Stability lecture 10

Stability lecture 10 - Hydrodynamic Stability Lecture 10...

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Hydrodynamic Stability Lecture 10 Convective (thermal) instability. Instability of a fluid layer heated from below. Basic Problem. 1. Assume initial instability is isentropic – heat transfer by conduction is relatively slow (steady state). 2. Temperature and density fluctuations are small. 0 0 1 TT T ± () ( ) 00 1 ρρ ρ α ==−−   T Taylor series expansion using these assumptions. 0 density at 0 z −= 0 0 const 0 T ρα =− = > - coefficient of volumetric expansion. Basic Backgroud state. () [ ] 0 ,1 x tz T B z ραβ =+ ² Only a small loss of entropy due to viscous effects. 1
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There is an entropy flux at boundaries, but the whole fluid doesn’t change state at all. Flow is incompressible. Incompressibility means 0 leads to 0 D Dt u ρ = ∇= i ± In the Boussinesq approximation we use u g t ² ± so that to a good approximation, for variable density flows, density differences only enter the momentum equation through buoyancy terms.
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Stability lecture 10 - Hydrodynamic Stability Lecture 10...

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