Stokes' second problem and reduction of inertia in active fluids

We study a generalized Navier-Stokes model describing the coherent thin-film flows in semiconcentrated suspensions of ATP-driven microtubules or swimming cells that are enclosed by a moving ring-shaped container. Considering Stokes' second problem, which concerns the motion of an oscillating boundary, our numerical analysis predicts that a periodically rotating ring will oscillate at a higher frequency in an active fluid than in a passive fluid, due to an activity-induced reduction of the fluid inertia. In the case of a freely suspended fluid-container system that is isolated from external forces or torques, active-fluid stresses can induce large fluctuations in the container's angular momentum if the confinement radius matches certain multiples of the intrinsic vortex size of the active suspension. This effect could be utilized to transform collective microscopic swimmer activity into macroscopic motion in optimally tuned geometries.

Figure 1

Sketch of the proposed active-fluid analog of the Andronikashvili experiment [33]. A rigid ring suspended on a torsional spring with the spring constant $k$ is filled with a thin-film active fluid that obeys Eq. (1). We study the fluid-ring interaction in three scenarios: boundary held fixed ($k=\infty$, Fig. 2), oscillatory motion of the ring induced by the torsional spring ($0<k<\infty$, Fig. 3), and the response of the ring solely to the fluid stresses ($k=0$, Fig. 5).

Figure 2

Typical flow and stress fields for an active fluid with vortex size $\mathrm{\Lambda }$ and wide vortex-size distribution ${\kappa }_w=1.5/\mathrm{\Lambda }$, confined to a planar disk geometry (radius $R=2.67\mathrm{\Lambda }$) with the boundary held fixed. The presence of the stress-free defects allows the stress director field (third panel) to develop complex configurations, enabling a nontrivial response to time-dependent boundary conditions (see Figs. 3 and 5). The effective Reynolds-number field [Eq. (2), last panel] can be locally as small as $O\left({10}^{-2}\right)$, but there exist regions where it can reach values of $O\left({10}^2\right)$. These localized high-Re domains arise when ${\parallel \mathbf{\nabla }·\mathbf{\sigma }\parallel }_2$ is small, which corresponds to partial cancellation of active and passive stresses. Averaging over space and time gives the effective mean Reynolds number $\langle Re\rangle =8.4$ for this simulation.

Figure 3

Stokes' second problem for an active fluid confined by a ring-shaped container pendulum coupled to a torsional spring. Response of active fluids with wide [(a) and (b) ${\kappa }_w=1.5/\mathrm{\Lambda }$] and small [(c) and (d) ${\kappa }_s=0.63/\mathrm{\Lambda }$] vortex-size distributions to oscillatory boundary conditions (see also movies 2 and 3, and Fig. 4). The boundary speed is sinusoidal with amplitude $A$ and angular frequency $\mathrm{\Omega }$. (b) and (d) Activity-induced relative change $\lambda$ in the effective inertia experienced by the ring pendulum. Negative values of $\lambda$ imply that the pendulum oscillates at higher frequency in an active fluid than in a passive fluid. The mean Reynolds numbers are (a) $\langle Re\rangle =9.3$ and (c) $\langle Re\rangle =3.3$.

Figure 4

Stokes' second problem for active fluids: additional example snapshots from simulations corresponding to selected parameter combinations in Figs. 3 and 3. (a) and (b) For a wider spectral driving range ${\kappa }_w\mathrm{\Lambda }>1$, a broader vortex-size distribution is observed and the bulk stress field becomes disordered already at small values of the driving frequency $\mathrm{\Omega }\tau \ll 2\pi$. (c) and (d) For a narrow spectral driving range ${\kappa }_s\mathrm{\Lambda }<1$, the vortices form more regular nearly periodic dynamic patterns and a higher driving frequency $\mathrm{\Omega }\tau >2\pi$ is required to disrupt the regular structure of the bulk stress field. In all panels, the driving amplitude is $A/U=1$ and the disk radius $R/\mathrm{\Lambda }=2.67$.

Figure 5

Geometrically quantized fluctuations in an active fluid with narrow vortex-size distribution ${\kappa }_S=0.63/\mathrm{\Lambda }$. When isolated, the fluid can significantly shake the enclosing container, a thin rigid ring of radius $R$. (a) The standard deviation ${\sigma }_L$ of the container angular momentum $L$ depends on the radius $R$ and the fluid-to-ring mass ratio $\alpha$. The fluctuations ${\sigma }_L$ are independent of $\alpha$ for heavy containers ($\alpha \ll 1$) but start to decrease monotonically with $\alpha$ when the containers become light ($\alpha \gg 1$). As $R$ varies, the fluctuations oscillate with the period set by the characteristic vortex scale $\mathrm{\Lambda }$ (see also Fig. 6). Black dots represent 323 simulated parameter pairs; the color code shows linear interpolation. (b) Representative time series of the container angular momentum for two different radii $R=3.67\mathrm{\Lambda }$ (movie 4) and $R=3.33\mathrm{\Lambda }$ (movie 5) and fixed mass ratio $\alpha =1$. (c) Standard deviation ${\sigma }_{\dot{\varphi }}\sim \alpha {\sigma }_L$ of the container angular speed $\dot{\varphi }$. (d) Horizontal cuts through (a) and (c) at constant radius $R=4\mathrm{\Lambda }$. In particular, to maximize both the angular momentum and velocity fluctuations, the fluid mass should match the container mass ($\alpha \sim 1$).

