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:
Is there a relation of the form $\sigma_{ij}+\sigma_{kl}=\sigma_{ik}+\sigma_{jl}$ (scalar or vectorial) between the surface tensions?
Are there any references or monographs which provide a good introduction to this study? I'm interested especially in surface tensions.