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Unformatted text preview: Chapter 2 Kinematics Kinematics is a study of the geometry of motion independently of applied forces. It consists of descriptive tools and various constraints that limit in important ways how the fluid can respond to the application of forces. We will learn about vectors, with velocity as the foremost example; about various algebraic, differential and integral operations on vectors; about such vectors as attributes of a particle (Lagrangian description) or of a point in space (Eulerian description); about rateofstrain and vorticity, maybe the most important new concept in the whole course; about mass balance and its consequences; and about flow decompositions, vortices as flow structures, and much more. Particularly relevant movies are: Klines Flow visualization, as back ground for all others; Lumleys Eulerian and Lagrangian descriptions, and Shapiros Vorticity. 2.1 Vectors Classical mechanics are firmly grounded in 3dimensional space, in which vectors are ubiquitous objects. Sure enough, scalar quantities such as energy and pressure are important, as seen in Bernoullis equation; but mechanics is dominated by the use of vectors for position, momentum, rotation, etc. Vectors are mathematical objects endowed with a direction and a mag nitude they may also have a point or line of application. In this course, we will use three equivalent notations for vectors, each with advantages de pending on the problem at hand. It is important that the student learn to 35 36 CHAPTER 2. KINEMATICS use these notations interchangeably: practice, practice... 2.1.1 Intrinsic notations The first notation, used in earlier chapters, makes no reference to any co ordinate system: the vectors exist independently of how they are described. This is the notation used in previous chapters, by default. A vector is de noted by an underline, as in velocity u . Operations on vectors include the multiplication by a number (scalar), e.g. u = u , (2.1) the dot product, which makes a scalar from two vectors taken in any order u 2 = u u (2.2) and the cross product, which makes a vector from two vectors, but reverses sign if the order of vectors is changed u = r = r . (2.3) The student is expected to know these operations from undergraduate classes, this presentation being included for the sake of notations and as a point of reference for alternative notations (below). Of a different nature, the vectoroperator is defined by its properties, although in many ways the component notation may be easiest to grasp. Assuming some background again, we simply note the gradient of a scalar function grad f = f, (2.4) the divergence of a vector div u = u , (2.5) the curl of a vector curl u = u , (2.6) and the Laplacian 2 = (2.7) (this last definition holds regardless of what the operator is applied to, vector or scalar). As an operator, only acts on the variables that follow it, therefore u 6 = u . (2.8) 2.1.2....
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 Spring '10
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