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Register a new userExistence and homogenization of the Rayleigh-Benard problem
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The Navier-Stokes equation driven by heat conduction is studied. As a prototype we consider Rayleigh-Benard convection, in the Boussinesq approximation. Under a large aspect ratio assumption, which is the case in Rayleigh-Benard experiments with Prandtl number close to one, we prove the existence of a global strong solution to the 3D Navier-Stokes equation coupled with a heat equation, and the existence of a maximal B-attractor. A rigorous two-scale limit is obtained by homogenization theory. The mean velocity field is obtained by averaging the two-scale limit over the unit torus in the local variable.
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The Rayleigh-Benard system with stress-free boundary conditions is shown to have a global attractor in each affine space where velocity has fixed spatial average. The physical problem is shown to be equivalent to one with periodic boundary conditions and certain symmetries. This enables a Gronwall estimate on enstrophy. That estimate is then used to bound the $L^2$ norm of the temperature gradient on the global attractor, which, in turn, is used to find a bounding region for the attractor in the enstrophy, palinstrophy-plane. All final bounds are algebraic in the viscosity and thermal diffusivity, a significant improvement over previously established estimates. The sharpness of the bounds are tested with numerical simulations.
This paper is concerned with the regularity theory of a transmission problem arising in composite materials. We give a new self-contained proof for the $C^{k,alpha}$ estimates on both sides of the interface under the minimal assumptions on the interface and data. Moreover, we prove the uniform Lipschitz estimate across a $C^{1,alpha}$ interface when the coefficients on both sides of the interface are periodic with independent structures and oscillating at different microscopic scales.
This contribution is concerned with the effective viscosity problem, that is, the homogenization of the steady Stokes system with a random array of rigid particles, for which the main difficulty is the treatment of close particles. Standard approaches in the literature have addressed this issue by making moment assumptions on interparticle distances. Such assumptions however prevent clustering of particles, which is not compatible with physically-relevant particle distributions. In this contribution, we take a different perspective and consider moment bounds on the size of clusters of close particles. On the one hand, assuming such bounds, we construct correctors and prove homogenization (using a variational formulation and $Gamma$-convergence to avoid delicate pressure issues). On the other hand, based on subcritical percolation techniques, these bounds are shown to hold for various mixing particle distributions with nontrivial clustering. As a by-product of the analysis, we also obtain similar homogenization results for compressible and incompressible linear elasticity with unbounded random stiffness.
The technique of periodic homogenization with two-scale convergence is applied to the analysis of a two-phase Stefan-type problem that arises in the study of a periodic array of melting ice bars. For this reduced model we prove results on existence, uniqueness and convergence of the two-scale limit solution in the weak form, which requires solving a macroscale problem for the global temperature field and a reference cell problem at each point in space which captures the underlying phase change process occurring on the microscale. We state a corresponding strong formulation of the limit problem and use it to design an efficient numerical solution algorithm. The same homogenized temperature equations are then applied to solve a much more complicated problem involving multi-phase flow and heat transport in trees, where the sap is present in both frozen and liquid forms and a third gas phase is also present. Our homogenization approach has the advantage that the global temperature field is a solution of the same reduced model equations, while all the remaining physics are relegated to the reference cell problem. Numerical simulations are performed to validate our results and draw conclusions regarding the phenomenon known as sap exudation, which is of great importance in sugar maple trees and few other related species.
Motivated by the observation that electrons in graphene, in the hydrodynamic regime of transport, can be treated as a two-dimensional ultra-relativistic gas with very low shear viscosity, we examine the existence of the Rayleigh-Benard instability in a massless electron-hole plasma. Firstly, we perform a linear stability analysis, derive the leading contributions to the relativistic Rayleigh number, and calculate the critical value above which the instability develops. By replacing typical values for graphene, such as thermal conductivity, shear viscosity, temperature, and sample sizes, we find that the instability might be experimentally observed in the near future. Additionally, we have performed simulations for vanishing reduced chemical potential and compare the measured critical Rayleigh number with the theoretical prediction, finding good agreement.
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