Three dimensional contractile droplet under confinement

Abstract

We numerically study the dynamics of a three-dimensional contractile fluid droplet in the bulk and under confinement. We show that varying activity leads to a variety of shapes and motile regimes whose motion is driven by an interplay between spontaneous flows and elasticity. In the bulk the droplet self-propels unidirectionally, acquiring either an almost spherical shape at intermediate activity or a peanut-like geometry for larger values. Under confinement, the droplet exhibits a previously unreported oscillating dynamics characterized by periodic hits against opposite walls of a microchannel while moving forward. These results could be of interest for the study of artificial microswimmers and their biological analogs, such as living cells.

Figures (4)

• Figure 1: (a) Equilibrium configuration of a passive fluid droplet ($\zeta=0$). Red arrows represent the polar field while the gray surface is a contour where $\phi\simeq 1$. These apply to all figures. (b-e) Steady state configurations of droplet and polar field for $|\zeta|=3\times 10^{-3}$ (b), $|\zeta|=3.7\times 10^{-3}$ (c), $|\zeta|=6\times 10^{-3}$ (d) and $|\zeta|=8\times 10^{-3}$ (e). For low values of $\zeta$ the droplet attains an ellipsoidal non-motile shape where ${\bf p}$ remains essentially uniform (b) while, for intermediate values, the droplet turns almost spherical and motile. Here ${\bf p}$ exhibits a splay deformation (c). Increasing $\zeta$ leads to a non-motile spherical shape where ${\bf p}$ displays a hedgehog configuration (d) while, for higher values, a peanut-like structure emerges, where ${\bf p}$ acquires an asymmetric splay (e). (f-h) Two dimensional section (in the y-z plane at $x=L_x/2$) of the polar field for $|\zeta|=3.7\times 10^{-3}$ (a), $|\zeta|=6\times 10^{-3}$ (b) and $|\zeta|=8\times 10^{-3}$ (c). Blue dots indicate topological defects of charge $+1$. (i) Phase diagram of steady state center of mass speed $vs$ activity. Color stripes are associated with the different regimes.
• Figure 2: Three dimensional structure of the velocity field in the surrounding of the droplet for (a) $|\zeta|=3\times 10^{-3}$, (b) $|\zeta|=3.7\times 10^{-3}$, (c) $|\zeta|=6\times 10^{-3}$ and (d) $|\zeta|=8\times 10^{-3}$. The color bar represents the magnitude of the velocity. Figures (e)-(f)-(g)-(h) represent the corresponding two dimensional section, taken at $x=x_{cm}$ in the $y-z$ plane. While at low values of $|\zeta|$ (a-e) the droplet is non-motile because of a symmetric four-roll mill collecting the fluid equatorially and expelling it longitudinally, at intermediate values of $|\zeta|$ (b-f) the velocity exhibits a double-vortex pattern propelling the droplet horizontally. Increasing $|\zeta|$ leads to either a non-motile state where the velocity displays a symmetric eightfold structure or, for higher values, to a motile state where a jet of fluid, located within approximately half of the droplet, drives the motion.
• Figure 3: Time evolution of an active droplet for $|\zeta|=9\times 10^{-3}$. Snapshots (a), (b), (c), (d), (e) and (f) show the fields $\phi$ and ${\bf p}$. The droplet acquires motion (for example upwards) following a dynamics akin to that of the peanut-like droplet in Fig.\ref{['fig1']}e. Afterwards, it bumps against the wall (a) and glides over it (b,c) before turning downwards (d) and colliding/gliding against the opposite wall (e,f). Finally, the process repeats leading to a periodic motion persistent over time. Note that here the polarization points inwards, thus yielding a $-1$ defect at the rear of the droplet.
• Figure 4: (a-d) Section of the fluid velocity along the $y-z$ plane (taken at $x=L_x/2$) for a droplet with $|\zeta|=9\times 10^{-3}$. Far from the walls (a,c), the flow consists of a bidirectional stream, resulting from the splay deformations of the polar field, along the backbone of the droplet. The integer topological defect located at the rear (with respect to the droplet motion) marks the change of direction of the stream. This one is accompanied by four stretched vortices, two large ones at the front and two small ones at the rear. Near the walls (b,d), the momentum sink considerably abates the vortices in their vicinity. This allows the droplet to temporarily glide over the wall and then turn towards the center of the channel. (e) Time evolution of the vertical component ($z$) of the position of the center of mass for four values of activity, $|\zeta|=7\times 10^{-3}$ (no motion), $|\zeta|=8\times 10^{-3}$ (persistent glide over the wall), $|\zeta|=8.7\times 10^{3}$, $|\zeta|=9\times 10^{-3}$ (periodic motion). (f-g) Time evolution of $y$ and $z$ components of the speed of center of mass. (h) Time evolution of the parameter $\Sigma=N_s/N_v$, where $N_s$ is the number of lattice points located at the droplet surface and $N_d$ is the number of lattice points located within, thus proportional to the volume.