Multi-scale permeability of deformable fibrous porous media

The contribution of fiber dynamics and clustering to the effective permeability in hierarchical fibrous media is poorly understood, due to the complex fluid–structure interactions taking place across fiber, yarn and textile scales. In this work, a two-dimensional model for fiber deformation subject to out-of-plane movement restrictions is derived for creeping flow conditions by analogy with non-Brownian suspensions of particles with confining potentials. This leads to a homogeneous Fokker–Planck equation in a phase space of fiber configurations, for the probability density function of the fiber displacements. A fiber clustering criterion is then defined using autoconvolution functions of the local probability densities, which yields the local change in fiber-scale permeability according to a topological description of the porous media instead of the typical geometric description. The resulting multi-scale hydrodynamic system is numerically solved by a coupled method, where the Stokes flow at yarn-scale is solved with a finite volume method and the mesoscopic model that recovers information from the fiber-scale is solved by a lattice Boltzmann method. The fiber-scale permeability is characterized in terms of porosity, dimensionless shear rate and dimensionless out-of-plane forces. When assessed in terms of a reduced viscosity related to Brinkman׳s closure for porous media, the mesoscopic model shows that deformable fibrous porous media qualitatively behave like dense particle suspensions. For low volume fractions a non-Newtonian reduced viscosity exhibiting shear-thinning and low- and high-shear plateaux is obtained. For high volume fractions and high shear rates the out-of-plane forces lead to shear thickening. The results on steady fiber-scale permeability are presented in the form of phase diagrams which show that in the typical range of parameters for textiles, the effective permeability of the deformable case can be up 60% lower than that of the rigid case due to the formation of fiber clusters.