1

Mathematics ReviewA.1 Vectors

A.1.1 Definitions

A.1.2 ProductsA.1.2.1 Scalar Products A.1.2.2 Vector Product

V ． W = ｜ v ｜ ｜ W ｜ cos(V,W) = vxwx + vywy + vzwz

V x W = ｜ v ｜｜ W ｜ sin(V,W) nvw

V = vx ex + vyey + vzez

ex ey ez

vx vy vz

wx wy wz

=

2

A.2 Tensors

A.2.1 Definitions A tensor (2nd order) has nine components, for example, a stress tensorcan be expressed in rectangular coordinates listed in the following:

A.2.2 Product The tensor product of two vectors v and w, denoted as vw, is a tensor defined by

xx xy xz

yx yy yz

zx zy zz

τ

x x x x y x z

y x y z y x y y y z

z x z y z zz

v v w v w v w

v w w w v w v w v w

v w v w v wv

vw

[A.2-1]

[A.2-2]

Explanation (Borisenko, p64)

3

The vector product of a tensor and a vector v, denoted by ． v is a vector defined by

xxx xy xz

yx yy yz y

zx zy zz z

x xx y xy z xz x yx y yy z yz

x zx y zy z zz

x y

z

v

v v

v

v v v v v v

v v v

τ

e e

e[A.2-3]

xx x x y x z

y x y y y z y

z x z y z z z

x x x x y y x z z y x x y y y y z z

z x x z y y z z z

x y z x x y y z z

x y

z

x y z

nv v v v v v

v v v v v v n

v v v v v v n

v v n v v n v v n v v n v v n v v n

v v n v v n v v n

v v v v n v n v n

vv n

e e

e

e e e

v v n [A.2-5]

The product between a tensor vv and a vector n is a vector

4

The scalar product of two tensors and , denoted as , is a scalar defined by

: :xx xy xz xx xy xz

yx yy yz yx yy yz

zx zy zz zx zy zz

xx xx xy yx xz zx yx xy yy yy yz zy

zx xz zy yz zz zz

σ τ[A.2-6]

: :x x x y x z xx xy xz

y x y y y z yx yy yz

z x z y z z zx zy zz

x x xx x y yx x z zx y x xy y y yy y z zy

z x xz z y yz z z zz

v w v w v w

v w v w v w

v w v w v w

v w v w v w v w v w v w

v w v w v w

vw [A.2-7]

The scalar product of two tensors vw and is

Physical quantity Multiplication sign Scalar Vector Tensor None X : ‧Order 0 1 2 0 -1 -2 -4

Table A.1-1 Orders of physical quantities and their multiplication signs

5

x y zx y z

e e e

x y zs s s

sx y z

e e e

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

The gradient of a scalar field s, denoted as ▽s, is a vector defined by

[A.3-1]

[A.3-2]

A.3.2 Products

The divergence of a vector field v, denoted as ▽‧v is a scalar .

[A.3-5]

v x y z x x y y z z

x y z

v v vx y z

v v v

x y z

e e e e e e

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

6

x y z x x y y z z

x y z

x y zx y z

a av av avx y z

av av avx y z

v v v a a aa v v v

x y z x y z

v e e e e e eSimilarly

[A.3-5]

For the operation of [A.3-7] a a a v v v

For the operation of s, we have ▽‧▽

[A.3-8]

[A.3-9]

2

2

2

2

2

2

)eee()eee(z

s

y

s

x

s

z

s

y

s

x

s

zyxs zyxzyx

In other words ss 2Where the differential operator▽2, called Laplace operator, is defined as

2

2

2

2

2

22

zyx

[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.

Lines of constant are streamlines of the flow for inviscid irrotational flow in the xy plane ▽2=0

7

x y z

x y z

e e e

x y z

v v v

v

x y z

x y zx y z

x y z

v v v

x x x xv v v

v v vy y y y

v v v

z z z z

v

The curl of a vector field v, denoted by ▽x v, is a vector like the vector product of two vectors.

[A.3-11]

[A.3-12]

Like the tensor product of two vectors, ▽v is a tensor as shown:

x y x z y xx y zv v v v v v

e e ey z z x x y

When the flow is irrotational, v = 0

8

[A.3-13]

Like the vector product of a vector and a tensor, ▽‧ is a vector.

xx xy xz

yx yy yz

zx zy zz

yx xy yy zyxx zx

yzxz zz

x y

z

x y z

x y z x y z

x y z

τ

e e

e

x x x y x z

y x y y y z

z x z y z z

x x x y x z x

y x y y y z y

z x z y z z z

v v v v v v

v v v v v vx y z

v v v v v v

v v v v v vx y z

v v v v v vx y z

v v v v v vx y z

vv

e

e

e

[A.3-14]

From Eq. [A.2-2]

[A.3-15]It can be shown that vv v v v v

9

A.4 Divergence Theorem

A.4.1 Vectors

Let Ω be a closed region in space surrounded by a surface A and n the outward-directed unit vector normal to the surface. For a vector v

Ad dA

v v n [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

A.4.3 Tensors

For a scale s

For a tensor or vv

Asd s dA

n

A

d dA n

A

d dA vv vv n

[A.4-2]

[A.4-3]

[A.4-4]

10

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.

Fig. A.5-1(a)

A.5.1 Cartesian Coordinates

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.

x

y

z

dx

dy

dzP(x,y,z) ex

ey

ez

11Fig. A.5-1(b)

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]

Fig. A.5-1(b)*

v = er vr + eθvθ + ezvz

rr r rz

r z

zr z zz

τ

and

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:

12

Fig. A.5-1(c)

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 cos-y = r sin sin[A.5-7]z = r cos-

[A.5-9]

rr r rz

r z

zr z zz

τ [A.5-10]

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 τcan be expressed as follows:

vvvrr eeev

Fig. A.5-1(c)*

θ

φ φ

θ

θ φ

θ

θ

13

A.5.3 Differential Operators

1

0

0

rz

r z

zr z zz

r zr

rr r

r

r rr z

τ e

e e e

1

0

0

r

r

r

r r

rr r

r

r rr

τ e

e e e

1r z

s s ss

r r z

e e e

1 1

sinr

s s ss

r r r

e e e

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

[A.5-11] [A.5-12]

In curvilinear coordinates, assumes different forms depending on the orders of ▽the physical quantities and the multiplication sign involved. For example, in cylindrical coordinates

Whereas in spherical coordinates,

[A.5-13]

[A.5-14]

14

The equations for s, ▽ ▽‧v, x▽ v, 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 φ

θ

θ φ

θ

θ