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 allow 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 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 unveil how the features of the probe particle dynamics depend on the oddity and activity of the viscoelastic medium in which it is immersed

A powerful method of probing the physical properties of a fluid is to observe the motion of a probe particle immersed in it [36][37][38].This is even more important for active odd viscoelastic materials.First, because the response contains signatures of the odd nature of the material, such as odd lift forces orthogonal to the probe's motion [39,40].Second, because the viscoelastic nature of the system makes the response in time to instantaneous external perturbations non-trivial [41].Third, because activity can give rise to sustained motion of the probe particle and, potentially, instability in the response to perturbations (see, e.g., Ref. [42]).
In this work, we shed light on these three points by looking at the motion of a passive probe particle in an active odd viscoelastic fluid (see Fig. 1).For symmetry reasons, isotropic odd fluids only exist in two dimensions [43], and we, therefore, focus on this case in this paper, although our methods could be applied in higher dimensions.Moreover, lift forces vanish in incompressible odd fluids [18] when no-slip boundary conditions hold [44], but can exist in compressible fluids [39].To assess odd viscoelastic responses in a generic framework, we therefore consider in the following a weakly compressible fluid, and odd lift responses will also be discussed.
First, we derive the material properties of such fluid from Onsager's theory and show that activity introduces interesting physics by breaking the Onsager-symmetry and thermodynamic inequalities that otherwise constrain the stress-strain relations.We then unveil the conditions that grant the stability of the motion of the probe particle when it is embedded in an active fluid.After confirming that a probe in a passive system has a stable trajectory, i.e. its velocity decays at long times, we show that, within our linear approach, there exists an intermediate regime where the probe motion through an active system is still stable.For larger activities, a probe that is subjected to an external force at an initial time becomes linearly unstable.We also focus on the signatures of oddity, which can be seen in the dynamics of the probe.We observe that odd viscoelastic terms promote long-lasting oscillations during the relaxation process after an instantaneous perturbation.These findings provide tools to assess the degree of oddity of a material by studying the motion of a probe inserted in it.
In this Section, we discuss the material properties of the two-dimensional odd viscoelastic fluid in which the probe particle moves.For this purpose, we start with the total stress tensor σ tot , which enters the momentum balance equation as We have introduced the momentum g = ρv with ρ and v i the fluid mass density and velocity, respectively, and f ext is an external force density.Mass density is conserved and obeys ∂ t ρ + ∇ • (ρv) = 0.The total stress σ tot can be decomposed into where ⊗ is the outer product and 1 the identity tensor.
The tensor σ s is the symmetric deviatoric stress, which incorporates viscoelastic contributions and σ a is the antisymmetric part of the stress [45].
For a passive odd viscoelastic fluid, the constitutive equations for the stress σ s and strain u tensors are found to be (see App.A for more details, also on σ a ): where D/Dt is the corotational derivative and v = [∇v + (∇v) ⊤ ]/2 is the symmetric part of the velocity gradient tensor.The elastic stress σ el results from the local strain u within the material and is obtained as σ el = δf /δu, where f denotes the free energy density.The objects ν p , η p and γ p in Eq. ( 3) are four-tensors that are isotropic and symmetric under the exchange of the first two as well as the last two indices, i.e. for a general four-tensor β we have β ijkl = β jikl = β ijlk .Such a general tensor can thus be characterized in two dimensions by a shear β s , bulk β b and odd β o coefficient (more details on the tensor notation can be found in App.B).The four-tensor η p in Eq. ( 3) corresponds to all viscous corrections, whereas γ p incorporates plasticity or strain relaxation.
As a consequence of Onsager symmetry (see App. C for its derivation for odd materials), the same four-tensor ν p relates the deviatoric stress to elastic stress and the strain rate to the velocity gradient.We emphasize that the odd coefficient ν p o is responsible for a passive and transient odd elasticity [3] (while steady-state odd elasticity cannot be passive [23]).Finally, to qualify as a passive material, the coefficients entering the four-tensors of Eq. (3) must satisfy [3] as dictated by the positivity of the entropy production rate.
