# 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 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.

- Authors:

- Aix–Marseille Université, UMR CNRS 7249, Centrale Marseille, Institut Fresnel, 13013 Marseille (France)
- Aix–Marseille Université, UMR CNRS 7258, UMR INSERM 1068, Centre de Recherche en Cancérologie de Marseille, Institut Paoli-Calmettes, 13009 Marseille (France)

- Publication Date:

- OSTI Identifier:
- 22488558

- Resource Type:
- Journal Article

- Journal Name:
- AIP Advances

- Additional Journal Information:
- Journal Volume: 5; Journal Issue: 5; Other Information: (c) 2015 Author(s); Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 2158-3226

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 36 MATERIALS SCIENCE; 97 MATHEMATICAL METHODS AND COMPUTING; ADVECTION; ANISOTROPY; CONCENTRATION RATIO; CONTROL; DIFFUSION; DIFFUSION EQUATIONS; FINITE ELEMENT METHOD; HEAT; THERMODYNAMICS; TIME DEPENDENCE

### Citation Formats

```
Guenneau, S., Petiteau, D., Zerrad, M., Amra, C., and Puvirajesinghe, T.
```*Transformed Fourier and Fick equations for the control of heat and mass diffusion*. United States: N. p., 2015.
Web. doi:10.1063/1.4917492.

```
Guenneau, S., Petiteau, D., Zerrad, M., Amra, C., & Puvirajesinghe, T.
```*Transformed Fourier and Fick equations for the control of heat and mass diffusion*. United States. doi:10.1063/1.4917492.

```
Guenneau, S., Petiteau, D., Zerrad, M., Amra, C., and Puvirajesinghe, T. Fri .
"Transformed Fourier and Fick equations for the control of heat and mass diffusion". United States. doi:10.1063/1.4917492.
```

```
@article{osti_22488558,
```

title = {Transformed Fourier and Fick equations for the control of heat and mass diffusion},

author = {Guenneau, S. and Petiteau, D. and Zerrad, M. and Amra, C. and Puvirajesinghe, T.},

abstractNote = {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.},

doi = {10.1063/1.4917492},

journal = {AIP Advances},

issn = {2158-3226},

number = 5,

volume = 5,

place = {United States},

year = {2015},

month = {5}

}