time simulations at the Cahn-Hilliard level. However, the problems concerning the third order derivatives of the composition field c make this task rather complicated, as well as the need for determination of the Lagrange multipliers in order to employ the full GENERIC procedure developed in this paper. 

## IV. CONCLUSIONS AND PERSPECTIVES 

In this paper, we have developed a systematic, thermodynamically guided method which establishes the coarse-grained Doi-Ohta model, used for rheological behavior, from the more detailed Cahn-Hilliard model of a phase separating binary fluid. The main contributions of the present work can be summarized as: (1) the introduction of the average interfacial width l as an additional structural variable; (2) the derivation of the coarse-grained Poisson operator L $^{(2)}$, Eq.(3.11), from its finer level analogue; (3) no new dissipative processes arise during level jumping apart from the known hydrodynamic and diffusive dissipation; (4) the derivation of an expression (3.21) for the Doi-Ohta relaxation times as a function of the elements of the friction matrix describing relaxation at the finer Cahn-Hilliard level. 

The crucial step in the coarse-graining procedure was to identify the relevant coarse- grained variables and find the appropriate mapping which expresses them in terms of the more microscopic variables. In order to capture the physics of the Doi-Ohta level, we in- troduced the interfacial width l ( r , t ) as an additional variable in the new model. In that way we could account for the stretching of the interface under flow and derive analytically the reversible (convective) behavior of the variables Q ( r , t ) and q ( r , t ), which recovers the already established phenomenological Doi-Ohta model. Introduction of the interfacial width l and its time evolution equation, as an addition to Q and q of the original Doi-Ohta model with zero-width interface, offers new possibilities for modeling of the rheological behavior of multiphase flows. In addition, the expression for the interfacial tension (3.8) in terms of the Cahn-Hilliard parameters follows as the direct result of the developed systematic coarse-graining procedure. 

Considering the irreversible dynamics on the Doi-Ohta level, it has been shown that the dissipative processes at this coarse-grained level are carried on from the more microscopic Cahn-Hilliard level. Although their analytical derivation is too complex and rich in structure, the way to connect the interface relaxation times of the Doi-Ohta model and the underlying 

