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

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 encom- pass 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, concen- trators and rotators is unaffected by their presence, whatever their shape or posi- tion. Finally, we propose simplified designs of layered cylindrical and spherical diffusion cloaks that might foster experimental efforts in thermal and biochem- ical metamaterials.
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Sebastien Guenneau, D Petiteau, Myriam Zerrad, Claude Amra, Tania Puvirajesinghe. Transformed Fourier and Fick equations for the control of heat and mass diffusion. AIP Advances, American Institute of Physics- AIP Publishing LLC, 2015, 5, pp.53404 - 400. ⟨10.1063/1.4917492⟩. ⟨hal-01451386⟩

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