On non-isothermal flows of dilute incompressible polymeric fluids

 In the first part of the talk, after revisiting some classical models for dilute polymeric fluids, we show that thermodynamically 
consistent models for non-isothermal flows of such fluids can be derived in a very elementary manner. Our approach is based on identifying the 
energy storage mechanisms and entropy production mechanisms in the fluid of interest, which in turn leads to explicit formulae for the Cauchy 
stress tensor and for all the fluxes involved. Having identified these mechanisms, we first derive the governing system of nonlinear partial 
differential equations coupling the unsteady incompressible temperature-dependent Navier–Stokes equations with a 
temperature-dependent generalization of the classical Fokker–Planck equation and an evolution equation for the internal energy. We then 
illustrate the potential use of the thermodynamic basis on a rudimentary stability analysis—specifically, the finite-amplitude (nonlinear) 
stability of a stationary spatially homogeneous state in a thermodynamically isolated system.

In the second part of the talk, we show that sequences of smooth solutions to the initial–boundary-value problem, which satisfy the 
underlying energy/entropy estimates (and their consequences in connection with the governing system of PDEs), converge to weak 
solutions that satisfy a renormalized entropy inequality. The talk is based on joint results with Miroslav Bulíček, Mark Dostalík, Vít Průša 
and Endré Süli.