An Introduction to Modeling and Simulation of Particulate Flows Part 2 pot

05 book

2007/5/15

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Chapter 1

Fundamentals

When the dimensions of a body are insignificant to the description of its motion or the action

of forces on it, the body may be idealized as a particle, i.e., a piece of material occupying

a point in space and perhaps moving as time passes. In the next few sections, we briefly

review some essential concepts that will be needed later in the analysis of particles.

1.1 Notation

In this work, boldface symbols imply vectors or tensors. A fixed Cartesian coordinate

system will be used throughout. The unit vectors for such a system are given by the mutually

orthogonal triad (e1, e2, e3). For the inner product of two vectors u and v, we have in three

dimensions

u · v =

3

i=1

viui = u1v1 + u2v2 + u3v3 = ||u|||v|| cos θ , (1.1)

where

||u|| =

u2

1 + u2

2 + u2

3 (1.2)

represents the Euclidean norm in R3 and θ is the angle between the two vectors. We recall

that a norm has three main characteristics for any two bounded vectors u and v (||u|| < ∞

and ||v|| < ∞):

• ||u|| > 0, and ||u|| = 0 if and only if u = 0,

• ||u + v|| ≤ ||u|| + ||v||, and

• ||γ u|| ≤ |γ |||u||, where γ is a scalar.

Two vectors are said to be orthogonal if u· v = 0. The cross (vector) product of two vectors

is

u × v = −v × u =

e1 e2 e3

u1 u2 u3

v1 v2 v3

= ||u||||v||sin θ n, (1.3)

where n is the unit normal to the plane formed by the vectors u and v.

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