Mixing in thermal convection

Viscoelastic RT turbulenceFluid model:Boussinesq+Oldroyd-B⎧∂ tu + u ⋅ ∇u = −∇p + νΔu − βgT + (2νη / τ p)∇ ⋅ σ⎪⎨∂ tT + u ⋅ ∇T = κΔT⎪∂ σ + u ⋅ ∇σ = (∇u) T ⋅ σ + σ ⋅ (∇u) − (2 / τ )(σ − 1)⎩⎪t pPolymer conformation tensorσ ij =/R 02Coil-stretch transitionE = P + K + Σ = −β g zT + 1 2u 2+ νητptr(σ ) − 3Scales in RT turbulencerL~t 2polymers become stretched(and active) when ∇u > 1 / τ pi.e. when τ pWi = > 1τ ηr L ~t 1/2η~t -1/4tLumley scaleτ p /τ(r L )=1Weissenber number grows as Wi≈t 1/2