Bacteria hinder large-scale transport and enhance small-scale mixing in time-periodic flows

Significance

How does the swimming motion of bacteria affect the mixing of passive scalars in chaotic flows? Answers to this question can lead to a better understanding of the formation of algal blooms in oceans and lakes, as well as potentially useful applications in vaccine and biofuel production. Our experiments show that the presence of swimming Escherichia coli in two-dimensional chaotic flows can hinder large-scale transport and reduce flow chaoticity. Locally, however, bacteria can significantly enhance small-scale mixing. Our simulations also reveal the potential mechanism for these phenomena being the transient accumulation and dispersion of bacteria near regions of large fluid stretching.

Abstract

Understanding mixing and transport of passive scalars in active fluids is important to many natural (e.g., algal blooms) and industrial (e.g., biofuel, vaccine production) processes. Here, we study the mixing of a passive scalar (dye) in dilute suspensions of swimming Escherichia coli in experiments using a two-dimensional (2D) time-periodic flow and in a simple simulation. Results show that the presence of bacteria hinders large-scale transport and reduces overall mixing rate. Stretching fields, calculated from experimentally measured velocity fields, show that bacterial activity attenuates fluid stretching and lowers flow chaoticity. Simulations suggest that this attenuation may be attributed to a transient accumulation of bacteria along regions of high stretching. Spatial power spectra and correlation functions of dye-concentration fields show that the transport of scalar variance across scales is also hindered by bacterial activity, resulting in an increase in average size and lifetime of structures. On the other hand, at small scales, activity seems to enhance local mixing. One piece of evidence is that the probability distribution of the spatial concentration gradients is nearly symmetric with a vanishing skewness. Overall, our results show that the coupling between activity and flow can lead to nontrivial effects on mixing and transport.

Swimming microorganisms often live in environments with fluid flows across a range of length scales, from natural habitats like oceans to industrial biofuel plants to human intestines (1). Flow can exert forces and torques on microorganisms, which can affect their locomotion and transport (2). Microorganisms, in turn, adapt their swimming directions and take advantage of flow gradients in order to forage and reproduce (3). This coupling between activity (i.e., swimming motion) and flow leads to intriguing physical phenomena that are not seen for passive particles, such as rheotaxis (4, 5), gyrotaxis (6, 7), and flow-induced chemotaxis (8, 9).

Even simple flows, such as shear flows, can profoundly alter the movement of microswimmers (10–12). Experiments show that motile bacteria can drift across streamlines out of the plane of shear (5), escape the low-shear regions, and get trapped in high-shear regions (10). Motile phytoplankton are found to deplete or accumulate in regions of different shear rates (11) and form intense cell assemblages called “thin layers” (12). Under quiescent flow condition, microswimmers can still induce fluctuating velocity fields in the fluid by moving collectively (13, 14). These self-induced flows can lead to unique properties in fluids laden with active particles, or, namely, “active suspensions.” Examples include enhanced Brownian diffusivity (15–17), active fluid transport and mixing (18–20), and even the possibility of work extraction (21, 22).

In more complex flows, the behavior of microswimmers shows rich dynamics, but is much less understood. In turbulent flows, gyrotactic swimmers are found to cluster in small-scale patches (6, 23) and gather in regions of positive velocity gradients (24). Numerical simulations in three-dimensional isotropic turbulence show that elongated swimmers preferentially align with flow velocity (25), and clustering and patchiness are greatly reduced (26). In chaotic flows, simulations show that rod-like swimmers can be trapped or expelled by elliptic islands (27), i.e., Kolmogorov–Arnold–Moser tori (28), depending on their shapes and swimming speeds. The trapping of particles in elliptic islands can even lead to a reduction in swimmer transport (29, 30). Recently, a study on steady flows in model porous media showed that bacteria align and accumulate with Lagrangian structures (31). However, how these dynamics affect mixing and transport in flows exhibiting chaotic advection are yet to be experimentally tested. Moreover, most simulations treat swimmers as self-propelled particles that do not feedback on the flow, and the effects of activity on flow are far less explored.

