Probe particles in odd active viscoelastic fluids: How activity and dissipation determine linear stability

Odd viscoelastic materials are constrained by fewer symmetries than their even counterparts. The breaking of these symmetries allows these materials to exhibit different features, which have attracted considerable attention in recent years. Immersing a bead in such complex fluids allows for probing their physical properties, highlighting signatures of their oddity and exploring the consequences of these broken symmetries. We present the conditions under which the activity of an odd viscoelastic fluid can give rise to linear instabilities in the motion of the probe particle, and we unveil how the features of the probe particle dynamics depend on the oddity and activity of the viscoelastic medium in which it is immersed.

Figure 1

Schematic picture of a probe particle moving with velocity $U$ in an odd viscoelastic fluid, experiencing a drag force $F_{\parallel }$ and a lift force $F_{\perp }$.

Figure 2

Circuit representation of the odd Jeffreys model given in Eq. (8) including dampers (${\mathit{\lambda }}_{1,2}$) and an elastic spring (${\mathit{E}}_1$). Each element of this representation has odd and even contributions, and can be represented as a rank-4 tensor (see the text for details). The relation between the rank-4 tensors appearing in Eq. (8) and the circuit rank-4 tensors is given by ${\mathit{\lambda }}_1=\mathit{\eta }, {\mathit{E}}_1=\mathit{\nu }\mathit{G}{\mathbf{\nu }}^{\prime }$, and ${\mathit{\lambda }}_2=\mathit{\nu }{\mathbf{\gamma }}^{-\mathit{1}}{\mathbf{\nu }}^{\prime }/2$.

Figure 3

Stability of a probe in an odd viscoelastic medium. (a,c) Stable (white and blue) and linearly unstable (red) regions in the parameter space $({\gamma }_{\mathrm{o}}{\eta }_{\mathrm{s}},{\gamma }_{\mathrm{s}}{\eta }_{\mathrm{s}})$ in panel (a) and $({\nu }_{\mathrm{o}},{\gamma }_{\mathrm{s}}{\eta }_{\mathrm{s}})$ in panel (c). The passive (blue) region indicates where the passivity constraint (6) is satisfied. (b) Drag response ${\eta }_{\mathrm{s}}{M}_{\parallel }$ as a function of the dimensionless time $\widehat{t}=t/{\tau }_0$ with ${\tau }_0=a^2{\rho }_0/{\eta }_{\mathrm{s}}$ for the points $A,B,C$ in the parameter space indicated in panel (a). (d) Linearly unstable (red) and passive (blue) regions in the three-dimensional parameter space $({\nu }_{\mathrm{o}},{\gamma }_{\mathrm{o}}{\eta }_{\mathrm{s}},{\gamma }_{\mathrm{s}}{\eta }_{\mathrm{s}})$. Importantly, the transformation $({\nu }_{\mathrm{o}},{\gamma }_{\mathrm{o}})\to (-{\nu }_{\mathrm{o}},-{\gamma }_{\mathrm{o}})$ is a symmetry of the system. Parameters used are as follows: for all panels, ${\nu }_{\mathrm{s}}=1$. (a) ${\nu }_{\mathrm{o}}=1.5$. (b) $A,B,C$: ${\nu }_{\mathrm{o}}=1.5, {\gamma }_{\mathrm{o}}{\eta }_{\mathrm{s}}=-2, G_{\mathrm{s}}{\tau }_0/{\eta }_{\mathrm{s}}=0.25$, and $A$: ${\gamma }_{\mathrm{s}}{\eta }_{\mathrm{s}}=1.3, B$: ${\gamma }_{\mathrm{s}}{\eta }_{\mathrm{s}}=0.75, C$: ${\gamma }_{\mathrm{s}}{\eta }_{\mathrm{s}}=0.2$. (c) ${\gamma }_{\mathrm{o}}{\eta }_{\mathrm{s}}=2$.

Figure 4

Drag response ${\eta }_{\mathrm{s}}{M}_{\parallel }$ of an odd incompressible viscoelastic fluid as a function of the dimensionless time $\widehat{t}=t/{\tau }_0$ with ${\tau }_0=a^2{\rho }_0/{\eta }_{\mathrm{s}}$. (a) In the absence of an odd elastic stress relaxation (${\gamma }_{\mathrm{o}}=0$) and for different values of ${\nu }_{\mathrm{o}}$. (b) In the presence of an odd elastic stress relaxation (${\gamma }_{\mathrm{o}}{\eta }_{\mathrm{s}}=8$) and for different values of ${\nu }_{\mathrm{o}}$. (c) Drag response for a varied odd elastic stress relaxation ${\gamma }_{\mathrm{o}}$ and a vanishing ${\nu }_{\mathrm{o}}$. (d) Drag response for a varied odd elastic stress relaxation ${\gamma }_{\mathrm{o}}$ and ${\nu }_{\mathrm{o}}=-1.2$. In all panels, the solid lines indicate passive fluids for which Eq. (6) is satisfied, while dashed lines are used for active fluids. The red dotted lines are unstable active drag responses for which Eq. (21) is violated. Such responses diverge at long time, and nonlinear mechanisms should be included to obtain the corresponding large-time behavior. Other parameters are ${\nu }_{\mathrm{s}}=1, {\gamma }_{\mathrm{s}}{\eta }_{\mathrm{s}}=0.3, G_{\mathrm{s}}{\tau }_0/{\eta }_{\mathrm{s}}=0.25$.

Figure 5

Drag and lift responses of an odd compressible viscoelastic fluid induced as a function of the dimensionless time $\widehat{t}=t/{\tau }_0$ with ${\tau }_0=a^2{\rho }_0/{\eta }_{\mathrm{s}}$. (a) Drag response ${\eta }_{\mathrm{s}}{M}_{\parallel }$ for different values of the dimensionless inverse compressibility $\widehat{\psi }=\psi {\eta }_{\mathrm{s}}^2/{\left(a{\rho }_0\right)}^2$. (b) Lift response ${\eta }_{\mathrm{s}}{M}_{\perp }$ for different values of the dimensionless inverse compressibility $\widehat{\psi }$. Other parameters are ${\nu }_{\mathrm{s}}={\nu }_{\mathrm{b}}=1, {\gamma }_{\mathrm{s}}{\eta }_{\mathrm{s}}=1, G_{\mathrm{s}}{\tau }_0/{\eta }_{\mathrm{s}}=1, {\eta }_{\mathrm{b}}/{\eta }_{\mathrm{s}}=0, {\gamma }_{\mathrm{b}}{\eta }_{\mathrm{s}}=0, G_{\mathrm{b}}{\tau }_0/{\eta }_{\mathrm{s}}=0, {\eta }_{\mathrm{o}}/{\eta }_{\mathrm{s}}=0.1, {\gamma }_{\mathrm{o}}{\eta }_{\mathrm{s}}=0.1, {\nu }_{\mathrm{o}}=0$.

Figure 6

Contour integration in the complex plane used to compute the inverse Laplace transform (F3). The dashed blue lines along the real axis indicate branch cuts of the integrand. To compute the integrals $G_i$ along the contours ${\mathcal{C}}_i$, the radius of the large circle $R$ and the width $2\varepsilon$ between the segments above and below the real axis are sent to $\infty$ and 0, respectively.