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An Introduction to Modeling and Simulation of Particulate Flows Part 2

Tham khảo tài liệu 'an introduction to modeling and simulation of particulate flows part 2', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | ọ ọ 2007 5 15 page 1 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 vu u1v1 u2v2 u3v3 u v cos0 1.1 where u J u2 u2 W3 1.2 represents the Euclidean norm in R3 and 0 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 TO and v to u 0 and u 0 if and only if u 0 u v u v and yu Ịy u where Y is a scalar. Two vectors are said to be orthogonal if u v 0. The cross vector product of two vectors is u X v v X u e U1 v1 C-2 U2 v2 e3 U3 v3 u v sin 0 n 1.3 where n is the unit normal to the plane formed by the vectors u and v. e e e 2007 57 5 page 2 e 2 Chapter 1. Fundamentals The temporal differentiation of a vector is given by d u t iuẹ e .7 .2 e u e u 2e2 u . 1.4 dt dt dt dt The spatial gradient of a scalar a dilation to a vector is given by V e dt e . dx ƠX2 dx 1.5 The gradient of a vector is a direct extension of the preceding definition. For example Vu has components of d . The divergence of a vector a contraction to a scalar is defined by V u fep- e2-d e - u e M2 2 u e fl 1 l 2 Ị Ầ . ox ƠX2 0X3 J ox 0X2 ox Ị 1.6 The curl of a vector is defined as V X u e e e ã a a dx dx2 dx .7 u U2 u 1.2 Kinematics of a single particle We denote the position of a point in space by the vector r. The instantaneous velocity of a point is given by the limit r t At - r t dr v limAt . .