Multiscale Mechanics and Thermodynamics of Suspensions

Fluids with internal structure that evolves in time on a scale that is comparable with the scale on which the macroscopic flow evolves exhibit a complex flow behavior and are therefore called complex fluids. The internal structure can be flow-induced (e.g. the structure emerging in turbulent flows) or it can also be a mesoscopic or microscopic structure of the fluids at rest. In this chapter, we investigate complex fluids in which the internal structure is the structure of suspended particles. The particles that we consider in this chapter are rigid and deformable fibers and lamellae, deformable ellipsoids, and rigid spheres. Our main objective is to present a systematic method allowing us to express our microscopic physical insight into a coherent mathematical formulation, called a rheological model, which provides a bridge between the microscopic properties and observed macroscopic flow and morphological behavior. The method is described in general terms in section 10.2 and is illustrated on four types of suspensions in sections 10.3–10.6. The governing equations of all the rheological models presented in this chapter arise as particular realizations of a single general model. We concentrate our attention on the translation of the microscopic and mesoscopic physics into mesoscopic governing equations. As for the passage from the governing equations to results of macroscopic rheological and morphological observations (i.e. passage from the governing equations to their solutions), we limit ourselves mainly to qualitative properties of solutions demonstrating agreement with qualitative experimental observations of the compatibility with mesoscopic models formulated on different scales (for instance, the compatibility with equilibrium thermodynamics). As for the comparison with specific and detailed rheological measurements, we present some results of this type but mainly we refer the readers to published papers where such details of solutions of the governing equations as well as their comparison with results of rheological observations can be found.