31!"

#$

C1111 C1122 C1133 C1112 C1123 C1131

C2211 C2222 C2233 C2212 C2223 C2231

C3311 C3322 C3333 C3312 C3323 C3331

C1211 C1222 C1233 C1212 C1223 C1231

C2311 C2322 C2333 C2312 C2323 C2331

C3111 C3122 C3133 C3112 C3123 C3131

!

"

%%%%%%%%

#

$

&&&&&&&&

! 11! 22! 332! 122! 232! 31

!

"

%%%%%%%%

#

$

&&&&&&&&

! 11

! 22

! 33

! 12

! 23

! 31

!

"

########

$

%

&&&&&&&&

=

C1111 C1122 C1133 C1112 C1123 C1131

C2211 C2222 C2233 C2212 C2223 C2231

C3311 C3322 C3333 C3312 C3323 C3331

C1211 C1222 C1233 C1212 C1223 C1231

C2311 C2322 C2333 C2312 C2323 C2331

C3111 C3122 C3133 C3112 C3123 C3131

!

"

########

$

%

&&&&&&&&

" 11" 22" 332" 122" 232" 31

!

"

########

$

%

&&&&&&&&

in which the square matrix is symmetrical. Therefore there are 21

independent elastic constants.

Isotropic linear elastic material

! = 1E

1+!( )" #! tr "( )I$% &'

! = E1+!( ) " + !

1# 2!tr "( )I$

%&'()

13

The trace, tr !( ) = gmn! mn and I is the unit tensor. E is Young’s

modulus and ν is Poisson’s ratio. These two equations are consistent

since

tr !( ) = 1E

1+!( )" 3!#$ %& tr '( ) = 1" 2!( )E

tr '( )

! = E1+!( )

1E

1+!( )! "! tr !( )I#$ %& +!

1" 2!1" 2!( )E

tr !( )I'()

*+,

= E1+!( )

1E

1+!( )! "! tr !( )I#$ %& +!Etr !( )I'

()*+, = !

In terms of components:

! mn =1E

1+"( )grmgsn !!grsgmn"# $%"rs

! ij = E1+"( )

!1! 2!

gijgmn + gimg jn"#$

%&' " mn

! x = " xx =1E

# x !$ # y +# z( )( ) = 1E 1+!( )! x !" ! x +! y +! z( )( )! y = " yy =

1E

# y !$ # z +# x( )( ) = 1E 1+!( )! y !" ! x +! y +! z( )( )! z = " zz =

1E

# z !$ # x +# y( )( ) = 1E 1+!( )! z !" ! x +! y +! z( )( )! xy = ! yx =

1+"( )! xy

E=! xy

2G

" yz = " zy =1+#( )! yz

E=! yz

2G

" zx = " xz =1+#( )! zx

E=! zx2G

Note the use of mathematical shear strain (one half the enfgineering

shear strain).

The shear modulus, G =E

2 1+ !( ) and the bulk modulus,

K =E

3 1! 2"( ) .

The above expression for stress in terms of strain can be rewritten

14

! ij = 2µgimg jn + "3gijgmn!

"#$%& # mn where ! and µ are the Lamé

parameters. Thus we still have two elastic constants to replace

Young’s modulus and Poisson’s ratio.

Two dimensions

Indices only in the range 1-2.

!i!ij = 0 is satisfied by ! ij = ! im! jn!m!n! = " im! jn !,mn "!,k#mn

k( )

since ! im! jn!m!n!i" = 0 . φ is the Airy stress function.

!im! jn" ij :mn =12 !

im! jn 12 vi: jmn + v j:imn( ) = 0 .

Plane strain

! 23 = ! 31 = ! 33 = 0 .

! ij =E

1 + "( )"

1# 2"gijgmn + gimg jn$

% & ' ( mn still applies but

! mn =1E

1+"( )grmgsn !!grsgmn"# $%"rs is replaced by

! mn =1+"E

grmgsn !"grsgmn( )! rs .