In this paper, we go beyond passive systems and include activity.This can be done in Onsager framework, in the spirit of active gel models, by introducing fuel consumption at the microscopic level (see, e.g., Refs.[45,46]).Fuel consumption can be represented by adding a thermodynamic flux, the scalar reaction rate r, and its associated thermodynamic force ∆µ, which is the chemical potential difference of a chemical reaction, to the description.Since it is an additional thermodynamic force, one should supplement equation (3) with terms proportional to ∆µ, where, by symmetry, the proportionality factor should be a rank-2 tensor.We thus add the active contributions ∆µ ζ (1) : σ el + 2ζ (2) : v to Eq. (3a) and ∆µ −ζ (3) : σ el + ζ (4) : v to Eq. (3b).In Onsager framework, the thermodynamic forces (here ∆µ, σ el and v) are considered small, and the expansion of the thermodynamic fluxes as a function of the forces is kept to the lowest order.This means that the ∆µσ el -and ∆µv-terms are second order, and one would therefore naively think that these terms should be neglected.However, as we will show in the rest of the paper, because these contributions are not bound by passivity constraints, even a small contribution can destabilize the odd viscoelastic model, and therefore to correctly capture the qualitative behavior of the active viscoelastic system we must retain them.
When discussing the stability of the probe in Sec.II and III, we will make use of the passive limit for the generalized tensor objects, which can be imposed by setting and where η s , η b , γ s , γ b ≥ 0 is always satisfied provided ∆µ is small.

B. Jeffreys model for the odd viscoelastic fluid
Having in mind the description of a probe particle embedded in a viscoelastic fluid, we now show how Eq. ( 5) leads to an odd Jeffreys model [47,48].We consider a material with linear elastic properties, and first use the identity where G is the four-tensor of elastic moduli.Since Eq. ( 7) is derived from free energy, it is not possible to have an odd elastic modulus [23] and thus G o = 0. Note however that odd linear couplings between the stress and the strain are present in Eq. ( 5), and correspond to nonequilibrium corrections derived following Onsager's theory.
Solving for the strain u, we find that the constitutive equations ( 5) can be rewritten as: where I is the rank-4 identity and where we have used the fact that the contraction of an odd isotropic rank-4 tensor with a rank-2 tensor commutes with taking the corotational time derivative.We have moreover defined: τ 1 = < l a t e x i t s h a 1 _ b a s e 6 4 = " i U f n X F x N t 1 x C q 6 y x d u K Z g h Circuit representation of the odd Jeffreys model given in Eq. ( 8) including dampers (λ1,2) and an elastic spring (E1).Each element of this representation has odd and even contributions, and can be represented as a rank-4 tensor (see text for details).The relation between the rank-4 tensors appearing in Eq. ( 8) and the circuit rank-4 tensors is given by λ1 = η, E1 = νGν ′ and λ2 = νγ −1 ν ′ /2.
and τ 2 = (η eff ) −1 τ 1 η, where four-tensor algebra is given in App.B. Equation ( 8) describes an odd active Jeffreys model, that is, a viscoelastic fluid with two (odd) dampers and one (odd) elastic spring, as depicted schematically on Fig. 2. Furthermore, one consequence of the thermodynamic constraint ( 4) is that an odd Maxwell model, for which τ 2 in Eq. ( 8) vanishes, cannot be passive [3].Similarly, the odd Kelvin-Voigt limit (γ s → 0, ν o ̸ = 0) is also forbidden by thermodynamics in the absence of fuel consumption.
In Laplace space, dropping the (nonlinear) vorticity contributions to the corotational time derivative, we can write the constitutive relation of the active odd Jeffreys model (8) as [36,49] where

contains the shear and odd components
where we have introduced the Laplace transform 0 dt e −st h(t) (using the same notation for the function and its transform).We will refer to the coefficients in Eq. ( 10) as Laplace viscosities, as they can be seen as viscoelastic generalizations of viscosity in the Laplace domain.