In this study, we experimentally investigate how swimming bacteria affect the transport and mixing of a passive scalar in a time-periodic chaotic flow. Dye experiments show that bacteria can significantly hinder large-scale transport and global mixing rates. These results are further characterized by computing the stretching fields from experimentally measured velocity fields, which show that bacteria attenuate large-amplitude fluid stretching. This leads to a lower mean finite-time Lyapunov exponent (FTLE), indicating that activity also decreases flow chaoticity. At small scales, however, bacteria activity can substantially increase local mixing. These two effects lead to a new balance in the dynamical system characterized by a delay in the formation of persistent structures that are overall more homogeneous. A simple numerical simulation reveals the potential mechanism for the experimental phenomena being the transient accumulation of swimming particles along the unstable manifolds of the flow.

We use the flow-cell setup (32, 33) to create a two-dimensional mixing flow with time-periodic magnetic forcing (Fig. 1A; see SI Appendix for details). As a sinusoidal voltage is imposed, the induced Lorentz forces in the fluid create a vortex array of alternating vorticity (Fig. 1B). The size of each vortex corresponds to the magnets’ spacing, $L$ = 6 mm. The flow is characterized by two dimensionless numbers. The first is the Reynolds number, $\mathrm{R}\mathrm{e}=ŪL/\nu$, where $Ū$ is the rms velocity, and $\nu$ is the fluid kinematic viscosity. The second is the path length, $p=Ū/Lf$, with $f$ being the driving frequency of the flow, which describes the normalized mean displacement of a typical fluid parcel in one period. To better contrast the effects of bacterial activity, we keep the $\mathrm{R}\mathrm{e}$ and $p$ at ∼14.0 and 2.3 in all our experiments, respectively. These conditions ($\mathrm{R}\mathrm{e}$, $p>1$) are known to produce chaotic mixing (33, 34), even though the flow preserves spatial and temporal symmetries.

Fig. 1.

(A) Schematic of the flow-cell apparatus. A thin layer of buffer solution with E. coli is placed above an array of magnets of alternating polarities (denoted by different colors). A sinusoidal current induces Lorentz force in the fluid to drive the periodic mixing. The right half of the fluid is labeled with fluorescent dye. (B) Vorticity field (color map) and velocity field (arrows) of the periodic mixing flow, corresponding to the first peak value in a time period (5 s). The data are measured in a region of 24 $\times$ 24 mm², at a Reynolds number of Re = 16.8. (C) Photographs of dye-mixing experiments in a 48 $\times$ 48 mm² region, for different bacteria volume fractions: ${\varphi }_b=0\%$ (Left) and ${\varphi }_b=0.9\%$ (Right). The data are taken after n = 300 periods of mixing at a frequency f = 0.2 Hz and Reynolds number Re = 16.8. (D) The magnitude of concentration gradient of the dye field in C, for ${\varphi }_b=0\%$ (Left) and ${\varphi }_b=0.9\%$ (Right), enlarged to a region of 24 $\times$ 24 mm² for better illustration. The data are normalized by the rms concentration gradient.

The effects of activity are investigated by adding swimming Escherichia coli to the buffer solution (Materials and Methods). The swimming speed of the bacteria ranges from 10 to 20 $\mu$m/s (35). The bacteria volume fraction ${\varphi }_b$ is adjusted from 0% (pure buffer) to 0.9%; experiments with nonmotile bacteria are performed at ${\varphi }_b=0.9\%$. This volume fraction range is considered dilute (36), without introducing large-scale collective motion. We note that the active Péclet number, $Pe=ŪL/D_{\text{eff}}$, is much larger than unity ($\sim 10^6$), where $D_{\text{eff}}$ is the effective diffusivity of E. coli (37). Passive dye-mixing experiments are performed by adding a minute amount of fluorescent dye to the fluid. Initially, the flow cell is partitioned by a solid barrier into two halves, one with and one without dye (Fig. 1C). The barrier is then lifted, and dye penetrates to the undyed portion with time or number of cycles $N$. The duration of each experiment is ∼30 min, or $N\approx 400$, over which the bacterial motility remains roughly constant (SI Appendix, Fig. S11). We also perform particle-tracking velocimetry (PTV) to obtain the velocity fields of the flow (Fig. 1B; Materials and Methods).

