# sandbox/benalessio/README

## Welcome

In this sandbox folder you can find the codes which I used to simulate diffusiophoresis in Turing patterns (Alessio and Gupta, 2023). These codes were originally inspired by (and draw heavily from) Basilisk’s Brusselator example where the key difference is the introduction of an advective-diffusive species which is propelled by the chemical gradients.

A diffusiophoretic colloidal species n can be modeled with an advective-diffusion equation:

\displaystyle \frac{\partial n}{\partial t} = \nabla\boldsymbol\cdot\left(D_n\nabla n - \boldsymbol v_\text{DP}n\right) where the diffusiophoretic velocity \boldsymbol v_\text{DP} depends on the gradients of the solutes c_i as \displaystyle \boldsymbol v_\text{DP} = \sum_i M_i\nabla c_i for non-electrolytes and \displaystyle \boldsymbol v_\text{DP} = \Gamma\nabla\log c for a single binary salt. More complicated forms can be found in the literature for mixtures of electrolytes and other effects. The gradients of the solutes c_i can come from diffusion out of a source or from reaction-diffusion instabilities (i.e. Turing patterns), and they may be modeled with a system of reaction-diffusion equations: \displaystyle \frac{\partial c_i}{\partial t} = D_i\nabla^2c_i + r_i where r_i is a reaction term coupling the various solutes. If r_i is sufficiently nonlinear, Turing patterns are permissible. The three codes I have here simulate the Brusselator model, a cell-cell interaction model, and the Gierer-Meinhardt model.

## References

[alessio2023] |
Benjamin M Alessio and Ankur Gupta. Diffusiophoresis-enhanced turing patterns. |

[shim2022] |
Suin Shim. Diffusiophoresis, diffusioosmosis, and microfluidics: surface-flow-driven phenomena in the presence of flow. |

[derjaguin1947] |
B.V. Derjaguin, G.P. Sidorenkov, E.A. Zubashchenkov, and E.V. Kiseleva. Kinetic phenomena in boundary films of liquids. |