Turbulent mixing and a generalized phase transition in shear-thickening fluids

Abstract

This paper presents a new theory of turbulent mixing in stirred reactors. The degree of homogeneity of a mixed fluid may be characterized by the Kolmogorov micro-scale. The smaller its value, the better homogeneity. The micro-scale scales inversely with the fourth root of the energy dissipation rate in the stirring process. The higher this rate, the smaller lambda, and the better the homogeneity in the reactor. This is true for Newtonian fluids. In non-Newtonian fluids the situation is different. For instance, in shear-thickening fluids it is plausible that high shear rates thicken the fluid and might strangle the mixing. The internal interactions between different fluid-mechanical and colloidal variables are subtle, namely due to the (until recently) very limited understanding of turbulence. Starting from a qualitatively new turbulence theory for inviscid fluids [Baumert, 2013], giving e.g. the Karman constant as $(2\pi)^{-1/2} = 0.40$ [the super-pipe in Princeton gave 0.40 p/m 0.02, Bailey et al., 2014], we generalize this approach to the case of viscous fluids and derive equations which in the steady state exhibit two solutions. One solution branch describes a state of good mixing, the other a state of strangled turbulence. The physical system cannot be in two different steady states at the same time. But it is physically admissible to switch between the two steady states by non-stationary transitions, maybe in a chaotic fashion.