  A consistent description of a shear flow, the accompanied viscous heating,
and the associated entropy balance is given in the framework of a deterministic
dynamical system, where a multibaker dynamics drives two fields: the velocity
and the temperature distributions. In an appropriate macroscopic limit their
transport equations go over into the Navier-Stokes and the heat conduction
equation of viscous flows. The inclusion of an artificial heat sink can
stabilize steady states with constant temperatures. It mimics a thermostating
algorithm used in non-equilibrium molecular-dynamics simulations.
