Modeling

To realise our artwork, we use an agarose gel to obtain the shape of a lemon. To push the resemblance to the extreme, we wish to have a crust in the edge of the lemon when we seperate it. In fact, we build bacteria who produce yellow/green color in presence of oxygen. Thus we must evaluate the penetration of the oxygen in the gel !

We use the following phenomenological law suggest by Adolphe Fick in 1855:

In an homogeneous and isotropic environment, containing particles distributed inhomogeneously,appears spontaneously a volumetric flow density vector particle $\overrightarrow{J}(M,t) $. In any point $M$ in space, this vector is proportional to the gradient of the particle density $n(M,t)$. Mathematicaly, this relationship take the form: \[ \overrightarrow{J}(M,t) = - D \times \overrightarrow{grad} n(M,t) \] where $D$ is the diffusion coefficient.

article ----> case of agarose $ D = 0{,}256 \times 10^{-8} m^2 s^{-1} $

• To simplify the problem, we consider that the diffusion of oxygen particle occurs only in one direction. So $\overrightarrow{J}(M,t) = J(x,t) \overrightarrow{e}_x $.
• The material's equation of conservation in lack of source:

\[ \frac{\partial n}{\partial t} (x,t) = - \frac{\partial J}{\partial x} (x,t) \]

By replacing (2) in (1), we obtain the following equation of diffusion \[ \bigg( \frac{\partial}{\partial t} - D \frac{\partial^2}{\partial x^2} \bigg) n(x,t) = 0 \]