This paper investigates creeping, two-phase flow of partially miscible components in submicron channels. A continuous gradient technique treats two-phase flow by coupling a single version of the Navier-Stokes equations to the modified Cahn-Hilliard equation (a form of the convective diffusion equation applicable to nonlinear diffusion). A body force based on gradient energy provides the coupling. Results for the cross-channel composition and velocity profiles are easily calculated at far downstream locations, including the case where the viscosity is a function of composition. The pressure drop versus flow rate in small channels is substantially different from that in large channels. Wall effects due to selective attractive forces are unimportant for the ranges of parameters studied. Two forms of entrance length problems are defined. In the simpler problem, the components are fed separately and form an equilibrium interface. The solution methodology for this problem can also be applied to fully miscible fluids that initially differ in composition and viscosity. The other problem envisions an initially homogeneous mixture that phase separates in the channel by spinodal decomposition. The necessary physical parameters are the properties of the pure components and an interaction parameter between the components (e.g., Flory-Huggins or regular solution theory). Potential applications include tertiary oil recovery and microreactors.