where is the external force acting on species , and is the external force acting on the fluid; the second order tensors suggest that the fluxes are not necessarily aligned with the concentration gradients. There will be additional thermal diffusion terms in Eq. 3.16 if higher terms in the fluxes are accounted for.
Still another type of diffusion can be described by the phase-space kinetic theory, namely the diffusion in the presence of nonhomogeneous velocity gradients. For a dilute solution of Rouse chains with N beads, the following generalization of Fick's second law has been obtained [5a]:
This equation allows for the description of the Uhlenhopp effect, which states that in a coaxial rotating viscometer, the solvent molecules will tend to diffuse toward the inner cylinder.
The kinetic theory of polymers suggests the existence of various cross effects:
The shaded areas on the diagram represent the main flux-force relations. Those marked with an "x" are the flux-force relations permitted in the classical thermodynamics