Energy dissipation of Oldroyd-B fluids in plane Couette flow

This paper establishes a rigorous upper bound on the infinite-time-averaged energy dissipation rate of Oldroyd-B fluids in plane Couette flow. The bound depends only on system parameters – the Reynolds number, Weissenberg number and viscosity ratio – and applies to all steady and unsteady solutions within a certain region in parameter space. The bound is proven by extending the ‘background-flow method’ to the case where the system energy is no longer a quadratic functional of the underlying flow fields, and is obtained by using a non-polynomial auxiliary functional related to the free polymeric energy. Within the range of the flow parameters in which the steady solution is known to be globally stable, the dissipation rate of the steady flow is recovered, and in the Newtonian limit the result reduces to the best-known bound for Newtonian Couette flow. Our analysis also identifies a range of parameters for which the total energy of the viscoelastic flow must be bounded, thus ruling out the possibility of energy blow-ups in these situations.