To illustrate the physical meaning of the Laplace viscosity tensor A(s), let us consider an even material with ν = ν ′ .In this case, A s turns into which corresponds to the usual Jeffrey's model, which is viscous at very short timescales with viscosity lim s→∞ Ãs (s) = η s , elastic on intermediate timescales and again viscous at long timescales with viscosity lim s→0 Ãs (s) = η s + ν 2 s 2γs .Note that the only role of activity, in this case, is to modify the values of the different parameters compared to their passive counterparts, while the physics is qualitatively unchanged.
From this simple discussion, we understand that the odd visco-elastic Jeffreys' model [Eqs.( 9) and ( 10)]: (i) is also characterized by a viscous, fluid-like behavior at short and long timescales; (ii) has three characteristic timescales in the shear and odd sectors, and two in the bulk sector, (iii) can have qualitative differences compared to the even Jeffreys' model since the sign of some effective parameters is not fixed (for instance the term

C. Probe particle
We now study the motion of a probe particle dragged in the viscoelastic fluid with constitutive equation (9).As discussed in the introduction, isotropic odd materials only exist in two dimensions, and we thus focus on this case in the following.A natural setting for a twodimensional viscoelastic fluid is to consider a thin layer of such a fluid at the interface between two bulk (even) fluids, for instance, water and air [39,40].For simplicity, we consider this layer to be flat and infinitely thin, such that the odd fluid can be described effectively as two-dimensional.
The probe particle velocity U(t) corresponding to an applied time-dependent force F(t) can be computed using Laplace transforms as: where Latin indices correspond to Cartesian coordinates and a summation over repeated indices is implied, and with L −1 [h(s)] (t) the inverse Laplace transform.The response matrix M ij (s) can be understood as a viscoelastic and therefore s-dependent generalization of the drag and lift coefficients for a purely viscous fluid.For simplicity, we consider only the translational degrees of freedom of the probe, noticing that in the incompressible limit and for rigid probes there is no odd viscosity-induced torque [18].
To obtain the response matrix M ij (s) of an odd viscoelastic fluid, we first linearize to first order in v i and δρ = ρ − ρ 0 the momentum and mass conservation equations.In order to discuss weakly compressible fluids, we need to specify an equation of state for the pressure, which at first order in δρ reads P = δρ ψρ0 where ψ is the the compressibility.Note that the incompressible case, for which the pressure is independent of density, is recovered by taking the limit ψ → 0 in the response matrix M ij (s).Then, using the constitutive equation ( 9), we obtain the following coupled linearized equations: Using a spatial Fourier transform with the convention , Eq. ( 13) can then be rewritten in matrix form : with G ij given by [39,40] where k = √ k i k i and ki = k i /k.Having formulated the linearized equations for a given Laplace frequency and Fourier wave vector, we can then use the shell localization approach [39,49,50] to obtain the linear response for a rigid disk-shaped probe particle moving in the viscoelastic fluid.When no-slip boundary conditions hold, the shell localization approach prescribes that, upon identifying the probe particle velocity U i (s) with v i (|x| = 0, s), the response matrix is given by (more details in App.D): where θ is the angle of the wave vector, a is the disk radius and J 0 (z) is the 0 th Bessel function of the first kind.
For concreteness, and before discussing in detail the drag and lift responses of a probe in an odd active viscoelastic fluid, let us first relate this expression, Eq. ( 16), to the well-known drag in an incompressible, even, purely viscous fluid in two dimensions.In this case, the response matrix is diagonal and can be written as ified Bessel function of the second kind.This expression can then be expanded in series of the Laplace frequency s, leading to: where γ EM is the Euler-Mascheroni constant.This last expression shows the expected divergence of the drag at vanishing frequency, a signature of the Stokes paradox in two dimensions.

