Passive scalars, moving boundaries, and Newton's law of cooling

Ian Tice,Juhi Jang

doi:10.3934/dcds.2016.36.1383

Abstract

We study the evolution of passive scalars in both rigid and moving slab-like domains, in both horizontally periodic and infinite contexts. The scalar is required to satisfy Robin-type boundary conditions corresponding to Newton's law of cooling, which lead to nontrivial equilibrium configurations. We study the equilibration rate of the passive scalar in terms of the parameters in the boundary condition and the equilibration rates of the background velocity field and moving domain.

Highlights

Introduction and motivationIn this paper we study the passive scalar equation∂tθ + u·∇θ = κ∆θ, (1.1)where θ = θ(t, x) ∈ R measures some scalar quantity at time t ≥ 0 and position x ∈ Ω(t), u(t, x) ∈ R3 is a given divergence-free velocity field, and κ > 0 is the diffusivity
Passive scalars are a popular model of turbulent diffusion, mixing phenomena, as well as pollution and combustion
The dynamics of the temperature of the fluid are described by the convection-diffusion equation (1.1) with boundary conditions of the form considered above:

Summary

Introduction and motivation

The periodic boundary conditions on θ play a role in the decay through the derivation of (1.5). Throughout the rest of this paper we will investigate how changes in these three features affect the decay of θ We will consider both static (and rigid) domains as well as moving domains with a range of boundary conditions for θ and u. The dynamics of the temperature of the fluid are described by the convection-diffusion equation (1.1) with boundary conditions of the form considered above: κ∇θ. We note that unlike in the case of a periodic box, the equilibrium temperature is in general not a constant because of the boundary conditions. The equilibrium is not unique; this is related to the fact that solutions converge (at least in the periodic case) to a constant determined by the initial data

Main results

L2-energy estimates

Coercivity of the dissipation

Proof of Main results