Homogenised balance equations for nematic liquid crystal flow in elastic porous media

We derive a new mathematical model for the macroscopic behaviour of a linear elastic porous medium weakly interacting with an incompressible, slowly flowing, nematic liquid crystal under the one elastic constant approximation and a simplified hypothesis concerning the fluid viscosities for which the stress tensor remains dependent on the nematic director, which is the average fluid molecular orientation, but is symmetric. In this situation, the angular momentum equation, which governs the dynamics of the nematic director, decouples from the linear momentum equations of the fluid. As such, whilst the nematic anisotropy affects the flow profile, the fluid flow no longer affects the configuration of the nematic director. We assume that the typical pore dimension (the microscale) is significantly smaller than the average size of the whole domain (the macroscale), and exploit this sharp length-scale separation in our use of the asymptotic homogenisation technique to derive new macroscale governing equations by upscaling the fluid-structure interaction problem between the porous elastic structure and the nematic liquid crystal fluid phase. The resulting novel anisotropic macroscale viscoelastic model describes the overall system representing a nematic liquid crystal flowing through an elastic porous solid and could therefore be termed an anisotropic poro-viscoelastic model. This system of partial differential equations incorporates the nematic director, its spatial variations, and the underlying microstructure through coefficients that are computed by solving appropriate microscale cell problems. The homogenised constitutive relationships account for the roles of the nematic director field, the elastic response of the solid phase, and their interplay with the underlying microstructural configuration. We then focus on the particular case in which the role of the elastic porous structure is negligible. In this case, the fluid flow is no longer affected by the deformations of the porous medium and is solely driven by a volume load, which depends on macroscale spatial variations of the nematic director and its interplay with the underlying microscale geometry. The resulting theoretical framework opens up new modelling possibilities for a wide range of potential applications for nematic liquid crystals encapsulated in a complex porous network.