variables x which fully describe the considered thermodynamic system. In addition to the hydrodynamic fields, the mass density ρ , the momentum density g , and the internal energy density ϵ , we use the composition c – the mass fraction of one component – in order to describe the configuration of the system. Taking into account the Cahn-Hilliard free energy of a phase separating binary system, we assume the total energy and entropy of the system, in terms of the variables x = ( ρ, g , ϵ, c ), to be where V is the total volume of the system. For simplicity, the gradient-squared term, which describes the contribution due to the presence of the interface, is included only in the energy functional (2.4a), although it can be separated between both, energy and entropy [44, 45]. The actual functional form of the entropy density s ( ρ, ϵ, c ) is not needed for the derivation of the Cahn-Hilliard time evolution equations in the GENERIC form. However, we use the assumption of local equilibrium, i.e., s ( ρ, ϵ, c ) has the same form as an equilibrium system at the corresponding state point. In the numerical simulations reported in this paper, we have used the popular c $^{4}$-structure for the uniform part of the entropy density, i.e., s ( ρ, ϵ, c ) = s$_{0}$ ( ρ, ϵ ) − ac $^{2}$/ 2+ bc $^{4}$/ 4, which presents the entropic contribution to the bulk free energy density [14, 15, 19, 20, 22–24]. Other possible functional forms include the one for the van der Waals systems, for example Eq. 4 in [44]. The functional derivatives of the above energy and entropy functionals with respect to the independent variables x are given with where we omitted the position dependence for simplicity. The Poisson operator L (1) determines the reversible contributions to the full evolution equations of the variables x , and hence models the convective behavior. For our particular choice of variables and in view of the functional derivatives (2.5) one arrives at the following 8 δS (1) δ x = ∂s ( ρ, ϵ, c ) ∂ρ , 0 , 1 T , ∂s ( ρ, ϵ, c ) ∂c ) T , (2.5b) δE (1) δ x = ( − 1 2 v $^{2}$, v , 1 , − κ$_{E}$ ∇ $^{2}$c ) T , (2.5a) E $^{(1)}$( x ) = ∫ V d $^{3}$r ( 1 2 g 2 ρ + ϵ + 1 2 κ$_{E}$ |∇ c | 2 ) , (2.4a) S $^{(1)}$( x ) = V d $^{3}$r s ( ρ, ϵ, c ) , (2.4b)