An example of the effects of the bacteria on dye mixing is shown in Fig. 1C, for pure buffer (${\varphi }_b=0\%$) and an active suspension (${\varphi }_b=0.9\%$). The snapshots are taken at $N=300$ (see Movies S1 and S2 and SI Appendix, Fig. S1 for other times and nonmotile data). A comparison between the two snapshots reveals that, in the presence of motile bacteria, the dye penetrates to the undyed region much slower than the buffer and nonmotile (SI Appendix, Fig. S1) cases. This indicates that bacterial activity is hindering large-scale (dye) mixing and transport. A closer look at the images show that finer dye structures within a vortex cells are also modified when motile bacteria are present; bacterial activity leads to smoother and less structured concentration fields. Normalized concentration gradient fields (Fig. 1D) show that, for both cases, gradients are steeper near the flow separatrices (see SI Appendix, Fig. S3 for nonmotile case). This suggests that the passive scalar is transported by regions of highest flow strain or material stretching (32, 38). However, the gradient fields in the active case (${\varphi }_b=0.9\%$) are broader and coarser, suggesting a higher diffusion with bacterial activity. These observations are consistent for all bacterial volume fractions ${\varphi }_b$ investigated here (see SI Appendix, Fig. S2 for other ${\varphi }_b$ and nonmotile data). These experiments indicate that the presence of swimming bacteria affect both large-scale transport and small-scale mixing of the passive scalar in the time-periodic flow.

The overall mixing rate can be characterized by the variance of the dye-concentration field, $⟨C^2⟩$, where $⟨\cdot ⟩$ denotes the spatial ensemble average. We find an exponential decay of the normalized $⟨C^2⟩$ with time or $N$ for all cases (Fig. 2A), a behavior that is consistent with observations in (time-periodic) chaotic flows (33, 39); solid lines are exponential fits to the data, $C_0\exp \left(-RN\right)$, for $20<N<250$. We note that the decay is slower as ${\varphi }_b$ is increased, which is quantified by a linear decrease in mixing rate $R$ with ${\varphi }_b$ (Fig. 2 A, Inset). We expect the linearity to break down for nondilute bacterial suspensions due to hydrodynamic interaction and collective behavior. The value of $R$ for the nonmotile case is nearly double that of the active case at the same ${\varphi }_b$. This indicates that the observed decrease in large-scale mixing is not simply a viscous effect, since the addition of passive rod-like particles to the fluid will only lead to an increase in viscosity of 3% (SI Appendix). We also observe that the mixing slows down and deviates from the exponential decay at later times ($N>300$). This slower decay mode or mixing rate is governed by the longest wavelength in the flow (33), and it sets in earlier for active suspensions. These results show that bacterial activity hinders large-scale transport and decreases the global mixing rates of a passive scalar in time-periodic flows.

Fig. 2.

(A) Decay of the normalized scalar variance $⟨C^2⟩$ as a function of time (in the unit of periods), for the buffer (${\varphi }_b=0$), active suspension (A), and nonmotile bacteria suspension (NM). All data are normalized to have an initial condition of unity. Solid curves are exponential fits to the data, $⟨C^2⟩\sim C_0\exp \left(-RN\right)$. A, Inset shows the mixing rate $R$, which decreases linearly with bacterial volume fraction ${\varphi }_b$. The errors in the estimation of the mixing rate $R$ from the exponential fits are less than $2.0\%$. (B) The backward (time) FTLE fields, related to the past stretching by $\mathrm{\Lambda }=\left(\log S\right)/\mathrm{\Delta }t$, for $\mathrm{\Delta }t=1.5\hspace{0.17em}\mathrm{P}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{o}\mathrm{d}\mathrm{s}$. (B, Upper) Buffer solution (${\varphi }_b=0$). (B, Lower) Active suspension (${\varphi }_b=0.9\%$). The magnitude of $\mathrm{\Lambda }$ is attenuated in the presence of bacteria, while the shape of the structures remains similar. (C) Probability distributions of the backward FTLE, showing attenuation at high amplitudes ($\ge 1\hspace{0.17em}{\mathrm{P}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{o}\mathrm{d}\mathrm{s}}^{-1}$) with increasing ${\varphi }_b$. C, Inset displays a decrease in mean FTLE value $⟨\mathrm{\Lambda }⟩$ as a function of ${\varphi }_b$; nonmotile bacteria suspension has a much higher $⟨\mathrm{\Lambda }⟩$ value compared to active suspension of the same ${\varphi }_b$.

