M. Grmela, WEAKLY NONLOCAL HYDRODYNAMICS, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 47(1), 1993, pp. 351-365
Anomalous viscoelastic, diffusion, and heat-transfer phenomena observe
d 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 deri
vatives with respect to the position coordinate. The governing equatio
ns 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 ex
pression of kinematics in the chosen state space. In weakly nonlocal h
ydrodynamics the Poisson structure remains the same as in local hydrod
ynamics 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 a
nd that their solutions agree with certain fundamental macroscopic obs
ervations (e.g., the observations constituting the basis of equilibriu
m thermodynamics).