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The main objective of this work is to demonstrate that non-local terms of the structure variable and shear-stress is a sufficient condition to predict multiple bands in rheologically complex fluids, i.e., shear-thickening fluids. Here, shear bands are considered as dissipative structures arising from spatial instabilities (Turing patterns) rather than the classical mechanical instability. In the present analysis, a monotonic relation between shear-stress and shear-rate holds. The formation of banded patterns is discussed for shear-thickening fluids with a model that consist of an upper-convected Maxwell-type constitutive equation coupled to an evolution equation for the structure variable, in which both non-local terms of the stress and of the structure variable are included (non-local Bautista-Manero-Puig model). The Turing mechanism is used to predict a critical point for primary instabilities (stable bands), while the amplitude formalism is used to predict secondary instabilities and marginal curves.

In this work, the dynamics of the bioconvection process of gravitactic microorganisms enclosed in a rectangular cavity, is analyzed. The dimensionless cell and energy conservation equations are coupled with the vorticity-stream function formulation. Then, the effects of the bioconvection Rayleigh number and the heating source on the dynamics of microorganisms are discussed. The results based in streamlines, concentration and temperature contours are obtained through numerical simulations considering eight different configurations of symmetrical and asymmetrical heat sources. It is concluded that microorganisms accumulate in the warmer regions and swim through the cooler regions to reach the surface. They form cells for each heat source, but at high concentrations, they form a single stable cell. The results presented here can be applied to control and to understand the dynamics of microorganisms with discrete heat sources.