A key property of flows exhibiting chaotic advection is the exponential divergence of nearby trajectories in real space, usually characterized by the largest FTLE $\mathrm{\Lambda }$ over a time interval $\mathrm{\Delta }t$. One can relate $\mathrm{\Lambda }$ to the stretching experienced by a fluid parcel by considering the deformation of an infinitesimal circular fluid element initially located at $\left(x,y\right)$. The stretching $S$ is defined as the ratio of the final major diameter (after $\mathrm{\Delta }t$) to its initial diameter and $\mathrm{\Lambda }=\left(\log S\right)/\mathrm{\Delta }t$. Here, stretching fields $S\left(x,y\right)$ are computed from experimentally measured velocity fields (see SI Appendix for details). Two different quantities are computed at each point, namely, past and future stretching. These quantities, past and future stretching, tend to be large on the unstable and stable manifolds of the hyperbolic fixed points of the Poincaré map, respectively (40). The past stretching fields for both the buffer and active suspension (${\varphi }_b=0.9\%$) are highly heterogeneous, being much larger along the flow separatrices near hyperbolic points (Fig. 2B; see SI Appendix, Fig. S5 for nonmotile case). While the structures of the field are relatively similar for all cases, stretching is clearly attenuated for the active case, particularly at regions of large stretching. Indeed, the probability distribution of stretching shows that bacteria systematically suppress large values of stretching (Fig. 2C). Note that the probability function shifts and decays at lower stretching values (than buffer and nonmotile cases) as the concentration of motile bacteria is increased. This behavior is captured by computing the spatially averaged FTLE, which decreases as bacterial concentration increases (Fig. 2 C, Inset). Similar to the mixing rate results, the nonmotile case shows a larger value of $⟨\mathrm{\Lambda }⟩$ compared to active case at the same ${\varphi }_b$. Overall, these results show that bacteria hinder large-scale transport and mixing by suppressing the stretching of fluid elements.

The effects of bacteria activity across different length scales are examined by the spatial power spectrum of the dye-concentration field, $E_c\left(k\right)$ (see SI Appendix for details). The spectra $E_c\left(k\right)$ characterize the fluctuation of the scalar field across wavenumbers (41, 42), whose total spectral power is the scalar variance $⟨C^2⟩={\int }_0^{\infty }{E}_c\left(k\right)\hspace{0.17em}dk$. Examples of $E_c\left(k\right)$ for the buffer, nonmotile, and active cases measured at $N=300$ are shown in Fig. 3A (see SI Appendix, Fig. S8 for $E_c\left(k\right)$ at other times). We observe a power law of $E_c\left(k\right)\sim k^{-2.0}$ that spans a substantial range of $E_c\left(k\right)$, including wavenumbers above $k_L$ and below $k_{\eta }$. Here, $k_L=2\pi /L$ is the energy injection scale, as shown by peaks in the spectra, and $k_{\eta }=2\pi /\eta$ is the viscous cutoff scale (43), estimated by setting the local Reynolds number $R{e}_{\eta }=u_{\eta }\eta /\nu \sim 1$. We find that the spectral power increases nearly uniformly with ${\varphi }_b$ in the range of $k_L<k<k_{\eta }$, while the change in logarithmic slope is more apparent at smaller scales, $k>k_{\eta }$. Note that the area under $E_c\left(k\right)$ is larger for active suspensions, which is consistent with a larger remaining scalar variance (Fig. 2A). The increases in the total spectral power with ${\varphi }_b$ suggest that the rate of transfer of scalar variance from large scale (low $k$) to small scale (high $k$) is hindered by bacterial activity compared to the buffer and nonmotile cases.

Fig. 3.

