Lagrangian transport characteristics of a class of three-dimensional inline-mixing flows with fluid inertia

Abstract

Laminar mixing by the inline-mixing principle is a key to many industrial fluids-engineering systems of size extending from micrometers to meters. However, insight into fundamental transport phenomena particularly under the realistic conditions of three-dimensionality (3D) and fluid inertia remains limited. This study addresses these issues for inline mixers with cylindrical geometries and adopts the Rotated Arc Mixer (RAM) as a representative system. Transport is investigated from a Lagrangian perspective by identifying and examining coherent structures that form in the 3D streamline portrait. 3D effects and fluid inertia introduce three key features that are not found in simplified configurations: transition zones between consecutive mixing cells of the inline-mixing flow; local upstream flow (in certain parameter regimes); transition/inertia-induced breaking of symmetries in the Lagrangian equations of motion (causing topological changes in coherent structures). Topological considerations strongly suggest that there nonetheless always exists a net throughflow region between inlet and outlet of the inline-mixing flow that is strictly separated from possible internal regions. The Lagrangian dynamics in this region admits representation by a 2D time-periodic Hamiltonian system. This establishes one fundamental kinematic structure for the present class of inline-mixing flows and implies universal behavior in that all states follow from the Hamiltonian breakdown of one common integrable state. A so-called period-doubling bifurcation is the only way to eliminate transport barriers originating from this state and thus is a necessary (yet not sufficient) condition for global chaos. Important in a practical context is that a common simplification in literature, i.e., cell-wise fully-developed Stokes flow (“2.5D approach”), retains these fundamental kinematic properties and deviates from the generic 3D inertial case only in a quantitative sense. This substantiates its suitability for (at least first exploratory) studies on (qualitative) mixing properties.

Scalar transport in inline mixers with spatially periodic flows

“…This Poincaré sectioning is a common technique for examining inline mixers. 6,9,11,[13][14][15] The Poincaré section of the Quatro mixer under current flow conditions (and Re = 0.5) is obtained by introducing 40 randomly distributed tracer particles at the mid-plane of the second mixing element of a periodic segment of the Quatro mixer. This is for consistency with the scalar field analysis in Section VI.…”

Topological equivalence between two classes of three-dimensional steady cavity flows: A numerical-experimental analysis

“…PLEASE CITE THIS ARTICLE AS DOI:10.1063/1.5126497 pattern or 'cat-eyes' flow structure. 17,24,42 In this case the streamline pattern is also symmetric about the planes x, z = 0 according to symmetries ( 7)- (8). Combination of these extra symmetries with S P results in the formation of three stagnation lines (corresponding to the three stagnation points in the symmetry plane) extending along y.…”

“…This is fundamental to the analysis of transport phenomena. [1][2][3][5][6][7][8][9] Flows in cavities are one of the canonical configurations to study Lagrangian transport and chaotic advection in three-dimensional (3D) laminar flows. 2 In this context, buoyancy-driven and lid-driven flows have been considered as archetypal configurations to study fundamental aspects of advection.…”

Designing a Finite-Time Mixer: Optimizing Stirring for Two-Dimensional Maps

“…For instance, one could introduce variable phases into the shears to control the locations of the fixed points at every step. We hope to use results gained from studying these simple, theoretical models to aid investigations of more physically applicable, three-dimensional models such as the partitioned pipe and rotated arc mixers [GAPM03,SDMC14].…”

“…For example, the partitioned-pipe mixer [GAPM03] and the rotated-arc mixer (RAM) [SDMC14] are finite-time mixers in which fluid is injected at one end of a cylindrical pipe, flows past a finite number of mixing elements, and is emitted at the other end. Other examples of this type of mixing include "making up rubber formulations on compounding rolls, using an extruder as a mixer, kneading bread dough, and pulling taffy" [SW51].…”