II. LINEAR STABILITY OF A PROBE PARTICLE IN AN ODD ACTIVE VISCOELASTIC FLUID A. General case
Active viscoelastic fluids can give rise to sustained motion.Therefore, the response of a probe placed into the medium after an initial perturbation does not necessarily decay to zero.At the level of linear response, such sustained motion is signaled by instability, which appears in the long-time behavior of the probe velocity given by Eq. (12).Without loss of generality, we consider an instantaneous perturbation applied at t = 0, such that F(t) ∝ δ(t), where δ(t) is the Dirac delta function.The velocity of the probe after this initial perturbation is given by: where m i (k, s) = M ij (k, s)F j and Γ is a real number such that the integral over s is convergent.To discuss stability, we need not focus on the precise expression of the velocity, but we remark the crucial point that m i (k, s) is a rational fraction in s.This implies that the longterm behavior of the velocity, and, therefore, the probe stability is determined by the sign of the real part of the poles in s of m i : where s * ℓ (k) is a pole of m i , see App.E for details.In the general case of an odd compressible viscoelastic fluid, the poles of m i are roots of a high-order polynomial, which makes further analytical discussion difficult.Nonetheless, Eq. ( 19) provides a simple and powerful criterion to determine the probe stability that can easily be used numerically.In the following, Sec.II B, we discuss the case of an incompressible fluid, where the analytical discussion regarding stability can be continued.
Note finally that in the regions of the parameter space where the inequality ( 19) is not satisfied, the motion of a probe is not linearly stable: the passive probe is therefore set into motion by the active viscoelastic fluid.In order to obtain physical solutions at longer times, nonlinear corrections must then be added to the description.The discussion of the form that these nonlinear corrections should take and their impact on the probe dynamics is left for future work.

B. Stability of probe motion in the incompressible limit
To continue our stability analysis while keeping the technicalities to a minimum, we now restrict to the incompressible case.In this simpler case, odd lift effects are absent [17,39], the tensor M ij is proportional to the identity, and the probe particle stability is determined by the poles of with τ 0 = ρ 0 a 2 /η s .As shown in App.E, these poles in the complex plane correspond to the roots of a thirdorder polynomial, and stability can be discussed analytically.We first discuss the simpler case where Onsager symmetry is enforced.
Onsager-symmetric fluid.We can take ν = ν ′ and stability is ensured iff and It is important to note that if the thermodynamics constraint (6) holds, all the conditions of Eq. ( 21) are satisfied and the system is always stable at long-time, as expected.Constraint (21b) coincides with the condition for the stability of a general odd viscoelastic fluid, which we derived in Ref. [3] by considering the stability of perturbative modes.We also note that the constraint (21c) depends on the mass density ρ 0 (through τ 0 ), whereas the other constraints Eqs.(21a) and (21b) only depend on viscoelastic coefficients.
For concreteness, we show in Fig. 3 stability diagrams of a probe immersed in an odd incompressible viscoelas-tic fluid.Figures 3(a), 3(c) and 3(d) display the unstable (red) regions of the parameter space and the passive (blue) regions, where the thermodynamics constraint ( 6) is satisfied.These two regions are disjoint, as expected from thermodynamics.Note that the three-dimensional view of the parameter space (ν o , γ o η s , γ s η s ) of Fig. 3(d) shows the symmetry (ν o , γ o ) → (−ν o , −γ o ) of the system.
As intuition would suggest, the friction γ s , which characterizes the elastic stress relaxation, has a stabilizing role and increasing it always leads to more stable systems.On the other hand, the odd coupling ν o is destabilizing the system as its magnitude is increased (provided its sign is kept constant).The odd elastic relaxation term γ o has a more subtle effect and may have a destabilizing or stabilizing influence [see Fig. 3(a)].
Figure 3(b) highlights the consequences of the odd active coefficients on the time-dependent (drag) response M ∥ (t) experienced by the particle along its velocity.In the linearly stable regions [points A and B in panel 3(a)], the drag decays to 0, indicating that the initial perturbation is eventually damped by viscous friction.On the other hand, the parallel response corresponding to parameters in the active unstable region [point C in panel 3(a)] increases with time, indicating that the probe motion is accelerated by the active fluid surrounding it.This acceleration must eventually be compensated by nonlinearities that we have not considered in the present analysis.
