Skew-symmetric couple-stress fluid mechanics

There can be no doubt as to the importance of vortical motion in fluid mechanics. Yet, very little attention is given typically to the balance law of angular momentum and to its role in defining the fundamental character of stress, which as a result is usually assumed as a symmetric tensor. Here, we allow for the possibility of couple-stresses, along with general non-symmetric force–stresses, and develop a self-consistent size-dependent theory within the context of classical continuum mechanics. This development relies upon the identification of the following key components for the dynamic response of three-dimensional fluid continua: (i) fundamental, uniquely defined kinematical measures of flow, (ii) an independent set of energy conjugate variables, (iii) the corresponding permissible natural and essential boundary conditions, and (iv) a non-redundant set of body-force and inertial contributions. Based upon this formulation, one can recognize that the previous couple-stress theory for fluids suffers from some inconsistencies, which may have restricted its applicability in the study of viscous flows. After presenting the general formulation of the new consistent theory, we specialize for incompressible viscous flow and consider the problem of generalized Poiseuille flow within this size-dependent fluid mechanics. Finally, we conclude that the theory presented here may provide a basis for a broad range of fluid mechanics applications and for fundamental studies of flows at the finest scales for which a continuum representation is valid.