Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1212.2940

Laminar boundary layers in convective heat transport

Christian Seis
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study Rayleigh-Benard convection in the high-Rayleigh-number and high-Prandtl-number regime, i.e., we consider a fluid in a container that is exposed to strong heating of the bottom and cooling of the top plate in the absence of inertia effects. While the dynamics in the bulk are characterized by a chaotic convective heat flow, the boundary layers at the horizontal container plates are essentially conducting and thus the fluid is motionless. Consequently, the average temperature exhibits a linear profile in the boundary layers. In this article, we rigorously investigate the average temperature and oscillations in the boundary layer via local bounds on the temperature field. Moreover, we deduce that the temperature profile is indeed essentially linear close to the horizontal container plates. Our results are uniform in the system parameters (e.g. the Rayleigh number) up to logarithmic correction terms. An important tool in our analysis is a new Hardy-type estimate for the convecting velocity field, which can be used to control the fluid motion in the layer. The bounds on the temperature field are derived with the help of local maximal regularity estimates for convection-diffusion equations.

Explore related subjects

math.APnlin.CDphysics.flu-dyn

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Christian Seis. 2012-12-12. Laminar boundary layers in convective heat transport. https://doi.org/10.1007/s00220-013-1826-7

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Blow-Up Dynamics for the $L^2$ critical case of the $2$D Zakharov-Kuznetsov equation

We study blow-up dynamics for the $L^2$-critical cubic Zakharov--Kuznetsov equation in two dimensions, \[ \partial_tu+\partial_{x_1}(Δu+u^3)=0 \qquad\text{on }\mathbb R^2. \] For a class of localized $H^1$ perturbations of the ground state $Q$, we establish a trichotomy near the soliton manifold: exit from a small $L^2$-tube, global asymptotic stability, or finite-time blow-up. In the stable blow-up regime, the solution concentrates a single bubble and \[ λ(t)\sim \ell_0(T-t)^{1/(3-c)}, \] where $\ell_0>0$ depends on the initial datum and $c\in(1,2)$ is an explicit constant determined by the transverse tail of the first-order approximate profile. Consequently, \[ \|\nabla u(t)\|_{L^2} \sim \frac{\|\nabla Q\|_{L^2}} {\ell_0(T-t)^{1/(3-c)}}. \] After subtraction of the concentrating soliton, the radiation converges strongly in $L^p(\mathbb R^2)$ for every $2\leq p<\infty$ to a common nonzero profile $u^*$, while \[ u^*\notin H^s(\mathbb R^2) \qquad\text{for every }s\geq\frac c2. \] The stable blow-up branch is open in the relative $H^1$ topology of the localized class. Finally, every non-soliton datum in this class with non-positive energy blows up in finite time. Interval-arithmetic computer-assisted proofs certify the numerical inputs to the virial coercivity argument. They also yield a rigorous enclosure of $c$, justifying the polynomial moment of order $21$ imposed on the initial data.

math.AP↗
arXiv2026

Propagation of wave packets close to conical intersections

In this paper, we study the propagation of wave packets close to conical intersections with respect to a system of two Schr{ö}dinger equations presenting a codimension 2 crossing. We focus on the dynamics that occur when the wave packets pass through an area close to the crossing, and our main results provide an explicit formula for the outgoing wave packet in terms of the incoming one, with a complete description of its phase and of the classical trajectories it follows, including a drift.

math.AP↗
arXiv2026

A Volterra Calculus for Lie Groupoids

A pseudodifferential Volterra calculus for inverting parabolic differential equations on Lie groupoids is introduced. This enables the study of fundamental solutions of various cases of heat flows on singular manifolds with corners with non-resonant boundary indicial symbols, such as the $b$-manifolds, as well as other geometric bisection covariant heat flows. We also establish the short time asymptotic expansion for the heat kernel of a positive, elliptic differential operator on a Lie groupoid that acts on suitable Sobolev Hilbert modules and is positive definite with respect to the appropriate $L^2$ inner product.

math.AP↗