Mathematical models for the non-isothermal Johnson–Segalman viscoelasticity in porous media: stability and wave propagation

We present a nonlinear model for Johnson–Segalman type polymeric fluids in porous media, accounting for thermal effects of Oldroyd-B type. We provide a thermodynamic development of the Darcy’s theory, which is consistent with the interlacement between thermal and viscoelastic relaxation effects and diffusion phenomena. The appropriate invariant convected time derivative for the flux vector and the stress tensor is discussed. This is performed by investigating the local balance laws and entropy inequality in the spatial configuration, within the single-fluid approach. For constant parame- ters, our thermomechanical setting is of Jeffreys type with two delay time parameters, and hence, in the linear/linearized version, it is strictly related to phase-lag theories within first-order Taylor approximations. A detailed spectral analysis is carried out for the linearized version of the model, with a scrutiny to some significant limit situations, enhancing the stabilizing effects of the dissipative and elastic mechanisms, also for retardation responses. For polymeric liquids, rheological aspects, wave propagation properties and analogies with other theories with lagging are pointed out.