# 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.

• @ 1. The closest I know regarding this is the Young equation for solid/liquid/gas contact lines. See: en.wikipedia.org/wiki/… Jan 30, 2012 at 19:16

• 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. Jan 30, 2012 at 19:46