On the role of initial velocities in pair dispersion in a microfluidic chaotic flow

Abstract

Chaotic flows drive mixing and efficient transport in fluids, as well as the associated beautiful complex patterns familiar to us from our every day life experience. Generating such flows at small scales where viscosity takes over is highly challenging from both the theoretical and engineering perspectives. This can be overcome by introducing a minuscule amount of long flexible polymers, resulting in a chaotic flow dubbed 'elastic turbulence'. At the basis of the theoretical frameworks for its study lie the assumptions of a spatially smooth and random-in-time velocity field. Previous measurements of elastic turbulence have been limited to two-dimensions. Using a novel three-dimensional particle tracking method, we conduct a microfluidic experiment, allowing us to explore elastic turbulence from the perspective of particles moving with the flow. Our findings show that the smoothness assumption breaks already at scales smaller than a tenth of the system size. Moreover, we provide conclusive experimental evidence that 'ballistic' separation prevails in the dynamics of pairs of tracers over long times and distances, exhibiting a memory of the initial separation velocities. The ballistic dispersion is universal, yet it has been overlooked so far in the context of small scales chaotic flows.Elastic turbulence, a random-in-time flow, can drive efficient mixing in microfluidics. Using a 3D particle tracking method, the authors show that the smoothness assumption breaks at scales far smaller than believed and the ballistic pair dispersion holds over much longer distances than expected.

Fig. 1

Pair dynamics example. The trajectories of two tracers are plotted in the left panels. The right panel shows a sub-sample of pair separation distances in the course of time. The figure outlines the analysis forming the ensembles of pairs, as well as demonstrates the chaotic nature of the flow as manifested by pairs; to develop the intuition and contrast with laminar flow, the reader is referred to, e.g., (Figs. 21–22 of ref. ); several features of the mean flow in our case are manifested in the Eulerian representation in Supplementary Fig. 2, particularly the striking differences from Poiseuille-like laminar flows. a A projection on to the plane of the camera, which is imaging the channel from the bottom side (gravity pointing out of the panel towards the reader), overlaid on a bright field image of the observation window (further technical details are provided in Supplementary Fig. 1 and in the Methods section). b A side projection; the vertical axis is aligned with that of gravity, as well as the channel depth, 0 μm marking the channel bottom plane; as the width of this panel spans a spatial range which is nearly six times longer than its height, for the sake of visualisation the vertical axis is stretched by 3/2; the colour code in the plot denotes time, which spans 4 s in this case. All pairs of tracers which were detected at some instant at a prescribed separation distance, R ₀ = 10 ± 0.5 μm in this particular example, are collected to form one ensemble. The event at which the pair separation was nearest to R ₀, marked by the red circles in the plot, is recorded as t ₀ for the specific pair for later analysis. Each R ₀ bin is 1 μm wide and centred at 6 through 50 μm, with sample sizes ranging from nearly 10⁴ to over 10⁶ pairs, respectively; sample size data are presented in Supplementary Fig. 6. c A sub-sample of pair separation distances R(δt) for 49 pairs belonging to the R ₀ = 10 μm ensemble, presented on a semi-logarithmic scale; for each pair, δt = t − t ₀ is the time elapsed since t ₀. The colour code denotes time, scaled separately for each curve

Fig. 2

Pair dispersion normalised by the initial separation. The plot shows the average squared pair separation distance, normalised by the initial separation, $⟨{{\left(R\left(\delta t\right)\mathrm{∕}{R}_0\right)}^2⟩}_{R_0}$ for various R ₀ between 6 and 50 μm; curves satisfying the asymptotic exponential pair dispersion $⟨R^2\left(\delta t\right)⟩=R_0^2\exp \left[2\xi \delta t\right]$, Supplementary Equation 3, would show-up on this semi-logarithmic presentation as straight lines, all sharing the same slope and, when extrapolated, hitting the origin, i.e., they should all collapse on a single linear relation. The insets present a zoom-in on the initial and intermediate temporal sub-intervals where the full range plot may seem to contain linear segments. Nevertheless, there is no unique slope which can be identified. Moreover, an exponential pair dispersion should extrapolate to the origin on this plot, which is clearly not the case here, and the curves do not merge asymptotically. The data show no supporting evidence for the exponential time dependence which follows Supplementary Equation 3. The un-normalised data ${⟨R^2\left(\delta t\right)⟩}_{R_0}$ can be found in Supplementary Fig. 5

