A new rheological framework for active polar fluids

Polar active fluids are an important topic in theoretical and applied physics, particularly in biophysics and the modeling of active materials such as living tissues. There are mainly two continuum hydrodynamic frameworks for polar active fluids in the literature, namely the active matter theory based on irreversible thermodynamics and the Onsager's variational principle based on the Rayleigh potential. In this article, we propose a new thermodynamic framework for the continuous modeling of active polar fluids, which also allows for the introduction of any rheology, such as nonlinear viscoelasticity or plasticity. To do so, we extend the general thermodynamic formalism recently developed by one of us, called YATE (Yet Another Thermodynamic Environment), to active polar fluids. To relax the symmetry assumption of the stress tensor, we additionally consider the classical couple stress theory, which introduces the effects of microrotations within continuum mechanics. We derive general constitutive equations that satisfy the second law of thermodynamics, or equivalently ensuring the positivity of the dissipation or the entropy production rate. This derivation is automatically obtained from a Helmholtz free energy and a dissipation potential, which allows for the introduction of any rheology. We show how to retrieve equations from the active gel theory and how to extend it with more complex rheologies, such as the FENE-P viscoelasticity or the Bingham plasticity. For each case, we also provide an entropy estimate, taking the form of a power balance between the rate of free energy, dissipation and power input.