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Unformatted text preview: Chapter 3 Dynamics Newton’s second law of mechanics is a monumental achievement. It vali dates the concept of point particles and shows the relevance of forces; maybe more importantly, it brings mathematics (calculus) to the center of our un derstanding of mechanical phenomena (a program initiated by Archimedes of Syracuse in his palimpsest, and evolving through much of modern mathe matical physics). Essential mathematical concepts are vectors and differentials; additional mechanical ideas include stresses (in common with solid mechanics, under the umbrella of continuum mechanics). 3.1 Newtonian dynamics of continua The extension of Newton’s second law from point particles to continua is basis for both elasticity theory and advanced fluid dynamics. Euler took the first step in this direction, by combining the use of material derivatives with the knowledge (from fluid statics) that a pressure gradient is an internal force. Dividing by the uniform density in an incompressible flow 1 ∂ t u + u · ∇ u = 1 ρ ∇ p (3.1) 1 The distinction between an incompressible flow, in which there are no significant effects of compressibility, and the flow of an incompressible fluid, is important. Even water is not incompressible, and a study of incompressible fluids would run headon into the second law of thermodynamics; on the other hand, the compressibility of air can be ignored in lowspeed aerodynamics: the weaker assumption of incompressible flow is sufficient to allow for simple solutions. See Section 3.7 for details. 83 84 CHAPTER 3. DYNAMICS Figure 3.1: Freebody diagram of an elementary volume, and a simpler 2D variant used in the analysis or ∂ t u i + u j ∂ j u i = 1 ρ ∂ i p (3.2) Euler’s equation represents a leap away from material particles to the velocity field in space, already discussed in the previous chapter. In hindsight, Euler’s equation is seen as ignoring the effects of viscosity: an approximation. The next step was the work of Cauchy, who adopted as the basic object of analysis the collection of particles included in a volume element specified in some cartesian coordinates as dV = dx dy dz . The Lagrangian description is implied for now. The freebody diagram (Fig. 3.1) entails body forces (such as gravity, or the Lorentz force) applied at the center of mass, and surface forces, applied at the center of each surface, that account for interactions with the neighboring elements. The body and surface forces can be decomposed into their cartesian com ponents, normalized by the size of the volume or surface to which they are applied. In the case of surface forces, taking the outward normal as one of the local axes, the normal component for force per unit area is a familiar concept: negative pressure 2 . The tangential forces per unit area on each surface are the components of stress, one of the cornerstones of continuum 2 The negative sign comes from the simple consideration that the force in the equation is the force applied...
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 Spring '10
 XX
 Fluid Dynamics, Equations, Force, Stress, vorticity

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