Self-organization of autophoretic suspensions in confined shear flows

Janus phoretic particles exploit chemical energy stored in their environment to produce mechanical work on the surrounding fluid and self-propel. These active particles modify and respond to their hydrodynamic and chemical environments, thus providing them with a sensibility to external flows and other particles. These chemical and hydrodynamic interparticle interactions are known to lead to nontrivial collective behavior within such biological or synthetic active suspensions (e.g., cluster formation of phoretic particles or bacterial swarming). Recent experiments and analysis have demonstrated that the response of active suspensions to shear flows is nontrivial and can, in fact, lead to significant reductions in viscosity due to the energy conversion at microscopic scales. In this work we numerically analyze using a continuum kinetic model the dynamics and response to shear of dilute and confined suspensions of chemotactic phoretic particles that reorient and drift toward the chemical solutes released by their neighbors. We show that a 1D transient steady distribution driven by the effect of confinement is a common feature for the confinement and shear rate intensities considered. This 1D state is stable for strong confinement and thus observed in the long-term dynamics in sufficiently narrow channels. For wider channels, the transient state becomes unstable to streamwise perturbations due to the chemotactic instability, leading to the formation of particle aggregates along the channel's walls. Their relative arrangements and dynamics are determined by the relative influence of shear intensity and chemotaxis and critically condition the suspension's dynamics and particle-induced flows. In a second step, the feedback effect on the flow and effective viscosity of the self-organized suspension is considered. We show that the induced flow and, consequently, its rheological behavior strongly depend on the self-organization regime, and therefore on the interplay of confinement, shear, and chemotaxis.

Figure 1

Problem schematic. A dilute suspension of Janus particle placed between two flat walls. The walls move opposite to each other to establish a simple shear flow.

Figure 2

Overview of self-organization of the current system. We identify three different regimes based on shear rate and confinement. Strong shear or strong confinement tends to stabilize the 1D state.

Figure 3

Short-term particle (top) and solute distribution (bottom) obtained for $\zeta =1,\gamma =0.25$. The black arrows (bottom) represent the local polarization direction and magnitude of the particles.

Figure 4

1D profiles of the solute and particle concentrations and streamwise and wall-normal components of the particle polarization in the transient regime ($\zeta =1,\gamma =0.25$).

Figure 5

Long-term dynamics of the chemotactic suspension for varying relative shear rate $\gamma$ and degree of confinement $\zeta ={\left(\widehat{H}\widehat{N}{\widehat{R}}^2\right)}^{-1}$. Colors indicate the nature of the particle distribution: 1D (blue) and 2D (orange). Different symbols are used for steady (square) and unsteady (triangle) regimes.

Figure 6

1D boundary layer chemotactic destabilization: a small disturbance in the particle distribution along the wall and the particles' activity introduce a local increase in solute concentration (green) and a small horizontal bias of the orientation of neighboring particles that start swimming toward and accumulating in this solute-rich region.

Figure 7

(Left) Long-term particle distribution for $\zeta =1$ and in the absence of external flow (top) and for $\gamma =0.025$ (bottom). (Right) Schematic representation of the opposite walls aggregates in the absence of background flow (top) and for weak shear flow (bottom). The black arrows represent the local polarization of the particles, white arrows show the direction of background advection, and the green arrow indicates the direction of chemotactic bias (reorientation).

Figure 8

(Left) Evolution of the particle density in time over a period of the relative motion of aggregates on opposite walls for $\gamma =0.125$ and $\zeta =1$. (Right) Corresponding schematic representation of the position of the particles' aggregates.

Figure 9

(Left) Evolution of the steady minimum offset between aggregates on opposite walls ($\mathrm{\Delta }x/\lambda$) with the shear intensity (with $\lambda$ representing the wavelength of the most unstable mode), in the limit of weak shear and $\zeta =1$. (Center) Time evolution of $\mathrm{\Delta }x\left(t\right)/\lambda$ for the unsteady regime at $\zeta =1$ and sufficiently large shear forcing ($\gamma =0.125$). (Right) Evolution with the shear rate intensity of the time period of the oscillations in the relative positioning of aggregates on the opposite walls for $\zeta =1$.

Figure 10

Particle distribution (left) and solute concentration (right) observed in the strongly confined regime ($\zeta =2$) for low and high shear rates.

Figure 11

Comparisson between the 1D steady states obtained by solving 1D version of the reduced order equations to the full simulation for $\zeta =1$ and $\gamma =1.5$.