Broken Onsager symmetry.We now continue the discussion in the case of an incompressible fluid with broken Onsager symmetry, that is, ν ′ ̸ = ν.We define ν ′ s,o = νs,o + e s,o and ν s,o = νs,o − e s,o , such that the Onsager symmetry is recovered for e s,o = 0. Breaking Onsager symmetry modifies the conditions for stability, although these new conditions take a form similar to those given in Eq. (21).They read: where γ ± = γ s (e 2 o − e 2 s ) ± 2γ o e s e o , and where we have defined ∆ o − e 2 s + 4γ s η s − ν2 o + ν2 s )/(τ 2 0 η s ).Importantly, note that e o cannot destabilize the system.This is expected, as this coefficient enters anti-symmetrically in the constitutive equations, which causes its contribution to the entropy production to vanish.In fact, taking e s = 0 in Eq. (22) shows that the constraints are more easily satisfied for e o ̸ = 0 and e o even has a stabilizing effect.
On the other hand, e s does add to the dissipation rate and therefore a non-vanishing e s can drive the system into an unstable regime, even if the symmetric equivalent of thermodynamic constraint 2η s γ s ≥ ν2 o is satisfied.If e s ̸ = 0 and γ o ̸ = 0, the odd deviation from Onsager symmetry e o can also play a role in destabilizing the system through the term γ o e s e o entering Eq. ( 22).These results show the complex interplay between oddity and the breaking of Onsager symmetry in determining the stability of a probe.

A. Drag response for an incompressible fluid
In this Section, we consider the response along the velocity direction [drag, M ∥ (s)] and perpendicular to it [lift, M ⊥ (s)], making the decomposition For simplicity, we enforce Onsager symmetry in this Section and take ν ′ = ν.We first consider the incompressible limit, so that M inc ⊥ (s) = 0, whereas the drag response coefficient reads: with τ 0 = ρ 0 a 2 /η s and K 0 (z) the 0 th modified Bessel function of the second kind.The corresponding drag response in time domain can be obtained numerically (see App. F for details), and the results are displayed in Fig. 4. In Figs.4(a) and 4(b), we first explore the role of the odd elastic coefficient ν o and of the odd elastic strain relaxation rate γ o .For a vanishing odd elastic strain relaxation rate γ o [Fig.4(a)], the drag response does not display oscillations, while a nonvanishing γ o [Fig.4(b)] leads to a long-lived oscillatory behavior, despite strain relaxation due to plasticity.This is a unique property that can signal odd viscoelasticity in experiments.Furthermore, we note that the amplitude of these oscillations is strongly amplified by the odd elasticity coefficient ν o .This finding highlights again the nontrivial interplay between the different odd coefficients, which was already observed in Sec.II B for the stability constraints.It shows that increasing γ o and ν o eventually induces instabilities.Specifically, the dotted red line in Fig. 4(a) (corresponding to ν o ≥ γ 2 s + ν 2 s ) corresponds to a diverging trajectory.Such a non-converging drag response should be regularized by nonlinear corrections.
In Figs.4(c) and 4(d), we compute the drag response for a varied γ o for ν o = 0 and ν o = 1, respectively.We find that an increasing γ o leads to an increasing oscillation frequency.In Fig. 4(d), the choice of ν o makes the viscoelastic fluid active, and it was shown in Sec.II B that this may induce instabilities for the motion of the probe particle.This is the case for the dotted red line, which violates the stability constraint of Eq. (21b).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: νs = 1, γsηs = 0.3, Gsτ0/ηs = 0.25.

B. Drag and lift response for a compressible fluid
Compressibility is a necessary condition for observing lift forces [18,39], a signature of odd fluids.In this Section, we thus discuss how a finite compressibility of the fluid can be considered within our framework.Although exploring systematically the role of finite compressibility for the response of a probe in an odd viscoelastic medium is beyond the scope of this paper, we illustrate here a few important features of finite compressibility.