(A) Concentration power spectra, $E_c\left(k\right)$, measured at n = 300 for the buffer, active suspension (A), and nonmotile suspension (NM). The inclined dashed line indicates a power law of $E_c\left(k\right)\sim k^{-2.0}$. Two vertical dashed lines mark the energy injection scale $k_L$ and the viscous cutoff scale $k_{\eta }$. (B) Spatial autocorrelation functions $R_c\left(r_y\right)$ of the dye concentration, measured at n = 300 for different ${\varphi }_b$. (C and D) Contour plots of the spatial–temporal autocorrelation function $R_c\left(r_y,r_t\right)$ for the buffer of ${\varphi }_b=0$ (C) and active suspension of ${\varphi }_b=0.9\%$ (D).

We now focus on large-scale structures in the scalar field by computing the spatial autocorrelation function, defined as $R_c\left(r_y\right)=⟨C\left(y\right)C\left(y+r_y\right)⟩/⟨C^2⟩$, where $y$ is the direction normal to the mean scalar gradient set by the initial condition (Fig. 1C). The correlation $R_c\left(r_y\right)$ is plotted for different ${\varphi }_b$, as well as buffer and nonmotile cases in Fig. 3B, which shows that (bacterial) activity leads to increasingly stronger spatial correlations, especially at $r_y<2L$ (within two vortex cells). The increase in the spatial correlation indicates the persistence of larger structures in the concentration field, even after a relatively long time at $N=300$. This result suggests that the stretching and folding mechanism responsible for creating thinner striations is hindered by bacterial activity, consistent with stretching field measurements. While we find long-range correlations and spatial symmetries in the scalar fields in the buffer and nonmotile cases (peaks at $r_y=2L$ and $4L$), these are nearly erased by bacterial activity (${\varphi }_b=0.9\%$). The absence of such peaks suggests that spatial periodicity of large-scale structures is broken by the bacterial activity.

Next, we examine the time evolution of dye structures by computing the spatial–temporal autocorrelation function $R_c\left(r_y,r_t\right)=⟨C\left(y,t\right)C\left(y+r_y,t+r_t\right)⟩/⟨C^2⟩$. For the buffer case, the contours of $R_c\left(r_y,r_t\right)$ quickly develop an invariant shape, and the decay of the correlation with time is nearly uniform in space for all $r_y$ (Fig. 3C). The initially formed invariant structures become decorrelated for $N>150$ due to diffusion. However, for the ${\varphi }_b=0.9\%$ case, we find that structures decorrelate much slower in time; correlations in the range $r_y<2L$ remain relatively high up to 200 periods. We believe that, for $r_y>2L$ (larger than two vortex cells), the structures are decorrelated due to local stochastic behavior of bacteria activity in each vortex cell (this will be discussed in more detail later). This slow decay in correlation shows the presence of larger structures in the concentration field, which are less susceptible to diffusion. Taken together, these results show that bacterial activity leads to a decrease in long-range spatial correlation, but an increase in temporal correlation. The structures created by the flow of these active fluids are coarser and more long-lasting than the passive case.

So far we have shown that bacterial activity can hinder large-scale transport and mixing by hindering the production of fine structures that decorrelate faster in time. To gain insights into mixing at small scales, we examine the scalar gradient fields (Fig. 1D). We further quantify the results by computing the probability density function (PDF) of the partial scalar gradient $\partial C/\partial x$ (Fig. 4A), whose skewness is an indicator of the local isotropy of scalar fields (44). Here, $x$ is defined as the direction of the mean scalar gradient (Fig. 1C). These PDFs have a non-Gaussian core with exponential tails at high gradient values. Remarkably, we find that the PDF for the active suspension (${\varphi }_b=0.9\%$) is nearly symmetric, while the PDFs for the buffer and nonmotile bacteria suspension are asymmetric. Further calculations show that the skewness values in the active suspension decrease by an order of magnitude relative to the buffer and nonmotile cases (Fig. 4 A, Inset). The active suspension has a vanishing steady-state skewness of 0.09, suggesting that the scalar field is almost isotropically distributed at small scales. This result is unexpected since local isotropy of scalar fields is rarely observed; the skewness of $\partial C/\partial x$ is at the order of unity (rather than zero), even at geophysical Reynolds number (44, 45). Stronger mixing at the “swimmer” scale was previously proposed (15–17), mostly by observing an increase in particle mean-square-displacement and effective diffusivity in quiescent flows. Here, in the presence of an imposed flow, we show that the swimmer–flow interaction can even lead to local isotropy in the scalar field.

