Entropy Production and Non-Equilibrium Hydrodynamics of Active Suspensions in Viscoelastic Fluids

Active suspensions navigating biological and technological environments frequently encounter viscoelastic media characterized by intrinsic memory and elasticity. While the kinematic consequences of active-viscoelastic coupling are increasingly well-documented, the fundamental thermodynamic costs of these interactions remain largely unexplored. In this study, we investigate the energetic landscape of a coarse-grained active polar suspension embedded in an incompressible Oldroyd-B solvent. To rigorously isolate the interplay between polymer relaxation, active orientational dynamics, and external forcing, the mathematical framework is intentionally restricted to a spatially homogeneous constitutive setting under imposed start-up and Small-Amplitude Oscillatory Shear (SAOS). Consequently, this model explicitly excludes spatial gradients, active turbulence, and near-field interactions. We verify the numerical solver by recovering the analytical benchmarks of the passive viscoelastic fluid. Upon introducing the established active polar field, the macroscopic mechanical power budget is thermodynamically decomposed into universally applicable channels (viscous dissipation, active power) and model-dependent channels (irreversible polymeric relaxation). Current central analytical prediction reveals a profound constitutive dissipation maximum: the macroscopic cycle-averaged entropy production is non-monotonically amplified, reaching a maximum when the external driving timescale overlaps precisely with the polymer relaxation time ( De∼ 1). We map the causal mechanics of this phase-matching, linking it to observable rheological predictions such as normal stress differences. The present work provides a theoretically consistent baseline for quantifying the energetic footprint of active stresses, offering predictive design guidelines for active microfluidics: dynamic regimes yielding the strongest elastic mechanical coupling simultaneously impose the severest thermodynamic penalty.