Surface tension of N non-mixing fluids

Question

I am a mathematician, not a physicist, so please be gentle with me if I write something wrong.

Consider a bounded, regular container $\Omega$, which is filled with the fluids $F_1,...,F_N$ which do not mix (i.e. $\bigcup_{i=1}^N F_i=\Omega$ and $F_i\cap F_j=\emptyset, \forall i\neq j$). Between two adjacent fluids $F_i,F_j$ there is a surface tension $\sigma_{ij}$ (which is eventually zero if $F_i$ and $F_j$ are not adjacent). The problem I want to study is given $F_i$ with volume $V_i$ and density $\rho_i$ then what is the final state in which the fluids will arrive.

There are three factors I have in mind:

• the interaction of $F_i$ and $F_j$ with $i\neq j$ by their surface tension;
• the interaction between $F_i$ and the boundary $\partial \Omega$ of the container;
• the action of gravity on each $F_i$.

I have two questions:

1. Is there a relation of the form $\sigma_{ij}+\sigma_{kl}=\sigma_{ik}+\sigma_{jl}$ (scalar or vectorial) between the surface tensions?

2. Are there any references or monographs which provide a good introduction to this study? I'm interested especially in surface tensions.

Answer 1

This is a very complex problem to solve so you will probably want to start with some simplifying assumptions such as N=2 to make it more tractable. You will be looking for a minimum energy solution where the energy is a combination of the gravitational potentials and the energy in the surface tension.

Depending on parameters there may be some meta stable solutions where energy is locally minimum but not globally. For example a state in which the fluids all form layers with horizontal separation boundaries ordered by density with the densest at the top would be at least metastable because any perturbation such as a distortion of the boundary would increase both types of energy. However, if one of the layers is sufficiently small in volume there may be a preference for the liquid in that layer to form a bubble between the layers above and below. This would depend on the surface tensions between the three layers. Even with just two fluids there may be bubbles formed for either the top or bottom layer. Working out the shape of the bubble to provide the minimum energy could be non-trivial.

With more liquids the number of odd arrangements you need to look at is going to grow. You have to consider that a heavy fluid may prefer to form a bubble above a lighter one if the surface tensions make that a lower energy configuration.

In all finding the optimal solution will be a mixture of working through large numbers discrete cases and then optimising the shape of the surface areas. You should perhaps be thinking in terms of how this can be done numerically on a computer rather than an analytical solution. You will want to think about whether the problem is NP hard or not.

Some simplifying assumptions may help but I cant see any physical reason why anything like your possible relationship between surface tensions should hold.