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In the context of a nonequilibrium statistical thermodynamics, based on anonequilibrium statistical ensemble formalism, a generalized hydrodynamics offluids under driven flow and shear stress is derived. At the thermodynamiclevel, the nonequilibrium equations of state are derived, which are coupled tothe evolution of the basic variables that describe the hydrodynamic motion insuch a system. Generalized diffusion-advection and Maxwell-Cattaneo advectionequations are obtained in appropriate limiting situations. This nonlinearhigher-order hydrodynamics is applied, in an illustration, to the case of adilute solution of Brownian particles in nonequilibrium conditions and flowingin a solvent acting as a thermal bath. This is done in the framework of suchgeneralized hydrodynamics but truncated up to a second order.

Nonlinear higher-order hydrodynamics: Fluids under driven flow and shear pressure