Applications of a stretch model to mixing, diffusion, and reaction in laminar and turbulent flows

Abstract

NOTATION AJAf = amplitude of surface center-line oscillation/amplitude of floor oscillation B = model parameter related to the amount of deformation at which the surface loses its memory (see Gardner, 1975) LITERATURE CITED Addison, Greek Letten model parameter model parameter dirac delta function surface shear rate magnitude steady shear surface shear viscosity complex surface viscosity dynamic surface viscosity (real part of 11') imaginary part of zero shear rate surface viscosity infinite shear rate surface viscosity relaxation time surface pressure = floor rotation rate, maximum allowable rotation rate frequency Applications of a Stretch Model Diffusion, and Reaction in Laminar and Turbulent FlowsIn a Lagrangian frame of reference based on lamina (fluid filament) thickness and in a warped time scale based on a single, flow dependent quantity, mixing, diffusion, and reaction can be described in a relatively simple way. Applications are presented for stretch and fold in taffy pull, egg beater and static mixer, shear stretch, stretch of laminae in a vortex, mixing with diffusion, reaction rate controlled by diffusion of reactant through a product layer, and very fast reactions in a turbulent flow. SCOPEMixing, diffusion, and reaction of soluble fluids have been analyzed primarily in a fixed frame of reference and in terms of the statistical descriptions and theories of fluid mechanics or in terms of variables of distributed value such as residence time. The objective here is to introduce in terse form a type of description and analysis which results in deterministic and predictive models capable of

Interfacial mass transport in steady three-dimensional flows in microchannels

“…Figure 12(a) shows the result of scaling the axial distance in the chaotically stirred flow in figure 1(a) by Pe 1/6 . This scaling provides notably better collapse than ln(Pe), that was expected due to Ranz (1979) and Villermaux et al (2008). Scaling by ln(Pe) causes the curves for various Pe to have similar shapes, i.e.…”

“…For a uniaxial flow, mixing occurs purely by diffusion and the mixing length is expected to scale linearly with Pe. For an efficiently mixed flow such as the chaotic flow in figure 1(a), the mixing length is expected to scale as either ln(Pe) (Ranz 1979, Villermaux and Duplat 2003, Villermaux et al 2008 or as a weak power law in Pe (Simonnet and Groisman 2005). The non-chaotic three-dimensional flow in figure 1(b) is expected to show scaling with Pe that falls somewhere between these two extremes.…”

Thermal delay of drop coalescence

“…Only on longer time scales does thermal diffusion become effective and the drop equilibrate thermally. A mathematical framework to study related advection-diffusion problems has been developed in the context of mixing by Ottino, Ranz & Macosko (1979) and Ranz (1979), and recently applied by Meunier & Villermaux (2003) to describe scalar mixing in an axisymmetric vortex flow. In the following, we apply this method to the case of thermal homogenization within the drop, assuming that a roughly symmetric process is also taking place in the bath (Savino et al 2003).…”

Chaotic mixing of a competitive–consecutive reaction

“…The chemical species diffuse (in the x-direction) and react; they are also advected by a straining flow (u, v) = (−µx, µy), which allows us to retain the effect of mixing in stretching the interface exponentially in time, but without the corresponding folding [2,12,13,[16][17][18][40][41][42]. The appropriate dimensionless governing equations are then…”

“…Since accurate two-dimensional simulations are computationally expensive, various reduced descriptions are routinely used to model the progress of the reactions [1,2,11,12]. The crudest of these ignore the segregation entirely, and model the system as becoming instantaneously well mixed [1].…”