Figure 6

Geometrically induced oscillatory behavior of fluctuations for an active fluid with narrow vortex-size distribution ${\kappa }_S=0.63/\mathrm{\Lambda }$. (a) Angular momentum fluctuations ${\sigma }_L$ as a function of the domain size for heavy containers obtained from Fig. 5 by averaging over $\alpha \in [0.01,0.1]$. (b) Angular speed fluctuations ${\sigma }_{\dot{\varphi }}$ as a function of the domain size for light containers obtained from Fig. 5 by averaging over $\alpha \ge 10$. (c) and (d) Close-up of the time series of the container's angular momentum (blue) calculated from Eq. (F1) shown in Fig. 5 for domain radii (c) $R=3.33$ and (d) $R=3.67$. Additionally, to illustrate the angular momentum conservation in the fluid-container system, we show the time series of the fluid's angular momentum (orange) calculated independently using the formula $L_{fluid}=\rho {\int }_0^{R}drr{\int }_0^{2\pi }d\theta r{v}_{\theta }$.

Figure 7

Stokes' second problem: driving protocol. (a) The driving amplitude [cf. Eq. (B1)] increases according to the prefactor $f\left(t\right)$ defined in Eq. (B2). (b) The mean kinetic energy time series for the simulation shown in Fig. 3 shows that the system relaxes well before the start time of the temporal averaging periods (vertical dashed lines).

Figure 8

The linear relation (5) between the torque $\mathcal{T}\left(t\right)$ and angular speed $\dot{\varphi }\left(t\right)$ approximately holds for active fluids. The formula becomes very accurate as the driving frequency $\mathrm{\Omega }$ and amplitude $A$ become larger than the corresponding active-fluid characteristic pattern formation parameters $2\pi /\tau$ and $U$, respectively. (a) and (d) Normalized power spectral density $|{\mathcal{T}}_{\omega }{|}^2/{\sum }_{{\omega }^{\prime }}{\left|{\mathcal{T}}_{{\omega }^{\prime }}\right|}^2$ of the (discrete) steady-state time series ${\mathcal{T}}_n$ for the two simulations shown in Figs. 3 and 3. The complex amplitudes ${\mathcal{T}}_{\omega }$ are obtained by applying the discrete Fourier transform to ${\mathcal{T}}_n$. The proportion of the energy concentrated (b), (c), (e), and (f) at the driving frequency $\mathrm{\Omega }$ as well as (insets) at the second most energetic frequency is shown as a function of (b) and (e) $\mathrm{\Omega }$ and (c) and (f) the oscillation amplitude $A$ for active fluids with the (b) and (c) wide and (e) and (f) small active bandwidths ${\kappa }_w$ and ${\kappa }_s$, respectively.

Figure 9

(a) and (d) Inertial and (b) and (e) dissipative response parameters $I_f$ and $\gamma$ as a function of the oscillating frequency $\mathrm{\Omega }$ and amplitude $A$ (insets) that appear in the relation (5) for active fluids with (a)–(c) wide ${\kappa }_w=1.5/\mathrm{\Lambda }$ and (d)–(f) small ${\kappa }_s=0.63/\mathrm{\Lambda }$ spectral bandwidths. The parameters were computed using Eq. (C1). (c) and (f) The average power input per unit length $\langle P\rangle$ in the steady state normalized by the value $\langle \mathcal{P}\rangle$ expected from the classical Stokes problem for a semi-infinite plate shows relative resonance at the characteristic frequency $2\pi /\tau$ of the active flow patterns. The markers indicate power input as computed from the full time series of the torque $\mathcal{T}\left(t\right)$, while the lines indicate the contribution derived from the linear relation (5).

Figure 10

A passive fluid (${\mathrm{\Gamma }}_2={\mathrm{\Gamma }}_4=0$) with viscosity ${\mathrm{\Gamma }}_0={10}^{-6}{m}^2/s$ confined to a disk domain of radius $R=200\mu m$ subject to oscillatory boundary conditions in Eq. (B1) with angular frequency $\mathrm{\Omega }=3.14rad/s$ and amplitude $A=628\mu m/s$ responds effectively as a rigid body. This is because for such parameters, typical for the active Stokes second problem presented in Fig. 3, the penetration depth $\delta$ of the passive fluid is much bigger than the domain size $R$. (a) Representative snapshot of the vorticity and flow fields illustrates the rigid-body-like response. (b) The corresponding power input (solid line) of the passive fluid driven according to the protocol described in Eq. (B1) follows accurately the formula (E3) for the power input of a rigid body rotating about the $z$ axis represented by a disk with mass equal to that of the fluid (dotted line).