Higher-grandient theories for fluids and concentrated effects

I consider the Virtual Power framework for Continuum Mechanics, which has recently gained considerable attention, mainly in connection with its applicability to non-classical models for materials. I introduce a geometrical approach to possibly infinite dimensional dynamical systems, based on the theory of Banach manifolds, which has not yet been fully exploited in Continuum Mechanics, though it has been used in some particular cases. This theory generalizes the Virtual Power framework, being even more flexible and allowing for the construction of continuum mechanical models on non-Euclidean domains. I studied the mathematical properties of a general linear isotropic incompressible second-gradient fluid. Constitutive prescriptions for these fluids are discussed, together with the constraints imposed by thermodynamical considerations. The key features of the analyzed model are the possibility of describing the adherence interaction of a three-dimensional fluid with one-dimensional structures immersed in it, and also of including concentrated interactions. A presentation of higher-gradient theories is provided, using the general framework proposed for dynamical systems. They turn out to be a particular class of continuum mechanical models, arising from precise assumptions on the kinematics of the descriptors of the system. Higher-order powers are defined as integral representations of elements of the cotangent bundle on the Banach manifold of the descriptors. Exploiting equivalent integral representations for powers of arbitrary order, the appearance of boundary interactions with a non-standard structure is described. The differential problems associated with the pressure-driven flow of a second-order linear liquid, which adheres to a one-dimensional structure, is considered. Existence and uniqueness of solution are established, also for the situation in which the one-dimensional structure drags the three-dimensional fluid, producing the motion. Finally, some examples are provided, in order to give explicit solutions, to show how the concentrated stresses, if present, can be computed, and to suggest possible interpretations for the physical meaning of the higher-order material parameters.