Plane stress

! 23 =! 31 =! 33 = 0 .

! mn =1E

1+"( )grmgsn !!grsgmn"# $%"rs still applies but

! ij = E1+"( )

!1! 2!

gijgmn + gimg jn"#$

%&' " mn is replaced by

! ij = E1+"( )

!1!!

gijgmn + gimg jn"#$

%&' " mn .

Therefore in plane stress

15

! im! jn!m!n" ij =! im! jn

E1+#( )grmgsn "#grsgmn#$ %&!m!n$

rs

= ! im! jn

E1+#( )grigsj "#grsgij( )! rp! sq!p!q!m!n%

= 1E1+#( )! im! jn! rp! sqgrigsj "#!

im! jn! rp! sqgrsgij( )!p!q!m!n%

= 1E1+#( )! im! jn! ir! jsg

rpgsq "#! ji!jn! sr!

sqgrpgim( )!p!q!m!n%

= 1E1+#( )gmpgnq "#gqpgnm( )!p!q!m!n%

so that gijgmn!i! j!m!n! = 0 .

This result applies for both plane stress and plane strain.

Plasticity

16

http://en.wikipedia.org/wiki/Von_Mises_yield_criterion

Tresca yield condition

! I !! II " Y

! II !! III " Y

! III !! I " Y

in which ! I , ! II ! III are the principal stresses and Y

is the yield stress in sumple tension.

von Mises yield condition

23Y 2 = ! I !

! I +! II +! III

3"#$

%&'2

+ ! II !! I +! II +! III

3"#$

%&'2

+ ! III !! I +! II +! III

3"#$

%&'2

= 23! I2 +! II

2 +! III2 ! ! I! II +! II! III +! III! I( )( )

=! I2 +! II

2 +! III2 ! 1

3! I +! II +! III( )2

= 13

! I !! II( )2 + ! II !! III( )2 + ! III !! I( )2()

*+

If ! is the stress tensor,

tr !( ) = gij! ij =! I +! II +! III

tr ! • !( ) = gingjm! ij! mn =! I2 +! II

2 +! III2

17

Hence

tr ! • !( )" 13tr !( )#$ %&

2=! I

2 +! II2 +! III

2 " 13! I +! II +! III( )2 = 2

3Y 2 .

In Cartesian coordinates tr !( ) = gij! ij =" x +" y +" z and

tr ! • !( ) = gingjm! ij! mn =" x2 +" y

2 +" z2 + 2! xy

2 + 2! yz2 + 2! zx

2 .

Hence

tr ! • !( )" 13tr !( )#$ %&

2

=! x2 +! y

2 +! z2 + 2! xy

2 + 2! yz2 + 2! zx

2 " 13! x +! y +! z( )2 = 23Y

2.

Fluid mechanics

Equations of motion

If no body forces,

! •" = ! #v#t

+ v •!v$%&

'()

!i"ij = ! #v j

#t+ vi!iv

j$%&

'()

!! x

!x+!" yx

!y+!" zx!z

= # !vx!t

+ vx!vx!x

+ vy!vx!y

+ vz!vx!z

"#$

%&'

!" xy

!x+!! y

!y+!" zy!z

= #!vy!t

+ vx!vy!x

+ vy!vy!y

+ vz!vy!z

"#$

%&'

!" xz

!x+!" yz

!y+!! z

!z= # !vz

!t+ vx

!vz!x

+ vy!vz!y

+ vz!vz!z

"#$

%&'

Strain rate

! = 12

"v + "v( )T( )! ij =

12

"iv j +" jvi( )

18

! x = " xx =!vx!x

= 12

!vx!x

+ !vx!x

"#$

%&'

! y = " yy =!vy!y

= 12

!vy!y

+!vy!y

"#$

%&'

! z = " zz =!vz!z

= 12

!vz!z

+!vz!z

"#$

%&'

" xy = " yx =12

!vy!x

+ !vx!y

"#$

%&'

" yz = " zy =12

!vz!y

+!vy!z

"#$

%&'

" zx = " xz =12

!vx!z

+!vz!x

"#$

%&'

Vorticity

! = 12

"v # "v( )T( )! ij =

12

"iv j #" jvi( )! k = 1

2" ijk"iv j

! is a small letter omega, ! .

