A modified Darcy's law for viscoelastic flows of highly dilute polymer solutions through porous media
Résumé
Viscoelastic flows of polymer solutions in complex geometries can generate a strong localization of stress within small regions of the fluid and the formation of birefringent strands. In porous media, these localized structures of stress drive preferential flow paths and increase global dissipation. Modeling the impact of such effects at Darcy or larger scales is a daunting task-one of the reasons being the lack of approaches using homogenization theories to help figure out both the correct form of the averaged transport equations and the relevant set of effective parameters. Here we homogenize the incompressible Oldroyd-B equations at zero Reynolds number to obtain a Darcy scale model that captures the effect of localized polymeric stress. This model consists of an advection-reaction transport equation for the average conformation tensor along with a form of Darcy's law that contains an additional drag term associated with structures of localized stress. The derivation is based upon a limit of high dilution, a regime where the Oldroyd-B model can be transformed into a sequence of linear problems using asymptotic developments. We validate our approach in test cases corresponding to flows in a channel and through arrays of circles. Besides providing a new model for viscoelastic flows in porous media, our work also shows that modelling viscoelastic flows through porous media is not simply a matter of determining an apparent permeability tensor-the homogenized model cannot be easily reduced to a simple form of Darcy's law-but rather requires the development of specific homogenized models that capture the coupling between the transport of the polymeric stress and momentum.
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