We prove that for nonlinear elastic energies with strong enough energetic control of the outer distortion of admissible deformations, almost everywhere global invertibility as constraint can be obtained in the $\Gamma$-limit of the elastic energy with an added nonlocal self-repulsion term with asymptocially vanishing coefficient. The self-repulsion term considered here formally coincides with a Sobolev–Slobodeckiĭ seminorm of the inverse deformation. Variants near the boundary or on the surface of the domain are also studied.

This work addresses the effects of different thermal sidewall boundary conditions on the formation of flow states and heat transport in two- and three-dimensional Rayleigh–Bénard convection (RBC) by means of direct numerical simulations and steady-state analysis for Rayleigh numbers ${\textit {Ra}}$ up to $4\times 10^{10}$ and Prandtl numbers ${\textit {Pr}}=0.1,1$ and $10$. We show that a linear temperature profile imposed at the conductive sidewall leads to a premature collapse of the single-roll state, whereas a sidewall maintained at a constant temperature enhances its stability. The collapse is caused by accelerated growth of the corner rolls with two distinct growth rate regimes determined by diffusion or convection for small or large ${\textit {Ra}}$, respectively. Above the collapse of the single-roll state, we find the emergence of a double-roll state in two-dimensional RBC and a double-toroidal state in three-dimensional cylindrical RBC. These states are most prominent in RBC with conductive sidewalls. The different states are reflected in the global heat transport, so that the different thermal conditions at the sidewall lead to significant differences in the Nusselt number for small to moderate ${\textit {Ra}}$. However, for larger ${\textit {Ra}}$, the heat transport and flow dynamics become increasingly alike for different sidewalls and are almost indistinguishable for ${\textit {Ra}}>10^{9}$. This suggests that the influence of imperfectly insulated sidewalls in RBC experiments is insignificant at very high ${\textit {Ra}}$ – provided that the mean sidewall temperature is controlled.

This work addresses the effect of travelling thermal waves applied at the fluid layer surface, on the formation of global flow structures in two-dimensional (2-D) and 3-D convective systems. For a broad range of Rayleigh numbers ($10^3\leq Ra \leq 10^7$) and thermal wave frequencies ($10^{-4}\leq \varOmega \leq 10^{0}$), we investigate flows with and without imposed mean temperature gradients. Our results confirm that the travelling thermal waves can cause zonal flows, i.e. strong mean horizontal flows. We show that the zonal flows in diffusion dominated regimes are driven purely by the Reynolds stresses and end up always travelling retrograde. In convection dominated regimes, however, mean flow advection, caused by tilted convection cells, becomes dominant. This generally leads to prograde directed mean zonal flows. By means of direct numerical simulations we validate theoretical predictions made for the diffusion dominated regime. Furthermore, we make use of the linear stability analysis and explain the existence of the tilted convection cell mode. Our extensive 3-D simulations support the results for 2-D flows and thus provide further evidence for the relevance of the findings for geophysical and astrophysical systems.

Classical and symmetrical horizontal convection is studied by means of direct numerical simulations for Rayleigh numbers $Ra$ up to 3 × 1012 and Prandtl numbers $Pr=0.1$, 1 and 10. For both set-ups, a very good agreement in global quantities with respect to heat and momentum transport is attained. Similar to Shishkina & Wagner (Phys. Rev. Lett., vol. 116, 2016, 024302), we find Nusselt number $Nu$ versus $Ra$ scaling transitions in a region $10^{8}\leqslant Ra\leqslant 10^{11}$. Above a critical $Ra$, the flow undergoes either a steady–oscillatory transition (small $Pr$) or a transition from steady state to a transient state with detaching plumes (large $Pr$). The onset of the oscillations takes place at $Ra\,Pr^{-1}\approx 5\times 10^{9}$ and the onset of detaching plumes at $Ra\,Pr^{5/4}\approx 9\times 10^{10}$. These onsets coincide with the onsets of scaling transitions.