To relax the incompressibility constraint while continuing analytic computations, we consider a weak oddity approximation, i.e. we consider η o , γ o and ν o small compared to their shear and bulk counterparts.We introduce ε o the corresponding small parameter (for instance, We discuss here the first nonvanishing correction, although the series expansion can be continued to higher orders.Starting from Eq. ( 15), we obtain the response coefficients that were defined in Eq. ( 23): where we have defined the dimensionless compressibility ψ = ψη 2 s /(aρ 0 ) 2 and As expected, the lift response vanishes in the incompressible limit ψ → 0. Note also the absence of odd corrections to the drag response [Eq.(25a)] at first order.Figure 5 displays the drag and lift responses for different values of ψ, illustrating the role of a finite compressibility.Increasing the compressibility ψ spreads out the response in time both for the drag and lift coefficients.For small values of the compressibility, we observe a short-time response, eventually converging to the incompressible limit [black line, Fig. 5(a)].The lift response, which is a hallmark of odd systems and thus vanishes in the absence of the odd coefficients, is shown in Fig. 5(b).Note the rapid sign change of the lift coefficient, which becomes more pronounced and symmetrical as the compressibility ψ is decreased, leading eventually to a vanishing lift response in the incompressible limit ψ → 0.

IV. DISCUSSION
In this work, we considered the stability of twodimensional odd viscoelastic fluids by looking at the motion of probe particles that experience an instantaneous push at t = 0. We considered contributions to the constitutive equation that are allowed for passive systems (as found in Ref. [3]), as well as active terms that rely on fuel consumption at a microscopic level.Such active terms modify the passive picture by breaking the two hallmarks of passive hydrodynamics: Onsager-symmetry and positive-definiteness of the Onsager matrices.To characterize the effect of activity, we first discussed the stability condition of a probe in a generic, compressible odd viscoelastic fluid, before focusing on the incompressible limit, where the discussion is simpler while the essential ingredients are kept.Importantly, although the lift force vanishes in this limit [17], odd viscoelastic effects manifest themselves spectacularly as they can destabilize the probe motion.We find that passive odd viscoelastic fluids, where viscoelastic coefficients are constrained by the Second Law of Thermodynamics, are always linearly stable.We find the converse statement that active odd viscoelastic fluids are always unstable not to be true.Specifically, we find that there is an intermediate region where the motion of probe particles is stable, despite the system being active and constantly consuming fuel.In this intermediate regime, there is a non-trivial interplay between the odd elasticity coefficient ν o and odd plasticity γ o , which is graphically displayed in Fig. 3.This interplay is even more surprising considering that odd plasticity is a non-dissipative phenomenon and therefore plays no role in determining the passivity of an odd viscoelastic fluid [3].Similarly, the non-dissipative shear elastic coefficient ν s is involved in stability constraints despite being non-dissipative.
The stability of viscoelastic models was considered in previous works by looking at the stability of perturbative modes.In Ref. [3], a small wave-vector analysis yielded a constraint identical to Eq. ( 21b), which is one of the three constraints obtained in this work for stable probe particle motion that are given in Eq. ( 21).In Ref. [25], odd viscoelastic materials were studied in the context of the formation of spatial-temporal patterns, whose presence indicates instability.The authors performed a numerical analysis of the perturbative modes to determine their stability for a range of wave vectors.The numerical findings of Ref. [25], showing that odd elasticity can destabilize the viscoelastic material, whereas shear elasticity, plasticity and viscosity tend to stabilize the viscoelastic material qualitatively agree with the analytic constraints derived in the present work.Such a qualitative finding is also consistent with the analytically derived stability constraint of Eq. (21b) found in Ref. [3].
