Pattern Formation in Active Fluids

We discuss pattern formation in active fluids in which active stress is regulated by diffusing molecular components. Nonhomogeneous active stress profiles create patterns of flow which transport stress regulators by advection. Our work is motivated by the dynamics of the actomyosin cell cortex in which biochemical pathways regulate active stress. We present a mechanism in which a single diffusing species up regulates active stress, resulting in steady flow and concentration patterns. We also discuss general pattern-formation behaviors of reaction-diffusion systems placed in active fluids.

Figure 1

The nonhomogeneous steady states with (a) one maximum and (b) two maxima for $\zeta \Delta \mu (c)=(\zeta \Delta \mu {)}_{0}c/(1+c)$, $\mathrm{Pe}=25$, $L/\ell =2\pi$, and $c_0=1$, with periodic boundary conditions. The total stress $\sigma$ and active stress $\zeta \Delta \mu$ are given in units of $(\zeta \Delta \mu {)}_0$.

Figure 2

(a) “Potentials” $V(u)$ for stable steady states with $\zeta \Delta \mu =(\zeta \Delta \mu {)}_{0}c/(1+c)$ and $c_0=1$. For small Pe, $V(u)$ is concave; for moderate Pe, $V(u)$ has a shallow minimum at $\ln c_0$; and for large Pe, $V(u)$ has a minimum at a $u<\ln c_0$ that is deep enough to support a nonhomogeneous steady state. Inset: Detail of the central extremum of $V(u)$. (b) Detail of relevant portions of potentials corresponding to the steady states depicted in Fig. 1 ($\mathrm{Pe}=25$). The thick lines depict the portions of the potentials explored by the steady state.

Figure 3

(a) Dispersion relation for $\zeta \Delta \mu =(\zeta \Delta \mu {)}_{0}c/(1+c)$ and $c_0=1$ for $\mathrm{Pe}=3$, 7, 10, and 25, corresponding to plots depicted in Figs. 1, 2, 4. The eigenvalue $\lambda (k)$ is nondimensionalized by the diffusive time scale ${\tau }_D={\ell }^2/D$. For $L/\ell =2\pi$ with periodic boundary conditions, as in Fig. 1, $k\ell$ may take only integer values (vertical gray lines). (b) Linear stability diagram for the same active stress function.

Figure 4

Illustration of dynamics for the same system as in Fig. 1, except with $L/\ell =3\pi$, such that the second mode grows fastest from the unstable homogeneous steady state. Time is nondimensionalized by ${\tau }_D={\ell }^2/D$.

Figure 5

Characterization of the ASDM. (a) Linear stability diagram in the absence of flow ($\mathrm{Pe}=0$) for ${\rho }_a=1$, assuming $L/\ell \gg 1$. Region I: $\lambda (k)$ is real and positive with $|k_{\max }|>0$. This is the pattern-forming region. Region II: $\mathrm{Re}[\lambda (k)]<0$ (stable homogeneous steady state). Region III: $\lambda (k)$ is real and positive with maxima at $k=0$ and some $k\ne 0$. Patterns may form in this region, depending on initial conditions. Region IV: $\mathrm{Re}[\lambda (k)]>0$, $\mathrm{Im}[\lambda (k)]\ne 0$, $k_{\max }=0$ (oscillatory regime). Region V: $\lambda (k)$ real and positive with $k_{\max }=0$. (b) Linear stability diagram for $\mathrm{Pe}=7$, ${\rho }_a=1$, and $\zeta \Delta \mu /(\zeta \Delta \mu {)}_0=a/(1+a)$. (c) Dispersion relation for the ASDM with $\lambda (k)$ nondimensionalized by ${\tau }_D={\ell }^2/D_s$. Solid lines are for parameters given by the “$\times$” in (b). For reference, the dashed line is for $\mathrm{Pe}=0$ and parameters given by the “$+$” in (a). (d) Nonhomogeneous steady state for $\mathrm{Pe}=7$ and parameters given by the “$\times$” in (b) with periodic boundary conditions. The activator and substrate concentrations are given by the solid black and gray curves, respectively. The velocity is given by the dashed curve.