Fig. 4.

(A) Probability distribution of the partial gradient $\partial C/\partial x$, measured at n = 300 for the buffer (${\varphi }_b=0$), active bacteria suspension (${\varphi }_b=0.9\%$; A), and nonmotile bacteria suspension (${\varphi }_b=0.9\%$; NM). A, Inset shows the skewness of these PDFs as a function of time. (B) Probability distribution of the concentration gradient magnitude $|\nabla C|$, normalized by the rms gradient ${⟨{\left(\nabla C\right)}^2⟩}^{1/2}$, measured at n = 300 for the buffer, active suspension (A), and nonmotile bacteria suspension (NM). B, Inset shows the differential entropy of the same PDFs versus time.

The effectiveness of the mixing process at the finer scales can be characterized by the PDF of the magnitude of scalar gradient $|\nabla C|$, normalized by ${⟨{\left(\nabla C\right)}^2⟩}^{1/2}$ to compensate for the decay of contrast (Fig. 4B). These PDFs reach invariant forms with time (SI Appendix, Fig. S9) that have a nonzero mean and a notably exponential tail. The invariant form is also known as a “persistent pattern” that is typical for time-periodic mixing (39). And an exponential distribution at high gradients is expected for a passive scalar subjected to a mean gradient undergoing random advection (46–48). Here, however, we show that the active suspension (${\varphi }_b=0.9\%$) reaches an invariant form distinct from the buffer and nonmotile cases, characterized by a longer and more pronounced exponential tail. It suggests that bacteria activity may be enhancing the randomness of the local advection of dye. Moreover, bacteria can also delay the formation of the invariant form by as much as 100 periods relative to the buffer and nonmotile cases, which is captured by the differential entropy of the PDFs (Fig. 4 B, Inset; see SI Appendix for the calculation). This is because stretching is adversely affected by the activity. The balance between these two effects—i.e., the decrease in stretching and the enhancement in diffusion—leads to a slower, yet better, small-scale mixing, as shown by a larger entropy in the active suspension (Fig. 4 B, Inset; $N>200$).

To gain further insight into these results, we perform numerical simulations of swimming particles in the flow (see Materials and Methods for details). Initially, swimmers are uniformly distributed in the simulation domain with random orientations. As the simulation begins, we see strong accumulation of particles along the flow unstable manifolds (Fig. 2B) at an early time of $N=50$; this is also shown by the number density pattern in Fig. 5A (see Movie S3 for the transient). The local particle number density ${\rho }_N$ at $N=50$ is an order of magnitude larger than the initial density ${\rho }_0$. At a later time of $N=150$, the particle accumulation reduces as the number density patterns grow wider and more disperse (Fig. 5 A, Right). This transient behavior is further quantified by the one-dimensional number density profile (Fig. 5C). Swimming particles, initially evenly distributed, form strong peaks at each multiple of $L$, where unstable manifolds are located. These peaks become weaker when particle distribution becomes more disperse at later times.

Fig. 5.

Numerical simulations of the transport of swimming particles. (A and B) Particle distribution in time-periodic flows of Re = 14.0 and P = 1.94; snapshots are taken at n = 50 (Left) and n = 150 (Right). Swimming particles are visualized as rods that are colored by particle number density ${\rho }_N$ normalized by initial density ${\rho }_0$ (A) and the cosine of particle swimming angles $\cos \left({\theta }_s\right)$ (B), respectively. (C) Horizontal profile of the vertically averaged number density ${⟨{\rho }_N⟩}_y$ normalized by ${\rho }_0$, showing accumulation of particles at the locations of the multiple of $L$. The accumulation becomes less intense at longer times. (D) The probability distribution of the swimming angle ${\theta }_s$ of all particles indicates the alignment of particles with the vertical and horizontal unstable manifolds by showing peaks at the multiple of $\pi /2$. The alignment becomes less strong at later times.