In this work, we furthermore considered the effect of several odd viscoelastic coefficients on probe particle motion.The obtained results provide a viscoelastic extension of Ref. [18] for the incompressible case and of Refs.[39,40] for the compressible case, which considered lift and drag force for an odd viscous fluid.When generalizing to viscoelastic fluids, one must consider inhomogeneities in time to see qualitatively different effects, as at vanishing frequencies only viscous effects remain.For this reason, we only considered the motion of a probe particle after a push at t = 0.A computational advantage of doing so is that there is no problem arising from the Stokes paradox [51], which for the steady case was resolved in Refs.[39,40] by regularizing the integral with momentum relaxation coming from the threedimensional bulk.We first considered the incompressible limit, which is a limit that is decoupled from passivity constraints and therefore a useful limit for considering active destabilizing effects.One key qualitative feature that we find is that the relaxing velocity becomes oscillatory for a non-zero odd plasticity coefficient γ o and that the amplitude of this oscillation is amplified by the odd elasticity coefficient ν o .This again highlights the non-trivial interplay between these two odd viscoelastic coefficients, and is illustrated in Fig. 4. We also discussed the role of compressibility in the response of the probe.In this case, we showed that the odd lift force, transverse to the direction of motion of the probe particle, that exists for viscous odd fluids, becomes time-dependent in viscoelastic systems.We discussed this case focusing on a small oddity limit, where analytical results can be obtained, and the results are displayed in Fig. 5.In the compressible regime, probes placed in a linearly unstable odd viscoelastic fluid are likely to perform limit cycles in the plane.Exploring these limit cycles, and more generally the probe motion in the linearly-unstable regime is an interesting aspect which requires introducing nonlinear terms whose symmetry and time-dependence need to be discussed.We leave this work for future publications.
Lastly, chiral biological systems such as cell assemblies or tissues are good candidates for displaying simultaneously odd, active, and viscoelastic behaviors.In this context, a layer of such cells is likely to exchange momentum with the three-dimensional bulk (for instance, an extracellular matrix in the case of an epithelial tissue, or sea water in the case of starfish oocytes [11]).In addition, cell number is usually not conserved in such systems since cell divisions, deaths or extrusions can occur.Although we have not considered finite momentum or density relaxation processes in this paper, taking them into account would be straightforward [52,53] in our formalism.The effects of such terms have been discussed for purely viscous odd fluids in Ref. [39].

Passive chiral viscoelastic fluid
In order to hydrodynamically describe a viscoelastic fluid we first formulate the conservation laws in absence of external forces and torques.Firstly, there is conservation of mass density ρ and momentum g i where σ tot ij is the total stress, v i is the fluid velocity.Momentum is related to fluid velocity as g i = ρv i .In addition, we have conservation of angular momentum, given by l ij the density of intrinsic angular momentum and M ijk is the angular momentum flux.Another property that is important for describing the state of viscoelastic fluids the object u ij .In the elastic limit, u ij is the elastic strain, however for general viscoelastic fluids u ij can undergo relaxation through plastic deformations [54,55].To study the evolution of u ij and the behavior the viscoelastic fluid in general, we start with formulating a local free energy [45,46]: where f (u ij ) the local free energy density, which depends on u ij .u ij is conjugate to the local elastic stress σ ij , such that σ el ij = ∂f /∂u ij .For an isotropic solid, the local free energy density at leading order in strain is given by To derive the constitutive equations for the viscoelastic fluid using the standard hydrodynamic approach, we decompose our stress into where σ s ij is the deviatoric stress, which is symmetric, and σ a ij is the anti-symmetric contribution contribution to the stress.The pressure P is obtained with the Euler relation with µ being the chemical potential and where the density n is given by n = ρ/m.If one considers the change of time of F , one finds upon plugging in the conservation laws of Eq. (A1) that the entropy production rate Θ is given by [45,46] T where we introduced the rotation as well as the corotational derivative which for a general two-tensor a ij is given by The presence of the corotational derivative in Eq. (A6) follows from the requirement that the free energy is invariant under rotations [45].To have a corotational derivative on the strain field requires the anti-symmetric stress to take the form σ a ij = u ik σ el kj − u jk σ el ki .Because of the conservation of angular momentum given by Eq. (A1c), there is no contribution coming from the anti-symmetric stress in the entropy production rate of Eq. (A6) [56].Instead, there is only a contribution from the deviatoric angular momentum flux M ijk , which will give contributions to the equations of motion