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expression for the Poisson operator 

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with 

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where δ = δ ( r − r ′ ) is Dirac’s δ -function. In addition to the usual hydrodynamic part of the Poisson matrix for the variables ρ , g , ϵ , element L (1 ) cg dictates the convection of the configurational variable c , which is a scalar, so the entry in the Poisson operator is as expected by [41]. The antisymmetry of the Poisson matrix (2.6) and the degeneracy requirement are satisfied by construction, and the Jacobi identity holds. At this point we mention that, in general, the symbol “ · ” in (2.2) implies not only summation over discrete indices. If field variables are involved the operators L and M are written in terms of two space arguments ( r , r ′ ), and an integration over r ′ must be performed when multiplied with a function of r ′ from the right. However, in the case of the field equations being local, one can express L and M in terms of a single variable r only [41], and no integration is implied when these operators are multiplied from the right. Such single variable notation is used below for the friction matrix M $^{(1 )}$. 

The irreversible effects in the Cahn-Hilliard model for a binary fluid under flow arise due to viscosity and heat conduction, as well as diffusion. Thus, to incorporate them through the friction matrix M $^{(1 )}$, one can write it as the sum of two matrices M (1 ) = M (1 ) , $^{hyd}$+ M (1 ) , $^{dif}$, where M (1 ) , hyd is the usual hydrodynamic friction matrix, and M (1 ) , dif contains all the transport coefficients related to diffusion of mass [41, 46]. The hydrodynamic part of the 

