  We present a thermodynamically consistent model of a ternary fluid
interacting with elastic membranes. Following a free-energy modelling approach
and taking into account the thermodynamics laws, we derive the equations
governing the ternary fluid flow and dynamics of the membranes. We also provide
the numerical framework for simulating such fluid-structure interaction
problems. It is based on the lattice Boltzmann method, employed for resolving
the evolution equations of the ternary fluid in an Eulerian description,
coupled to the immersed boundary method, allowing for the membrane equations of
motion to be solved in a Lagrangian system. The configuration of an elastic
capsule placed at a fluid-fluid interface is considered for validation
purposes. Systematic simulations are performed for a detailed comparison with
reference numerical results obtained by Surface Evolver, and the Galilean
invariance of the proposed model is also proven. The proposed approach is
versatile, and a wide range of geometries can be simulated. To demonstrate
this, the problem of a capillary bridge formed between two deformable capsules
is investigated here.
