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Unformatted text preview: Lecture 34: Hydrodynamics II (20 Nov 09) A. Review: density and momentum 1. Consider small volume element fixed in space, the change within, and the flows across the bounding surface. 2. Equation of continuity for mass density and stream velocity v : 0 = t + ~ v = t + v ~ + ~ v d dt + ~ v where d/dt is the hydrodynamic derivative following a fluid element. 3. Newtons law for acceleration of a fluid element by pressure gradient and body force f per unit mass: d v dt = v t + v ~ v = f- 1 P v ~ v = 1 2 v 2- v ( ~ v ) The latter identity on the derivative of v shows the simplification in cases with no circulation ( ~ v = 0). 4. Conservation law for momentum: momentum density in volume ele- ment dV is v , the change in the momentum in that element that arises from the net outgoing flux through the bounding area is:- Z A d ~ A v ( v ) =- Z V X j x j ( v j v ) dV 5. Define the stress tensor T ij = P ij + v i v j (both terms have dimensions of force/area = energy/volume). The momentum conservation law, including the change in momentum from gradients of the pressure (directed inward on the surface of the volume...
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