Simulation results also show that swimmers align with the unstable manifolds, as characterized by the cosine of their swimming angle ${\theta }_s$, as shown in Fig. 5B (see SI Appendix, Fig. S11 for $\sin {\theta }_s$). Nearly all particles accumulated near vertical unstable manifolds are swimming upward or downward (green; $\cos {\theta }_s=0$), and almost all particles near horizontal manifolds are swimming leftward or rightward (blue/red; $\cos {\theta }_s=\pm 1$). The alignment of particles becomes weaker at later times ($N=150$) as the particle distribution broadens (Fig. 5 B, Right). This is characterized by the probability distribution of the swimming angle (Fig. 5D). While initially, swimming angle is uniformly distributed, at later times, we find a decay in all peaks as more particles diffuse to intermediate angles. While the steady-state accumulation of active particles along unstable manifolds have been observed (30, 31), here, however, we show that this accumulation may be transient and nonmonotonic in time (SI Appendix, Fig. S12). Taken together, these simulation results suggest that the accumulation and alignment of swimmers along the unstable manifolds may be responsible for the decrease in fluid stretching. We believe that this accumulation may lead to an increase in local extensional viscosity or resistance of stretching along the manifolds, leading to a decrease in local FTLE (Fig. 2C). This causes a decrease in mixing rate (Fig. 2A) and an increase in average structure size (Fig. 3B).

We can estimate the increase in extensional viscosity (${\mu }_{\epsilon }$) due to local bacteria accumulation. For dilute suspensions of rod-like pusher swimmers, the extensional viscosity in the limit of high Péclet number can be estimated as ${\mu }_{\epsilon }=3{\mu }_0\left(1+{\varphi }_b{\alpha }^2\pi /\left(18\ln 2\alpha \right)\right)$, where ${\mu }_0$ is the shear viscosity of the suspending fluid, and $\alpha$ is the particle aspect ratio (49) (see SI Appendix for details). Using this expression, we estimate that nonmotile bacteria suspension of ${\varphi }_b=0.9\%$ will only lead to a uniform increase of 2.7% in extensional viscosity. However, an active suspension of same ${\varphi }_b$ has a much higher local volume fraction (${\varphi }_b=10.8\%$) due to bacteria accumulation near the manifolds, leading to an increase in ${\mu }_{\epsilon }$ of 33.3% (SI Appendix). We believe that this relatively large increase in local extensional viscosity along the unstable manifolds is responsible for the attenuation of stretching and the observed hindrance in large-scale mixing.

Fluid mixing is an important phenomenon that occurs in diverse natural situations (e.g., lakes, rivers, and atmosphere). Here, we explored how swimming microorganisms affect mixing and transport in flows exhibiting chaotic advection. We find that swimming E. coli can hinder large-scale transport of a passive scalar (i.e., dye) in time-periodic flows, even though small-scale (local) mixing is enhanced. Bacteria activity can adversely affect large-scale mixing by suppressing the stretching and folding mechanism, a hallmark of chaotic advection. This was demonstrated explicitly by measuring the flow-stretching fields; regions of large stretching were attenuated by the presence of bacteria, which resulted in a lower mean of FTLE as bacteria concentration was increased. In other words, bacterial activity reduces the chaoticity of the flow. The attenuation of stretching results in coarser and lasting structures, as shown by the increase in both spatial and temporal autocorrelations with swimming E. coli. This is in contrast to the strong local mixing produced by bacteria at later times, which can even lead to nearly locally isotropic dye-concentration fields. This strong local mixing coupled with attenuation in stretching leads to a dynamical system in which scalar field-invariant behavior is delayed; this invariant field is, however, subjected to higher randomness at the finer scales for ${\varphi }_b>0$, as shown by the longer exponential tail in Fig. 4B. Simulation results suggest that the attenuation of stretching and mixing at early times may come from an increase in local extensional viscosity due to the accumulation of swimmers along unstable manifolds. Overall, our results provide insights into how the nonlinear coupling between flow and activity work together to transport and homogenize scalars such as impurities and temperature.

Materials and Methods

Bacterial Suspension Preparation.

