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# p09fl34 - Lecture 34 Hydrodynamics II(20 Nov 09 A Review...

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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 ρ and stream velocity v : 0 = ∂ρ ∂t + ~ ∇ · ρ v = ∂ρ ∂t + v · ~ ∇ ρ + ρ ~ ∇ · v ≡ dρ dt + ρ ~ ∇ · v where dρ/dt is the hydrodynamic derivative following a fluid element. 3. Newton’s 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 ρ ∇ 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 ρ 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 ( ρ v ) =- Z V X j ∂ ∂x j ( v j ρ v ) dV 5. Define the stress tensor T ij = Pδ ij + ρv 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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p09fl34 - Lecture 34 Hydrodynamics II(20 Nov 09 A Review...

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