Transformed Fourier and Fick equations for the control of heat and mass diffusion

D. Petiteau,C. Amra,T. Puvirajesinghe,M. Zerrad,S. Guenneau

doi:10.1063/1.4917492

Abstract

We review recent advances in the control of diffusion processes in thermodynamics and life sciences through geometric transforms in the Fourier and Fick equations, which govern heat and mass diffusion, respectively. We propose to further encompass transport properties in the transformed equations, whereby the temperature is governed by a three-dimensional, time-dependent, anisotropic heterogeneous convection-diffusion equation, which is a parabolic partial differential equation combining the diffusion equation and the advection equation. We perform two dimensional finite element computations for cloaks, concentrators and rotators of a complex shape in the transient regime. We precise that in contrast to invisibility cloaks for waves, the temperature (or mass concentration) inside a diffusion cloak crucially depends upon time, its distance from the source, and the diffusivity of the invisibility region. However, heat (or mass) diffusion outside cloaks, concentrators and rotators is unaffected by their presence, whatever their shape or position. Finally, we propose simplified designs of layered cylindrical and spherical diffusion cloaks that might foster experimental efforts in thermal and biochemical metamaterials.

Highlights

There is currently a keen interest in the control of heat flux using thermal metamaterials in steady,[1,2,3,4,5,6,7] transient[8,9,10,11,12] and periodic[13] regimes
Let us note that cloaking for diffusion processes[19,20] is more subtle than for waves[21,22,23,24,25,26,27,28] wherein the field vanishes in the invisibility region irrespective of its material constituent, time and its distance of the source. This appears to be in sharp contrast with thermal cloaks wherein temperature inside the invisibility region appears to depend on its diffusivity, upon time, and the distance from the source
We find that the transformed convection-diffusion equation has the form: ρ(x′)c(x′)det(J)

Summary

INTRODUCTION

There is currently a keen interest in the control of heat flux using thermal metamaterials in steady,[1,2,3,4,5,6,7] transient[8,9,10,11,12] and periodic[13] regimes. We discuss the functionality of diffusion cloaks for heat and mass, as well as concentrators and rotators,[14] via transformed Fourier[8,15] and Fick[16,17,18] equations. Such designs are based upon the extension of metamaterials designed using tools of transformation optics to the fields of thermodynamics and biochemistry.

TRANSFORMED CONVECTION-DIFFUSION EQUATION

Diffusion cloak

Diffusion concentrator

Diffusion rotator

Three-dimensional diffusion cloak of a complex shape

A MULTILAYERED CLOAK WITH SIMPLIFIED ISOTROPIC PARAMETERS

DIFFUSION INVISIBILITY CARPET

Three-dimensional carpets

CONCLUDING REMARKS