How does the concentration of two dyes change in a fluid ? is there an analytical solution?

Question

I was thinking of how we can model two or more dye concentrations in the same fluid. is the answer just a system of two PDEs where the concentration becomes a vector to is there an interaction between both concentrations making the problem difficult to solve analytically?

Answer 1

This depends on the nature of the dyes. If the diffusing entities (I'm using this for a more general situation than dyes) have some sort of reaction --- if you're diffusing baking soda and vinegar in water --- then when they meet, the concentrations of both may change radically (with baking-soda and vinegar, they change into CO2 and something like sodium acetate; the CO2 bubbles out of the solution, and the sodium acetate perhaps precipitates out --- I can't recall).

If the interactions are more complex, wonderful things can happen. Turing (of the Turing machine and many other things) studied this, and you can see more recent work by searching for "reaction-diffusion equations". This work has gotten more attention in recent (sort of) years; one example is this paper by Kass and Witkin on reaction-diffusion textures: https://history.siggraph.org/learning/reaction-diffusion-textures-by-witkin-and-kass/

Answer 2

I don’t study this kind of PDEs, so I might be saying something completely obvious. As a first example, you can still consider a convection-diffusion equation $$ \frac{\partial c}{\partial t}=\nabla \cdot (D\nabla c) -\nabla(vc)+R $$ (with notations and meanings as in Wikipedia’s page), but instead of $c$ being a scalar, you have $c\in\mathbb R^2$, where each component of $c$ represents the density/saturation level of one of the two dyes. Looking at the above equation, it is somewhat clear that the two dyes do not interact directly (although, if $v$ satisfies a PDE on its own, they could still both interact with $v$ and their evolution be not completely decoupled). Starting from here you might start modifying the setting adding more constraints: for example, you might assume there is a saturation level that depends on all the dyes’ densities, like $$ c_1(x,t)+c_2(x,t)\leq c_{\max}, $$ you might add a repulsion coefficient to counter the convection when the concentration of one dye becomes too high, you could modify the equation with $D$ being a matrix instead of a scalar,… You can do many things, and often you will not even be able to write a single nice PDE for the vector $c$, but this is essentially what you need mathematically: your concentration $c$ becomes a vector when you have more that one dye, so your PDE becomes a system (which can still be written as a single vector valued PDE in some easy cases). This can immediately be generalized to an arbitrary number of dyes by taking $c\in\mathbb R^d$, $d\in\mathbb N$.