What formulas should I use to realistically model the diffusion of a drop of ink in a water?

Question

I am a mathematician and am originally from the math side of stackexchange. I want to model the behaviour of a drop of ink diffusing in water. I dont want to simply use the diffusion equation $u_t(\mathbf{x},t)=D \triangledown^2u(\mathbf{x},t) $ because firstly it will produce a diffusion of the ink completely symmetrical in the $x$, $y$ and $z$ direction, secondly, it does not take into account the gravity producing a force (say in the $z$-direction) and lastly, it does not take into account the velocity of the moving ink particles and the different pressure at each point.

Now I would like (if possible) to create a program that will give a result that looks similar to the sort of chaotic diffusion we see in real life, possible by creating a non-symmetrical initial disturbance in the form of an inital velocity in the ink. What formulas should I be looking at in this case? Can I ignore some of the things I've mentioned above and still get a realistic result? Is it possibly true that the problems I've mentioned can be fixed by not taking $D$ to be constant but rather a function of the velocity, density and pressure and then using the formulas from fluid dynamics to find these at each position and time?

As I said, I am a mathematician and I apologize in advance for the possibility that there are silly errors in my question or that my limited understanding of physics makes this a non-sensical question all together. Any help would be greatly appreciated though!

Answer 1

If the drop is very much static (in still water) and of similar fluid properties to the water around it (so that the ink just labels some initial region), then this is the correct equation to use. If, however, you want to treat the ink as having distinct properties from the water, then you want the Navier-Stokes equations. Since you are interested in gravity, I assume you have a different density in mind for the ink, and probably a different viscosity as well.

Certainly turbulent fluids mix much faster than diffusion predicts. Generally, the mechanism by which this enhanced diffusion takes place is this: First, turbulent fluid flow, via nonlinear coupling term $(\mathbf{v}\cdot\nabla)\mathbf{v}$, creates smaller and smaller scale structures, i.e., fine layers of ink and water. Second, once these scales are small enough, diffusion is effectively fast, having only very small length scales to mix together. This depends on your system being unstable to perturbations, which depends a great deal on the geometry of your ink drop and the ink properties.

As a starting point, you could treat the two fluids as immiscible, asking how pure parcels of fluid disperse due to shear or the Rayleigh-Taylor instability. Many Navier-Stokes solvers have been written; this page provides a non-exhaustive list.

A more complex picture allows for true mixing, i.e., by diffusion. This can also be done, although it involves keeping track of the "ink density" and having a means of computing properties like density and viscosity of dilute ink. A simplistic approach might be to, at each time step, advance the fluid code, then apply a separate diffusion step, and then repeat, while keeping track of the ink density and the viscosity at each grid point.