Experimental and computational studies of mixing in complex Stokes flows: the vortex mixing flow and multicellular cavity flows

Abstract

A complex Stokes flow has several cells, is subject to bifurcation, and its velocity field is, with rare exceptions, only available from numerical computations. We present experimental and computational studies of two new complex Stokes flows: a vortex mixing flow and multicell flows in slender cavities. We develop topological relations between the geometry of the flow domain and the family of physically realizable flows; we study bifurcations and symmetries, in particular to reveal how the forcing protocol's phase hides or reveals symmetries. Using a variety of dynamical tools, comparisons of boundary integral equation numerical computations to dye advection experiments are made throughout. Several findings challenge commonly accepted wisdom. For example, we show that higher-order periodic points can be more important than period-one points in establishing the advection template and extended regions of large stretching. We demonstrate also that a broad class of forcing functions produces the same qualitative mixing patterns. We experimentally verify the existence of potential mixing zones for adiabatic forcing and investigate the crossover from adiabatic to non-adiabatic behaviour. Finally, we use the entire array of tools to address an optimization problem for a complex flow. We conclude that none of the dynamical tools alone can successfully fulfil the role of a merit function; however, the collection of tools can be applied successively as a dynamical sieve to uncover a global optimum.

Laminar mixing and heat transfer for constant heat flux boundary condition

“…the rods located at the midpoint between the center of the tank and its boundary, was found to be close to the optimum in our previous study [12]. This value is also near to the optimum of eccentricity found by Jana et al [20] for a radii ratio R 3 /R 1 = 4 and for a fixed tank and alternately rotating rods. …”

“…It is similar to the geometry of the two-roll-mills studied in the literature by Price et al [17] or Young et al [18] and Chiu et al [19]. This geometry was studied in details by Jana et al [20] under the name of "Vortex mixing flow". They used numerical and experimental tools to study the mixing of a scalar (dye in the experiments) by rotating alternately the two rods in a fixed tank (the outer cylinder).…”

Mixing in Stokes flow in an annular wedge cavity

“…In spite of their complicated expressions, the terms in series (27) and their first partial derivatives decay exponentially with ν. Only a few first terms (usually, ν 5) with obvious limits for the term with ν = 1 at the boundary are enough to provide an excellent accuracy of calculations of the velocity field in the whole cavity.…”

“…The traditional approach, based upon the presentation of the dyed blob as a collection of N points uniformly distributed over the area S b of the blob, can provide a reasonable treatment of mixing with excellent correspondence with the experiments, even in complex domains (Jana et al [27]). For long time evolutions, however, this approach provides only a qualitative general picture of mixing (see Liu et al [28] for several examples).…”

Two-dimensional Stokes flow driven by elliptical paddles

“…The first, the iterated mapping plot, 6 gives qualitative information about the long-time behavior of fluid particles, and indicates regions of regular and chaotic Lagrangian particle paths. Although its relevance to shorttime mixing may be disputed, 13 the iterated mapping does provide a strong visual indication of the extent, if any, of the chaotic regions. The second diagnostic is the rate of stretch of a finite material line.…”