Weakly nonlocal hydrodynamics

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Abstract
Anomalous viscoelastic, diffusion, and heat-transfer phenomena observed in spatially inhomogeneous simple and complex fluids are analyzed in this paper in the setting of weakly nonlocal hydrodynamics. Governing equations of this generalized hydrodynamics involve higher-order derivatives with respect to the position coordinate. The governing equations are obtained on the basis of the following consideration. The time evolution in both local and weakly nonlocal hydrodynamics is generated by a thermodynamic potential in a state space equipped with a Poisson structure and a dissipative potential. The Poisson structure is an expression of kinematics in the chosen state space. In weakly nonlocal hydrodynamics the Poisson structure remains the same as in local hydrodynamics but the potentials are generalized. The potentials are allowed to depend on higher-order derivatives of the hydrodynamic fields that are chosen as state variables. This way of introducing the governing equations guarantees that the equations are intrinsically compatible and that their solutions agree with certain fundamental macroscopic observations (e.g., the observations constituting the basis of equilibrium thermodynamics).

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