sandbox/benalessio/README

Welcome

My name is Ben Alessio and I am a PhD student in mechanical engineering at Stanford University. In my sandbox you can find (so far) codes for diffusiophoresis-enhanced Turing patterns and simple microfluidic flows. Please feel free to write me at balessio@stanford.edu if you have any questions.

Diffusiophoresis-Enhanced Turing Patterns

In this section 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.

Viscous flow in a microchannel with a solute

In this section you can find the code(s; soon to be more) for simulating viscous Stokes flow in a microchannel with an advective-diffusive solute. This simple example demonstrates a custom aspect-ratio microchannel with a solute that disperses from a thin puck into a diffuse cloud.

Population migration with chemotaxis

In this section you can find an example code for simulating population dynamics from the Fisher-KPP equation modified to include chemotaxis (Alessio and Gupta, 2024). The equations are:

\displaystyle \frac{\partial n}{\partial t} = \nabla\boldsymbol\cdot \left(\kappa_n\nabla n - v n \right) + r_n(n) \displaystyle v = \chi\nabla\phi \displaystyle \frac{\partial \phi}{\partial t} = \nabla\boldsymbol\cdot\left(\kappa_\phi\nabla\phi\right) + r_\phi(\phi)

for population density n and attractant \phi, with diffusivities \kappa_n and \kappa_\phi, growth rates r_n and r_\phi, and chemotactic mobility \chi.

References