Figure 12

(Left) Linear growth rate $\left[\text{Re}\right(\sigma \left)\right]$ as a function of background shear and degree of confinement. (Right) Frequency $\left[\text{Im}\right(\sigma \left)\right]$ of the most unstable mode as a function of background shear and degree of confinement. The black curve is the neutral stability curve separating the stable region from the unstable region.

Figure 13

Comparison of the phase plot obtained via the linear stability analysis (SA) with the phase plot obtained using the full simulation (FS) Red squares represent unstable 1D state, and the blue circles represent the stable 1D state.

Figure 14

(Left) Steady-state particle distribution and (right) real part of the eigenmodes of the particle density and solute distribution corresponding to the most unstable mode.

Figure 15

Streamlines of the induced flow for 2D steady state for $\gamma =0$ (top) and $\gamma =0.0167$ (bottom).

Figure 16

Evolution in time of particle density distribution and induced flow streamlines over a period of the unsteady regime, (a) $t=0$, (b) $t\sim T/4$, (c) $t\sim T/2$, and (d) $t\sim 3T/4$ with $\gamma =0.125,\zeta =1$.

Figure 17

(Left) Induced flow field for the 1D symmetric flow for a 1D steady-state regime with $\gamma =0.125,\zeta =1.33$; inset shows the driving force due to active stress exerted by the particles.

Figure 18

Illustration of the fluid flow for 2D symmetric case. The white arrows show the direction of horizontal fluid forcing, and the black arrow shows particle polarization. The fluid forcing changes direction at a point $t$ equidistant from the aggregates in the same wall due to a change in horizontal polarization.

Figure 19

Schematic showing asymmetrical horizontal forcing, which leads to two vortices with the same direction of rotation. Green arrows show the relative dominant flow forcing resulting in the two vortex flows shown in green. The relative increase in the flow forcing is due to the orientation bias created due to the presence of aggregate on the opposite wall.

Figure 20

Time evolution of effective viscosity (${\eta }_r={\eta }_{\mathrm{hydro}}+{\eta }_{\mathrm{act}}$) for 2D unsteady (top) state, 2D steady state (center), and 1D steady state (bottom) for parametric values $\gamma =0.125,\zeta =1$ (top), $\gamma =0.025,\zeta =1$( center), and $\gamma =2,\zeta =1$.

Figure 21

${\langle u_d\rangle }_y$ (average disturbance velocity profile) for the 2D particle distribution with top aggregate displaced right (left) of bottom aggregate and (right) top aggregate displaced to the left of bottom aggregate for $\gamma =0.125,\zeta =1$ (unsteady regime).

Figure 22

Net flow (averaged along the flow direction) compared to the imposed flow for (left) shear rates 0.125 (unsteady regime) and (right) 0.025 (steady regime) and $\zeta =1$ at viscosity minima for the 2D unsteady state.

Figure 23

(Left) Variation of effective viscosity with respect to background shear rate for different degrees of confinement ($\zeta$). (Right) Long-term effective viscosity $\left({\eta }_r\right)$ on shear rate ($\gamma$) confinement space ($\zeta$). The range of the color axis is modified such that the effective viscosity below $-1$ is all colored identically as the data points are highly skewed for weak shear rates and weak confinement. Rough boundaries are drawn for the different long-term regimes for respective shear rate and strength of confinement. Golden boundary indicates long-term 1D steady state, orange boundary indicates long-term 2D steady state, and brown boundary indicates long-term 2D unsteady state.

Figure 24

Time evolution of spatial mean polarization magnitude and maximum $|{\mathbf{\nabla }}_{x}C|$ for the weak confinement case. The plateau region corresponds to the 1D transient state, and the long-term periodic behavior corresponds to the long-term unsteady 2D state. The plots are for $\gamma =0.125, \zeta =1$.

Figure 25

Comparison between disturbance velocity field (top) total active stress and (bottom) only the pusher signature of active stress for $\gamma =0.125,\zeta =1,t=1325$. The color bar represents the particle density in the domain.

Figure 26

Horizontal and vertical fluid forcing for the two terms of Eq. (31) for $\gamma =0.125,\zeta =1,t=1500$. The forcing field is shown on equally spaced grid points instead of the Chebyshev-Fourier grid so that the regions of strong forcing (very close to the walls) are visible. Both effects have the same pattern of contribution to the driving force.