We inoculated a strain of E. coli (wild-type K12 MG1655) into Luria–Bertani broth (Lennox, Sigma-Aldrich) liquid medium. The medium was then incubated at 37 $°$C and 135 rpm overnight (12 to 14 h) to attain a stationary phase of approximately ${\mathrm{10}}^9$ cells per mL cell density (35). We centrifuged the stationary-phase culture at 5,000 rpm for 3.5 min and resuspended the pellet into 20 mL of buffer solution. The buffer was an aqueous solution of 2 wt% KCl, 1 wt% NaCl, which does not inhibit cell viability (50). The bacterial volume fraction ${\varphi }_b$ was adjusted from 0.5 to 0.9% within the same buffer. Nonmotile bacteria suspension was made by killing the E. coli with 6% NaClO solution (1:20 to the culture) and then repeating the aforementioned centrifuging–resuspending protocol.

Dye-Mixing Experiments.

We labeled half of the fluids with 250 $\mathrm{\mu }$L of dye (2.5 $\times \hspace{0.17em}{\mathrm{10}}^{-3}$ M fluorescein salt sodium aqueous solution). The dye was then stirred and dispersed uniformly (6.25 $\times \hspace{0.17em}{\mathrm{10}}^{-5}$ M) in the labeled fluid, during which a barrier placed in the middle prevented the labeled and unlabeled fluid from mixing. After the barrier was lifted, we imposed an AC voltage of 4 V and 0.2 Hz to the fluid layer to drive the mixing. Images were taken in the 36-cm² center region using a complementary metal–oxide–semiconductor camera (IO Industries, Flare 4M180), operating at 5 frames per second (fps) and 2,000²-pixel resolution. The dye field was illuminated under black light (USHIO, F8T5/BLB). The black light (peak emission at 368 nm) was in the ultraviolet (UV) range, but 90% of energy was in the range of long-wave UVA-I (340 to 400 nm), which does not harm the cells (51). A filter (TIFFEN, Yellow 12) was used to cut off UV, such that the light intensity was linearly proportional to dye concentration.

PTV.

We dispersed 100-$\mathrm{\mu }$m-large polyethylene fluorescent particles in the bacterial suspension. Particle positions were recorded by the aforementioned camera, this time operating at 30 fps and ${\mathrm{1},\mathrm{140}}^2$-pixel resolution in a 12.96-cm² region. By using particle-tracking software (52), we extracted particle trajectories and measured particle velocities from 6^th-order polynomial fitting. The velocities were then phase-averaged and interpolated on a spatial grid to obtain the velocity map.

Simulations.

The swimmers were modeled as noninteracting axisymmetric ellipsoids with a swimming speed v_s in the direction of $\mathbf{n}$. The positions of the swimmers $\mathbf{x}$ are governed by:

$$\dot{\mathbf{x}}={\mathbf{v}}_f\left(\mathbf{x},t\right)+v_s\mathbf{n},$$ [1]

where ${\mathbf{v}}_f$ is the fluid velocity, and $v_s=20\hspace{0.17em}\mu$m/s for E. coli. The orientations of the swimmers can be modeled by using Jeffery’s dynamics (53):

$$\dot{\mathbf{n}}=\left[\mathbf{\Omega }\left(\mathbf{x},t\right)+\gamma \mathbf{D}\left(\mathbf{x},t\right)\right]\mathbf{n}-\gamma \left[\mathbf{n}\cdot \mathbf{D}\left(\mathbf{x},t\right)\mathbf{n}\right]\mathbf{n},$$ [2]

where $\mathbf{D}$ and $\mathbf{\Omega }$ are the symmetric and skew parts of the velocity gradient tensor $\nabla {\mathbf{v}}_f$. Here, $\gamma =\left(1-{\alpha }^2\right)/\left(1+{\alpha }^2\right)$ is a shape factor, with $\alpha$ being the swimmer aspect ratio; the factor is roughly $\gamma \approx 0.88$ for E. coli. The flow in the simulation is defined by a Taylor–Green type stream function:

$$\psi =\left(UL/\pi \right)\sin \left(\pi x/L\right)\sin \left(\pi y/L\right)\sin \left(2\pi ft\right),$$ [3]

where $U$ is the maximum velocity, related to rms velocity by $Ū=U/\sqrt{8}$. The simulation domain is ${\left(6L\right)}^2$ in size, with 10,000 swimming particles in it and periodical boundary conditions imposed on all boundaries.

Data Availability

All study data are included in the article and/or supporting information.