# Surface tension of N non-mixing fluids

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.

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 @ 1. The closest I know regarding this is the Young equation for solid/liquid/gas contact lines. See: en.wikipedia.org/wiki/… – Bernhard Jan 30 '12 at 19:16

 Thank you for your answer. The assumption from my first question would allow me to use a good framework which allows me to write the energy functional in a better form from which it can be proved that an optimal condition exists, and a numerical approach can be formulated. It was a long guess, but I said that I have to ask this to some physicians before give it up. – Beni Bogosel Jan 30 '12 at 19:27 The fact that you are considering such possible relationships suggests that you have already considered at least as much as I have mentioned above. A relationship such as $\sigma_{ij} = \sigma_{ik} \pm \sigma_{kj}$ could imply yours and this could be true at least as an approximation. – Philip Gibbs Jan 30 '12 at 19:46