Tumbling dynamics of inertial inextensible chains in extensional flow

This paper investigates the effect of inertia on the dynamics of elongated chains to go beyond the overdamped case that is often used to study such systems. For that purpose, numerical simulations are performed considering the motion of freely jointed bead-rod chains in an extensional flow in the presence of thermal noise. The coil-stretch transition and the tumbling instability are characterized as a function of three parameters: the Péclet number, the Stokes number, and the chain length. Numerical results show that the coil-stretch transition remains when inertia is present and that it depends nonlinearly on the Stokes and Péclet numbers. Theoretical and numerical analyses also highlight the role of intermediate stable configurations in the dynamics of elongated chains: chains can indeed remain trapped for a certain time in these configurations, especially while undergoing a tumbling event.

Figure 1

An elongated chain is approximated as a bead-rod Kramers chain. The beads ${\mathit{X}}_i$'s are linked together by infinitesimal rigid rods of length ${\ell }_{\mathrm{K}}$. The orientation of the links is given by the unitary vectors ${\mathbf{\xi }}_i=({\mathit{X}}_i-{\mathit{X}}_{i-1})/{\ell }_{\mathrm{K}}$.

Figure 2

Real part of the most unstable eigenvalue ${\mu }_{\mathcal{M}}$ (in units of the fluid shear rate $\sigma$) as a function of the Stokes number St for the fully stretched chain (solid line, filled symbol) and the completely folded case (dashed line, empty symbols). The lines show the predictions (see text), while symbols are numerical evaluations for various chain lengths, as labeled.

Figure 3

Stable configurations of a chain with $N=20$ and $\mathrm{St}=0$ in the $(\mathcal{F},\mathcal{L})$ plane. The filled circles stand for the real part of the less stable eigenvalue ${\mu }_{\mathcal{M}}$ (in units of $\sigma$). Two intermediate stable configurations are singled out at $\mathcal{L}\approx 1/3$, $\mathcal{F}=(N-5)/(N-1)$, where the chain is approximately folded in three equal pieces, and at $\mathcal{L}=1/5$ and $\mathcal{F}\approx 0.15$, for which the chain alternates between stretched and folded segments.

Figure 4

End-to-end length and orientation of a freely jointed chain ($N=40$) at two values of Péclet numbers showing the chain in both the coiled state ($\mathrm{Pe}=0.05$) and the stretched state ($\mathrm{Pe}=0.3$). (a) Snapshots of a chain in the coiled and stretched states. The insets display the histograms of $\mathcal{L}$ and $\mathcal{F}$ for both states. (b) Map showing the probability of occurrence for each discrete state in the $(\mathcal{F},\mathcal{L})$ plane around the coiled (hot color) and stretched state (cold color).

Figure 5

Chain end-to-end length and orientation as a function of $\mathrm{Pe}$ showing the subcritical phase transition for $N=10,20$, and 40. Subcritical phase transition visible when plotting the average over several realizations of the flow ${\langle \left|\mathcal{L}\right|\rangle }_{\mathrm{ens}}$. Inset showing the average orientation over time ${\langle \mathcal{F}\rangle }_t$.

Figure 6

Average chain end-to-end length over time ${\langle \mathcal{L}\rangle }_t$ in the $(\text{Pe},N)$ plane. The red solid line corresponds to the critical Péclet number ${\mathrm{Pe}}^{★}$ at which the chain end-to-end length exceeds a threshold ${\langle \left|\mathcal{L}\right|\rangle }_t>L^{★}=0.9$. This shows the existence of an asymptotic regime for sufficiently long chains.

Figure 7

Time-averaged extension ${\langle \mathcal{L}\rangle }_t$ in the $(\mathrm{St},\mathrm{Pe})$ plane. The dotted vertical line shows the critical value $\mathrm{St}=1/8$. The solid and dashed lines display the prediction ${\mathrm{Pe}}^{★}\left(\mathrm{St}\right)={\mathrm{Pe}}^{★}\left(0\right)/(1+2\mathrm{St})$ and the asymptotic behavior ${\mathrm{Pe}}^{★}\left(\mathrm{St}\right)={\mathrm{Pe}}^{★}\left(0\right)/\left(2\mathrm{St}\right)$, respectively, with ${\mathrm{Pe}}^{★}\left(0\right)=0.4$.

Figure 8

Evolution of the chain orientation for $N=20$ near the CS transition (here $\mathrm{Pe}=0.14$) in the overdamped case. (a) Chain orientation $\mathcal{L}\left(t\right)$ as a function of time. Inset showing a zoom on a selected tumbling event. (b) Probability of occurrence for each discrete state over the whole simulation in the $(\mathcal{F},\mathcal{L})$ plane. (c) Analogous to (b) but over a selected tumbling event.

Figure 9

Probability density function (PDF) of the persistence time for $N=20$ in the overdamped case for various values of the Péclet number (slightly above the CS transition). The dotted lines correspond to the exponential law $p\left({\tau }_t\right)=\text{exp}(-{\tau }_t/{\tau }_t^{\text{avg}})/{\tau }_t^{\text{avg}}$.

Figure 10

Average persistence time $\langle {\tau }_t\rangle$ as a function of the normalized Péclet number ${\mathrm{Pe}}^{+}=(\mathrm{Pe}-{\mathrm{Pe}}^{★})/{\mathrm{Pe}}^{★}$. Inset showing the evolution of the persistence time $\langle {\tau }_t\rangle$ with the chain length $N$.

Figure 11

Average persistence time $\langle {\tau }_t\rangle$ as a function of the normalized Péclet number ${\mathrm{Pe}}^{+}=(\mathrm{Pe}-{\mathrm{Pe}}^{★})/{\mathrm{Pe}}^{★}$ for various Stokes number ($N=20$).