Fig. 3

Initial relative velocity dependence on the separation distance. The second moments of the relative velocity ${⟨u^2⟩}_{R_0,t_0},$ (blue left-triangles) and the separation velocity ${⟨u_l^2⟩}_{R_0,t_0},$ (green right-triangle), where u _l = u⋅R/R, are plotted in the inset (right axis values are half the left ones) as function of the initial separation distance R ₀; both ensemble averages are taken at the initial time t ₀, when the pairs separation distance is closest to R ₀. Rescaling these data by the squared initial separation $R_0^2$ reveals the deviation from the commonly applied assumption of linear velocity field, as presented on a logarithmic scale in the main plot (right axis values are one order of magnitude smaller than the left ones). Had 〈u ²〉_R ∝ R ² held, the rescaled curves would have remained constant; this is clearly not the case. Indeed, the ${⟨u_l^2⟩}_{R_0,t_0}\mathrm{∕}{R}_0^2,$ data level off as R ₀ approaches the smaller distances, providing supporting evidence for the linearity of u _l with R at scales smaller than 12 μm. However, this does not hold beyond a tenth of the channel depth. A linear flow regime is not supported by the rescaled relative velocity data ${{⟨u^2⟩}_{R_0,t_0}R}_0^2$, which values keep increasing even for the smallest R ₀ values explored here. Further note that ${⟨u^2⟩}_{R_0,t_0}$ and ${⟨u_l^2⟩}_{R_0,t_0}$ (inset) are empirical estimators for the second order structure functions of the velocity and the longitudinal velocity, correspondingly; the former is the coefficient of the quadratic term in Eq. (2). The error bars in the inset (smaller than the marker) indicate the margin of error based on a 95% confidence interval

Fig. 4

Relative pair dispersion forward and backwards-in-time evolutions. a Forward-in-time $⟨{{∥\mathbf{R}\left(\delta t\right)-{\mathbf{R}}_0∥}^2⟩}_{R_0}$ for various initial separations (inset) between 6 and 50 μm, collapse initially on a single curve which follows a power-law δt ², once rescaled by the average squared relative velocity at the initial time, $⟨{u^2⟩}_{R_0,t_0}$. A significant deviation from δt ² is noticed after 2–3 s, indicating the time beyond which higher order terms should be considered. b Backwards-in-time relative pair dispersion $⟨{{∥\mathbf{R}\left(-\delta t\right)-{\mathbf{R}}_0∥}^2⟩}_{R_0}$ for the same initial separations (inset), showing the same initial scaling collapse as the forward in time

Fig. 5

Relative pair dispersion time asymmetric terms and dimensionless form. a Taking the difference between the data sets plotted in the insets of Fig. 4, $\frac{1}{2}⟨{{∥\mathbf{R}\left(\delta t\right)-{\mathbf{R}}_0∥}^2-{∥\mathbf{R}\left(-\delta t\right)-{\mathbf{R}}_0∥}^2⟩}_{R_0}$, exposes the contribution of the time asymmetric terms, odd powers in δt, presented here in the inset (sign inverted). Rescaling by the empirical estimator for ${⟨\stackrel{°}{\mathbf{u}}\cdot \mathbf{u}⟩}_{R_0,t_0}$, these data collapse on δt ³ initially; the data sets of R ₀ ≤ 10 μm (grey in the legend) are omitted from the main figure due to the scatter of the estimator; Supplementary Fig. 3. The plot shows a deviation from δt ³ at times shorter than 300 ms, indicating the dominance of higher order (odd) terms at early times and that the δt ³ term alone does not trivially explain the deviation from δt ², observed in Fig. 4 after more than 2 s. b Rescaling the relative pair dispersion data (inset of Fig. 4a) by $⟨{u^2⟩}_{R_0,t_0}{\left(\delta t_{R_0}^{*}\right)}^2$ (see Eq. (2)), results in a dimensionless form, plotted here against dimensionless time, δt rescaled by $\delta t_{R_0}^{*}=\mid {⟨u^2⟩}_{R_0,t_0}\mathrm{∕}{⟨\stackrel{°}{\mathbf{u}}\cdot \mathbf{u}⟩}_{R_0,t_0}\mid$; the empirical estimators of $\delta t_{R_0}^{*}$ can be found in Supplementary Fig. 4. The data sets indeed collapse onto a single curve ${\left(\delta t\mathrm{∕}\delta t_{R_0}^{*}\right)}^2-{\left(\delta t\mathrm{∕}\delta t_{R_0}^{*}\right)}^3,$ (dashed black line) for $\delta t\mathrm{∕}\delta t_{R_0}^{*}$ ≲ 0.2. The zoom-in (inset) emphasises the behaviour as $\delta t\mathrm{∕}\delta t_{R_0}^{*},$ approaches unity and the first two terms cancel out each other