Spatially resolved stretching-rotation-stretching sequence in ﬂow topology as elementary structure of ﬂuid mixing Spatially resolved stretching-rotation-stretching sequence in ﬂow topology as elementary structure of ﬂuid mixing

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Spatially resolved stretching-rotation-stretching sequence in ﬂow topology as elementary structure of ﬂuid mixing

Spatially resolved stretching-rotation-stretching sequence in ﬂow topology

as elementary structure of ﬂuid mixing

Ankush G. Kumar, P. Vishal, V. Meenakshi, and R. Aravinda Narayanan†

Department of Physics, Birla Institute of Technology and Science Pilani - Hyderabad Campus, Hyderabad - 500 078,

INDIA

(*Corresponding author: †raghavan@hyderabad.bits-pilani.ac.in)

(Dated: 25 October 2022)

We performed two-dimensional numerical simulations of passive stirring of two liquids which generated two spatially

distinguishable paradigmatic velocity ﬂows, viz. few large extended vortices and rapid oscillations. Using the Okubo-

Weiss criterion, we mapped regions of stretching and rotation in the ﬂow ﬁeld. We ﬁnd that an elementary oscillatory

sequence of stretching-rotation-stretching induces spatial redistribution and steepening of concentration gradients, and

the aggregation of such sequences determines ﬂuid mixing at a downstream position. Furthermore, we quantify the

paradigms of ‘stretching’ and ‘folding’ by developing measures based on ﬂow topology which would facilitate the

design of stirring protocols.

This letter concerns the question of elementary structure of

mixing in homogeneous ﬂuids, which has been discussed con-

siderably in the literature in the last few decades1–4. When a

drop of ink is added to water, the speed at which the state of

uniform concentration is reached depends on whether water

is stationary or moving; the velocity ﬁeld determines the rate

of mixing. Insights on the mixing process can been gained

by either following the velocity ﬁeld or the passively coupled

concentration ﬁeld5. By picturizing a mixture as discrete in-

teracting sheets of varying concentration proﬁles shaped by

ﬂuid stirring and diffusion, it was shown that mixing is caused

by an aggregation of these effects3,4.

In a ﬂuid stirred through a spatially non-uniform velocity

ﬁeld, idealized ﬂuid elements are shear stretched in one di-

rection (compressed in the other) or rotated, bringing adjacent

ﬂuid elements closer to each other, producing steeper concen-

tration gradients to enhance mixing6; in experiments, this is

visualized as stretching and folding of material lines7–9. Re-

cently, ﬂuid deformations delineated through geometric mea-

sures showed that mixing occurs in two steps separated in time

scales10. Initial linear deformations are dominated by stretch-

ing which builds up to cause non-linear deformations associ-

ated with folding.

In this Letter, we quantify stretching and folding by devel-

oping measures that are directly based on ﬂuid velocity defor-

mations. The advantage of this approach is that mixing arising

out of speciﬁc stirring protocols can be distinguished. This

gains importance, as ﬂow ﬁelds, in particular, spatiotemporal

chaotic ﬁelds11–14 are being designed through active and pas-

sive geometries15,16, to overcome diffusion-limited mixing in

low Reynolds number (Re) micro-scale ﬂows. In our study, we

characterize two prototypical ﬂuid ﬂow topologies using the

Okubo-Weiss parameter (Q), which spatially resolves stretch-

ing and rotation components in a velocity ﬁeld17,18. We dis-

cover that ﬂuid stirring, and thereby, local mixing, is im-

printed in a speciﬁc stretching dominated sequential oscilla-

tion of Q. This elementary structure of mixing causes resdis-

tribution and sharpening of concentration gradients; the num-

ber of such sequences depends on the stirring scheme which

aggregates to determine the mixing state at a downstream lo-

cation.

A close connection exists between the state space of a

Hamiltonian system and a two-dimensional incompressible

ﬂow ﬁeld19,20. Hamiltonian dynamics are controlled by two

types of critical points, namely, elliptic (centre) and hyper-

bolic (saddle)21. Ouellette and Gollub showed that by con-

verting ﬂow trajectories of advected particles into curvature

ﬁelds, the type and spatial location of critical points can be

extracted22 using the Q parameter, which is deﬁned as fol-

lows:

Q=1

2(kΩk2− kSk2),(1)

where Ω=1

2(∇u−∇uT)is the vorticity, and S=1

2(∇u+

∇uT)is the rate-of-strain tensor: The two quantities are the

anti-symmetric and symmetric parts of the velocity gradient

tensor (∇u), respectively. They further demonstrated that hy-

perbolic (Q−:Q<0) and elliptic (Q+:Q>0) critical points

corresponding to stretching and rotation dominated regions of

the ﬂow, at low Re, forms an ordered square lattice: As Re

increases, the lattice gradually loses order, reﬂecting the de-

formation of the ﬂow ﬁeld22. Thus, it can be gleaned that crit-

ical points in state space govern the conﬁguration space ﬂow

dynamics23.

Chaos theory helps translate ideas from state space to real

ﬂows that have stretched and rotational character: A signa-

ture of chaotic mixing is the exponential separation of ﬂow

trajectories24. Such ﬂow separation is understood through two

types of iterated maps - Bakers maps (iterative stretch and

fold) and Twist map (whirling vortices), which help visual-

ize distinct ways to increase the concentration gradient and

thereby shorten the diffusion distances5. For our study, we

chose steady pressure-driven laminar ﬂow for the ease of con-

trol of the velocity ﬁeld, which carries the passive concentra-

tion ﬁeld.The prototypical passive mixing conﬁgurations, the

pair of bafﬂes (BF) and the teardrop (TD) shaped obstacle25,

simulated in this study, mimic the two ﬂow topologies (Fig.1).

Two-dimensional numerical simulations were performed

using COMSOL Multiphysics®26 software’s microﬂuidics

arXiv:2210.12171v1 [physics.flu-dyn] 21 Oct 2022

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Spatiallyresolvedstretching-rotation-stretchingsequenceinowtopologyaselementarystructureofuidmixingSpatiallyresolvedstretching-rotation-stretchingsequenceinowtopologyaselementarystructureofuidmixingAnkushG.Kumar,P.Vishal,V.Meenakshi,andR.AravindaNarayananDepartmentofPhysics,BirlaInstituteofTech...

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Spatially resolved stretching-rotation-stretching sequence in ﬂow topology as elementary structure of ﬂuid mixing Spatially resolved stretching-rotation-stretching sequence in ﬂow topology as elementary structure of ﬂuid mixing

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