The thermodynamic temperature

Equilibrium thermodynamics provides a very useful connection between mechanical and thermal properties of fluids and solids. The predicted relationships between different quantities measured under different thermodynamic conditions are a fundamental consequence of thermodynamics. It is natural to attempt to develop a similar thermodynamic treatment of non-equilibrium systems, at least for steady states. At present, there are a number of different treatments: the extended irreversible thermodynamics (Jou et al., 2001); the approach to microscopic relaxation processes (Öttinger, 2005); and the approach that we follow here. It is fair to say that, at present, there is no consensus on the correctness of any of these approaches, and indeed some debate about whether it is even possible to define the usual thermodynamic quantities for a nonequilibrium system. Clearly then, it is necessary to limit the types of nonequilibrium processes to which we apply thermodynamics. As an example of a system where a thermodynamic treatment may be successful, consider a steady-state Poiseuille flow system where we can define a local temperature and local shear rate at each point in the fluid. There will be gradients in both the shear rate and the temperature that determine the local streaming velocity profile and the conduction of heat to the boundary.