Theoretical developments to model microstructural effects on the rheology of complex fluids
Faroughi, Salah Aldin
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Many processes in engineering and science involve multiphase flow, and the rate at which they operate is controlled by multiphase rheology. A rheology model with predictive capability over a wide range of physical and flow conditions is, therefore, of interest to quantify and optimize these processes. Complex fluids, a family of multiphase fluids, are ubiquitous across a broad range of industrial setting, including Pharmaceutical products, Food & Dairy, Oil & Gas, Mining & Coating, Cement & Concrete, Cosmetics, Composite & Powder, and Polymers and Biological fluids. Complex fluids include one continuous phase and one or more external phases such as suspended solid particles, droplets of liquid, bubbles of gas that can be rigid or deformable, and can stick together or repel each other. The interactions between constituents cause the overall behavior of these fluids to be different from pure fluids and solids. The rheology of complex fluids can be highly non-linear. At very low volume fraction of the dispersed phase (volume fraction of $\psi < 0.1$), the relationship between the relative viscosity of the complex fluids and $\psi$ is simple and linear, i.e. it does not depend on the particle size distributions and the deformation history. Under these conditions, experimental data agree well with each other and allow the development and application of simple rheology models. The situation becomes more challenging as the volume fraction of particles increases and hydrodynamic interactions between particles become more important. For instance, as particle volume fractions approach the maximum packing limit, the particle shape, deformation and size distributions, the formation of a microstructure or fabric (including the orientation of particles, their arrangement along shear planes) and the shear conditions under which the fluid is flowing play a more crucial role. This complexity is illustrated by the non-linear response of complex fluids to deformation and especially the lack of self-similarity observed experimentally. This loss of self-similarity, however, might be an artifact resulting from ignoring several important processes and key variables describing the dependence of complex fluids rheology on the rate of deformation. We present a theoretical framework to build a generalized rheological model for non-Brownian complex fluids subjected to a steady straining flow at low Reynolds number. We first consider the effect of a single deformable fluid particle on the ambient velocity and stress fields to constrain the rheological behavior of dilute mixtures. We then extend the solution to the rheology of concentrated complex fluids using an incremental differential effective medium theory operating in a fixed and finite volume. These analyses provide a framework to include non-linear microstructural processes in the description of the shear deformation of complex fluids, e.g., particle shape and size distribution, particle deformation and breakage, and particle alignment and rearrangement. The proposed framework is based on new state variables and is designed to restore self-similarity by providing a microscopic parameterization for the influence of the shearing condition on the rearrangement and evolution of microstructures. It forms the basis for the derivation of a predictive model for the effective viscosity of concentrated complex fluids.