  We develop a systematic coarse-graining procedure which establishes the
connection between models of mixtures of immiscible fluids at different length
and time scales. We start from the Cahn-Hilliard model of spinodal
decomposition in a binary fluid mixture under flow from which we derive the
coarse-grained description. The crucial step in this procedure is 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 introduce the interfacial width as an
additional variable at that level. In this way, we account for the stretching
of the interface under flow and derive analytically the convective behavior of
the relevant coarse-grained variables, which in the long wavelength limit
recovers the familiar phenomenological Doi-Ohta model. In addition, we obtain
the expression for the interfacial tension in terms of the Cahn-Hilliard
parameters as a direct result of the developed coarse-graining procedure.
Finally, by analyzing the numerical results obtained from the simulations on
the Cahn-Hilliard level, we discuss that dissipative processes at the Doi-Ohta
level are of the same origin as in the Cahn-Hilliard model. The way to estimate
the interface relaxation times of the Doi-Ohta model from the underlying
morphology dynamics simulated at the Cahn-Hilliard level is established.
