Aspect Ratio Dependence of Heat Transfer in a Cylindrical Rayleigh-Bénard Cell.

While the heat transfer and the flow dynamics in a cylindrical Rayleigh-Bénard (RB) cell are rather independent of the aspect ratio Γ (diameter/height) for large Γ, a small-Γ cell considerably stabilizes the flow and thus affects the heat transfer. Here, we first theoretically and numerically show that the critical Rayleigh number for the onset of convection at given Γ follows Ra_{c,Γ}∼Ra_{c,∞}(1+CΓ^{-2})^{2}, with C≲1.49 for Oberbeck-Boussinesq (OB) conditions. We then show that, in a broad aspect ratio range (1/32)≤Γ≤32, the rescaling Ra→Ra_{ℓ}≡Ra[Γ^{2}/(C+Γ^{2})]^{3/2} collapses various OB numerical and almost-OB experimental heat transport data Nu(Ra,Γ). Our findings predict the Γ dependence of the onset of the ultimate regime Ra_{u,Γ}∼[Γ^{2}/(C+Γ^{2})]^{-3/2} in the OB case. This prediction is consistent with almost-OB experimental results (which only exist for Γ=1, 1/2, and 1/3) for the transition in OB RB convection and explains why, in small-Γ cells, much larger Ra (namely, by a factor Γ^{-3}) must be achieved to observe the ultimate regime.

Energetics and dynamics of global integrals modeling interaction between stiff filaments.

The attractive and spacing interaction between pairs of filaments via cross-linkers, e.g. myosin oligomers connecting actin filaments, is modeled by global integral kernels for negative binding energies between two intersecting stiff and long rods in a (projected) two-dimensional situation, for simplicity. Whereas maxima of the global energy functional represent intersection angles of 'minimal contact' between the filaments, minima are approached for energy values tending to -infinity, representing the two degenerate states of parallel and anti-parallel filament alignment. Standard differential equations of negative gradient flow for such energy functionals show convergence of solutions to one of these degenerate equilibria in finite time, thus called 'super-stable' states. By considering energy variations under virtual rotation or translation of one filament with respect to the other, integral kernels for the resulting local forces parallel and orthogonal to the filament are obtained. For the special modeling situation that these variations only activate 'spring forces' in direction of the cross-links, explicit formulas for total torque and translational forces are given and calculated for typical examples. Again, the two degenerate alignment states are locally 'super-stable' equilibria of the assumed over-damped dynamics, but also other stable states of orthogonal arrangement and different asymptotic behavior can occur. These phenomena become apparent if stochastic perturbations of the local force kernels are implemented as additive Gaussian noise induced by the cross-link binding process with appropriate scaling. Then global filament dynamics is described by a certain type of degenerate stochastic differential equations yielding asymptotic stationary processes around the alignment states, which have generalized, namely bimodal Gaussian distributions. Moreover, stochastic simulations reveal characteristic sliding behavior as it is observed for myosin-mediated interaction between actin filaments. Finally, the forgoing explicit and asymptotic analysis as well as numerical simulations are extended to the more realistic modeling situation with filaments of finite length.