Shear flow , viscous heating , and entropy balance from dynamical systems

– 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. Introduction. – In the last years, there has been an increasing interest in modeling transport phenomena by low-dimensional, deterministic dynamical systems [1–13]. Multibaker maps [4–13] appeared to be the simplest models of this approach. They provide an opportunity to derive the equations of non-equilibrium thermodynamics from an underlying dynamics without using the concept of particles. The strongly-chaotic mixing properties of these two-dimensional maps seem to be sufficient to ensure consistency with the entropy balance equation of thermodynamics provided that a properly chosen coarse-grained entropy and a macroscopic limit are taken [7,10,11]. Previous work successfully described the phenomena of diffusion [4,6], conduction in an external field [5, 7, 9], chemical reactions [8], thermal conduction [10], and cross effects due to the simultaneous presence of an external field and heat conduction [11] by means of multibaker maps. Not only stationary, but also transient states could be addressed [7, 11, 12]. Here we add to this list the phenomenon of shear flows and the accompanying viscous heating. The interest of this is to clarify how the shear rate enters the expression for the irreversible entropy production. After all, by the definition of local equilibrium the macroscopic flow profile does not appear in the entropy balance. Thermodynamic averages contain only deviations from the average streaming velocity.