A.1 Vectors
Mathematics Review
V = vx ex + vyey + vzez
A.1.1 Definitions
A.1.2 Products
A.1.2.1 Scalar Products
A.1.2.2 Vector Product
V x W = ｜v｜｜W｜sin(V,W) nvw
=
V．W = ｜v｜ ｜W｜cos(V,W)
= vxwx + vywy + vzwz
ex
ey
ez
vx
vy
vz
wx
wy
wz
1
A.2 Tensors
A.2.1 Definitions
A tensor (2nd order) has nine components, for example, a stress tensor
can be expressed in rectangular coordinates listed in the following:
  xx  xy  xz 


τ   yx  yy  yz 


 zx  zy  zz 
[A.2-1]
A.2.2 Product
The tensor product of two vectors v and w, denoted as vw, is a tensor defined by
 vx 
 vx wx vx wy vx wz 
 


vw   v y  wx wy wz   v y wx v y wy v y wz 
[A.2-2]
v w v w
v 
vz wz 
z y
 z x
 z


Explanation (Borisenko, p64)
2
The vector product of a tensor  and a vector v, denoted by ．v is a vector defined by
  xx  xy  xz   vx 

  
τ  v   yx  yy  yz    v y 

  
 zx  zy  zz   vz 
 ex  vx xx  v y xy  vz xz   ey  vx yx  v y yy  vz yz 
ez  vx zx  v y zy  vz zz 
[A.2-3]
The product between a tensor vv and a vector n is a vector
 vx vx

vv  n   v y vx
v v
 z x
vx v y
vy vy
vz v y
v x v z   nx 
  
v y vz    n y 
vz vz   nz 
 ex  vx vx nx  vx v y n y  vx vz nz   ey  v y vx nx  v y v y n y  v y vz nz 
ez  vz vx nx  vz v y n y  vz vz nz 
  exvx  eyv y  ezvz  vx nx  v y n y  vz nz 
 v  v n
[A.2-5]
3
The scalar product of two tensors s and , denoted as s:, is a scalar defined by
 s xx s xy s xz    xx  xy  xz 

 

σ : τ   s yx s yy s yz  :  yx  yy  yz 
[A.2-6]
s
 

 zx s zy s zz    zx  zy  zz 
 s xx xx  s xy yx  s xz zx  s yx xy  s yy yy  s yz zy
s zx xz  s zy yz  s zz zz
The scalar product of two tensors vw and  is
 vx wx vx wy vx wz    xx  xy  xz 

 

vw :    v y wx v y wy v y wz  :  yx  yy  yz 
 v w v w v w     
 z x z y z z   zx zy zz 
 vx wx xx  vx wy yx  vx wz zx  v y wx xy  v y wy yy  v y wz zy
[A.2-7]
vz wx xz  vz wy yz  vz wz zz
Table A.1-1 Orders of physical quantities and their multiplication signs
Order
Physical quantity
Scalar Vector Tensor
0
1
2
Multiplication sign
None
X
‧
0
-1
-2
:
-4
4
A.3 Differential Operators
A.3.1 Definitions
The vector differential operation , called “del”, has components similar to those
of a vector. However, unlike a vector, it cannot stand alone and must operate on
a scalar, vector, or tensor function. In rectangular coordinates it is defined by



  ex  ey  ez
x
y
z
[A.3-1]
The gradient of a scalar field s, denoted as ▽ s, is a vector defined by
s  ex
s
s
s
 ey  ez
x
y
z
[A.3-2]
A.3.2 Products
The divergence of a vector field v, denoted as ▽‧v is a scalar .
 

 
  v   ex  ey  ez    exvx  eyvy  ezvz 
y
z 
 x
vx vy vz



x y z
[A.3-5]
Flux is defined as the amount that flows through a unit area per unit time
Flow rate is the volume of fluid which passes through a given surface per unit time
5

Similarly    av    ex




 
 ey  ez    exavx  eyavy  ezavz 
x
y
z 



 avx    avy    avz 
x
y
z
[A.3-5]
 vx vy vz   a
a
a 
 a



v
x

v
y

v
z
 

y
z 
 x y z   x
For the operation of   av   a   v   v a
[A.3-7]
For the operation of ▽‧▽s, we have



s
s
s
2s 2s 2s
  s  (e x  e y
 e z )  (e x  e y
 ez )  2  2  2
x
y
z
x
y
z
x y
z
In other words
[A.3-8]
  s   2 s
[A.3-9]
Where the differential operator▽2, called Laplace operator, is defined as
2
2
2



2  2  2  2
x y z
[A.3-10]
For example: Streamline is defined as a line everywhere tangent to the velocity
vector at a given instant and can be described as a scale function of f.
Lines of constant f are streamlines of the flow for inviscid irrotational flow
in the xy plane ▽2f=0
6
The curl of a vector field v, denoted by ▽ x v, is a vector like the vector product
of two vectors.
ex ey
 
 v 
x y
vx vy
ez
  ex  vx  vy   ey  vx  vz   ez  vy  vx 






 z x 
z
 y z 
 x y 
vz
When the flow is irrotational,
[A.3-11]
 v = 0
Like the tensor product of two vectors, ▽v is a tensor as shown:
  
 vx
 


x
 
 x
  
 vx
v     vxvyvz   
y
y
 

 
 vx
 

 z 
 z
vy
x
vy
y
vy
z
vz 

x 
vz 
y 

vz 

z 
[A.3-12]
7
Like the vector product of a vector and a tensor, ▽‧ is a vector.
  xx  xy  xz 
     

