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

International audienceWe review recent advances in the control of diffusion processes in thermodynamicsand life sciences through geometric transforms in the Fourier and Fick equations,which govern heat and mass diffusion, respectively. We propose to further encom-pass transport properties in the transformed equations, whereby the temperatureis governed by a three-dimensional, time-dependent, anisotropic heterogeneousconvection-diffusion equation, which is a parabolic partial differential equationcombining the diffusion equation and the advection equation. We perform twodimensional finite element computations for cloaks, concentrators and rotators ofa complex shape in the transient regime. We precise that in contrast to invisibilitycloaks for waves, the temperature (or mass concentration) inside a diffusion cloakcrucially depends upon time, its distance from the source, and the diffusivity ofthe invisibility region. However, heat (or mass) diffusion outside cloaks, concen-trators and rotators is unaffected by their presence, whatever their shape or posi-tion. Finally, we propose simplified designs of layered cylindrical and sphericaldiffusion cloaks that might foster experimental efforts in thermal and biochem-ical metamaterials