  We study a one-dimensional equation arising in the multiscale modeling of
some non-Newtonian fluids. At a given shear rate, the equation provides the
instantaneous mesoscopic response of the fluid, allowing to compute the
corresponding stress. In a simple setting, we study the well-posedness of the
equation and next the long-time behavior of its solution. In the limit of a
response of the fluid much faster than the time variations of the ambient shear
rate, we derive some equivalent macroscopic differential equations that relate
the shear rate and the stress. Our analytical conclusions are confronted to
some numerical experiments. The latter quantitatively confirm our derivations.