τ  




yy
yz 
  yx
 x y z  

  zx  zy  zz 
  xx  yx  zx 
  xy  yy  zy 
 ex 




  ey 


x

y

z

x

y
z 



  xz  yz  zz 
ez 




x

y
z 

[A.3-13]
From Eq. [A.2-2]
  vxvx  vxvy  vxvz 
     

    vv   
    vyvx  vyvy  vyvz 
 x y z    vzvx  vzvy  vzvz 


 



 ex    vxvx     vyvx     vzvx  
y
z
 x

 



ey    vxvy     vyvy     vzvy  
y
z
 x

 



ez    vxvz     vyvz     vzvz  
y
z
 x

It can be shown that
   vv   v    v    v v
[A.3-14]
[A.3-15]
8
A.4 Divergence Theorem
A.4.1 Vectors
Let Ω be a closed region in space surrounded by a surface A and n the outwarddirected unit vector normal to the surface. For a vector v


  vd    v  ndA
A
[A.4-1]
This equation , called the gauss divergence theorem, is useful for converting from a
surface integral to a volume integral.
A.4.2 Scalars
For a scale s


sd    sndA
A
[A.4-2]
A.4.3 Tensors
For a tensor  or vv
   d   A   ndA
[A.4-3]
  vv d     vv  ndA
[A.4-4]

A
9
A.5 Curvilinear Coordinates
For many problems in transport phenomena, the curvilinear coordinates such as
cylindrical and spherical coordinates are more natural than rectangular coordinates.
A point P in space, as shown in Fig. A.5.1, can be represented by P(x,y,z) in
rectangular coordinates, P(r, θ,z) in cylindrical coordinates, or P(r, θ,ψ) in
spherical coordinates.
A.5.1 Cartesian Coordinates
ey
y
For Cartesian coordinates, as shown
in A.5-1(a), the differential increments
of a control unit in x, y and z axis are
dx, dy , and dz, respectively.
dy
P(x,y,z)
ez
dz
dx
ex
x
z
Fig. A.5-1(a)
10
A.5.1 Cylindrical Coordinates
For cylindrical coordinates, as shown in A.5-1(b), the variables r, θ, and z are
related to x, y, and z.
x = r cosθ [A.5-1]
y = r sinθ [A.5-2]
z=z
[A.5-3]
The differential increments of a control unit, as shown in Fig. A.5-1(b)*, in r, ,
and z axis are dr, rd , and dz, respectively. A vector v and a tensor τcan be
expressed as follows:
v = er vr + eθvθ + ezvz
and
  rr  r  rz 


τ    r     z 


 zr  z  zz 
Fig. A.5-1(b)
Fig. A.5-1(b)*
11
A.5.2 Spherical Coordinates
For spherical coordinates, as shown in A.5-1(c), the variables r, θ, and ψ are
related to x, y, and z as follows
x= r sin cosf
A.5-6
y = r sin sinf
[A.5-7]
z = r cos
A.5-8
Fig. A.5-1(c)
The differential increments of a control
unit, as shown in Fig. A.5-1(c)*, in r, θ,
and φ axis are dr, rdθ , and rsinθdφ ,
respectively. A vector v and a tensor τ
can be expressed as follows:
v  e r vr  e v  ef vf
  rr  r  rz 


τ    r     z 


 zr  z  zz 
[A.5-9]
θ φ
θ
θ
θ
θ
[A.5-10]
φ
Fig. A.5-1(c)*
φ
12
A.5.3 Differential Operators
Vectors, tensors, and their products in curvilinear coordinates are similar in form
to those in curvilinear coordinates. For example, if v = er in cylindrical coordinates,
the operation of τ．er can be expressed in [A.5-11], and it can be expressed in
[A.5-12] when in spherical coordinates
  rr  r  rz  1 

  
τ  er    r     z    0 

  
 zr  z  zz   0 
 er rr  e  r  ez zr
[A.5-11]
  rr  r  r  1 

  
τ  er     r        0  [A.5-12]

  
  r       0 
 er rr  e  r  e  r
In curvilinear coordinates, ▽ assumes different forms depending on the orders of
the physical quantities and the multiplication sign involved. For example, in cylindrical
coordinates
s  e r
s
1 s
s
 e
 ez
r
r 
z
[A.5-13]
Whereas in spherical coordinates,
s  er
s
1 s
1 s
 e
 ef
r
r 
r sin  f
[A.5-14]
13
The equations for ▽s, ▽‧v, ▽ xv, and ▽2s in rectangular, cylindrical, and
spherical coordinates are given in Tables A.5-1, A.5-2, and A.5-3, respectively.
θ φ
θ
θ
θ
φ
θ
φ
14