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Stability lecture 32 - Hydrodynamic Stability Lecture 32...

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Hydrodynamic Stability Lecture 32 Inviscid instability of steady rotating flows Coordinate system – cylindrical ( ) ( ) ( ) , , , , or , , r z r z u v w v v v θ θ Incompressible, Inviscid Governing Equations 2 1 1. radial momentum 1 1 2. Asmuthial momentum 1 3. axial momentum 4. 1 5. 0 Du v p Dt r r D uv p V Dt r r Dw p Dt z D v u w Dt t r r z u u v w r r r z ρ ρ θ ρ θ θ = − + = − = − = + + + + + + = 1
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Conservation of mass. B.C.’s – assume steady state. ( ) ( ) 0, , u v r w r =  ± 1-5 reduce, for the mean flow to 2 1 v p r r ρ = − . A balance between centripital force and pressure force. Kinematic B.C. is satisfied for concentric cylinder geometry. Rayleigh’s Heuristic Argument: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) with 0 rewrite 2 1 1 or 1 0 for axisymmetric flow. is angular momentum. 0 says Angular momentum/unit mass is conser w r rv rv v v v v uv p r u w rv u rv w t r r z r t r r z D p rv Dt rv D rv Dt 1 θ θ ρ ρ θ = + + + + = + + +
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