! xy = !! yx =12

"vy"x

! "vx"y

#$%

&'(

! yz = !! zy =12

"vz"y

!"vy"z

#$%

&'(

! zx = !! xz =12

"vx"z

!"vz"x

#$%

&'(

Continuity Equation

!!!t

+ v •"! + !" • v = 0

!!!t

+ vi"i! +"ivi! = 0

!!!t

+ vx!!!x

+ vy!!!y

+ vz!!!z

+ ! !vx!x

+!vy!y

+!vz!z

#$%

&'(= 0

Viscous fluid

! = "pI + 2µ# " 23µtr #( )I

! ij = "gij p + 2µgimg jn" mn "23µgijgmn" mn

! x = "p + 2µ# x "23µ # x + # y + # z( )

! y = "p + 2µ" y "23µ " x + " y + " z( )

! z = "p + 2µ" z "23µ " x + " y + " z( )

! xy = ! yx = 2µ" xy

! yz = ! zy = 2µ" yz

! zx = ! xz = 2µ" zx

19

The trace, tr !( ) = gmn! mn .

Incompressible flow of viscous fluid

gij! ij = !ivi = 0

! ij = !gij p + 2µgimg jn! mn = !gij p + µ g jk"kvi + gik"kv

j( )

!i!ij = !gij"i p + µ g jk!k!iv

i + gik!k!ivj( ) = !gij"i p + µgik!k!iv

j

Therefore the equations of motion are

!gij"i p + µgik"k"ivj = ! #v j

#t+ vi"iv

j$%&

'()

which can also be written

!" j p + µgik"k"iv j = !#vj#t

+ vi"iv j$%&

'()

.

These equations, together with !ivi = 0 are the incompressible

Navier-Stokes equations.

Differentiating,

! jmn!m "! j p + µgik!k!iv j( ) = !" jmn!m

#vj#t

+ vi!iv j$%&

'()

µgik" jmn!m!k!iv j = !" jmn ##t

!mvj( )+!mvi!iv j + v

i!m!iv j$%&

'()

"2µgik!k!i!n = " "2 #!

n

#t+ # jmn!mv

i!iv j " 2vi!i!

n$%&

'()

.

However,

! jmn!mvi!iv j = ! jmn!iv

i!mvj " !jmn!rv

i!svj! imq!rsq

= 0 "!rvj!svj!

rsn +!rvn!svj!

rsj

= 2!rvn" r

so that

!! n

!t+ vi"i!

n = µ!gik"k"i! +"rv

n! r

D#Dt

= µ!"2# +# •"v

.

This is the vorticity transport equation for an incompressible

viscous fluid.

20

Appendix: Green-Gauss divergence theorem

In all the following the coordinate system is assumed to be

stationary – i.e. not convected.

! •F( )dVV" = F • dA

!V" where

F is a vector field,

!V is the surface

forming the boundary of the volume

V . The vector dA is an

element of the boundary in the direction of the outwards pointing

normal. See

http://mathworld.wolfram.com/DivergenceTheorem.html

Cartesian coordinates

!Fx! x

+!Fy! y

+!Fz!z

!"#

$%&dxdydz

V' = FxdAx + FydAy + FzdAz( )

!V'

= Fxdydz + Fydzdx + Fzdxdy( )!V'

Curvilinear coordinates

g j • Figi( ), j dVV! = Figi •g

j dAj!V! so that !iF

i dVV" = Fi dAi

!V" .

Second order tensors

g j • Qikgigk( ), j dVV! = g jdAj( )• Qikgigk( )

!V!

!iQikgk dV

V" = Qikgk dai

!V"

! •QdVV" = dA •Q

!V"