Description of mixing with diffusion and reaction in terms of the concept of material surfaces

The essence of fluid-mechanical mixing of diffusing and reacting fluids can be traced back to kinematics, connectedness of material volumes and transport processes occurring across deforming material surfaces. Descriptions based on kinematics of homoeomorphic deforming material surfaces (tracers) are restricted solely to continuous motions and conveniently analysed by transport equations in Lagrangian frames. Connectedness of material volumes restricts the mixing topology and generates bicontinuous structures characterized by intermaterial-area and striation-thickness distributions. Upper bounds for area generation and material-line elongation are related to mean values of viscous dissipation and govern the average reaction rate in diffusion-controlled reactions. Two concepts are introduced: micromixing, related to local flows, rate of stretching and local viscous dissipation, and macromixing, associated with connectedness of isoconcentration surfaces, vorticity and average viscous dissipation. Several small-scale flows can be used to typify the interplay between fluid mechanics, mass and energy transport, and chemical reactions: elliptically symmetrical stagnation flows, vortex decay, and swirling flow with uniform stretching. It is proposed that complex fluid motions might be interpreted in terms of integrated behaviour of populations of small-scale flows distributed in space and time to simulate mixing behaviour. The objective of this work is to present the foundations of a continuum mixing description making reference to earlier approaches to demonstrate computational applicability and practical significance.