that are two orders higher in gradients, and these contributions will therefore be omitted.Note that because odd viscoelastic fluids are often introduced by chiral agents that draw arbitrary amounts of angular momentum from the environment, the conservation of angular momentum tends not to hold.In the absence of angular momentum conservation, there are ways in which the anti-symmetric stress can get hydrodynamic corrections.However, we will omit these anti-symmetric contributions for simplicity by upholding angular momentum conservation.Having the entropy production rate of Eq. (A6) at our disposal, we are now ready to construct the Onsager matrix with containing the most general entries in the deviatoric currents.An important difference compared to the isotropic achiral viscoelastic fluids in two dimensions is that, in addition to the identity, the fully anti-symmetric two-dimensional Levi-Civita tensor ε ij can be used to construct entries in the constitutive equations.As we will discuss below, these tensors can feature components that are odd under the exchange of indices.This can for example give rise to odd viscosity [6,[57][58][59].Such terms signal parity-breaking, and for passive two-dimensional systems, one often finds that this breaking of parity is accompanied by a breaking of time-reversal symmetry, as is for example the case for parity-breaking induced by a background magnetic field.We therefore uphold symmetry under the simultaneous transformation of parity and time-reversal when deriving the passive constitutive equations, which modify the Onsager relations.The details of this are provided in App. C. From the expression of entropy production (A6) and the Onsager relations, we obtain the following constitutive relations: What we see in Eq. (A8) are the shear, bulk and odd viscosity given by the η-terms, as well as γ-terms representing shear and bulk plasticity, as well as a coefficient that could be called "odd plasticity".This odd plasticity was first considered in Ref. [60].Then, there are ν-terms which are non-equilibrium corrections representing shear, bulk and odd elasticity, the latter of which was first considered as an active term in Ref. [23].Using the compact notation described in App.B, Eq. (A8) can be rewritten as To see the dissipative properties of the entries in the constitutive equations of Eq. (A9), we plug Eq. (A8) back into Eq.(A6), which yields Requiring the first term on the right hand side of Eq. (A10) to be non-negative is equivalent to requiring that the matrix is semi-definite positive, which is satisfied if all its eigenvalues have non-negative real part.As was found in Ref. [3], this condition is guaranteed if, in addition to η p s , γ p s , η p b , γ p b ≥ 0, we have which is the thermodynamic constraint given in Eq. (4).

Active chiral viscoelastic fluid
The passive model we have introduced above can be generalized by considering activity induced by microscopic fuel consumption of microscopic agents [46].The rate of fuel consumption can be represented by ∆µ, which is the chemical potential difference of a chemical reaction, multiplied by a scalar reaction rate r.Because of this fuel consumption, the entropy production rate of Eq. (A6) is generalized to This new contribution opens up a wide range of new possible entries in the constitutive equations, which are given by σ s = ν p : σ el + 2η p : v + ∆µ ζ (1) : σ el + 2ζ (2) : v , The same correlation function can also be computed using time reversal.We have: where ϵ α is the signature of Φ α under time reversal and the overline • • • indicates that the sign of some parameters (for instance magnetic field) has to be changed.We thus deduce the Onsager matrix symmetry: It yields in particular: which means that an equilibrium odd viscosity needs to change sign under time reversal to be non-vanishing.This is consistent for instance with odd viscosity [6,[57][58][59], which is proportional to the magnetic field and therefore changes sign upon time reversal.Assuming a sign change upon time-reversal for all odd coefficients thus leads to the following structure for the Onsager matrix: where η p , ν p and γ p are isotropic rank-4 tensors that include odd and even parts.This yields Eq. (A9).a, s), which can be done averaging over the boundary line between the fluid and the probe particle [50].We instead consider the fluid velocity localized at |x| = 0, i.e. we relate as has also been done in Refs.[39,49,61].As shown in Ref. [50], averaging over velocities at |x| = a would yield an additional factor J 0 (ak) in the integrand of Eq. ( 16).However, this modification only leads to minor quantitative effects, and we thus ignore it for mathematical simplicity.
Appendix E: Computing the probe particle stability for active odd fluids

General case
In this Appendix, we detail the stability analysis of a probe particle in an active odd viscoelastic fluid, as discussed in Sec.II.Such stability is determined by the long-term velocity of the probe particle.This velocity follows from Eq. ( 12), which can be rewritten as where Γ is a real number so that the contour path of integration is in the region of convergence of the integrand.
As discussed in the main text, we consider a force perturbation along the eigendirections of M ij , and furthermore consider an instantaneous perturbation applied at t = 0, such that F j (t) ∝ δ(t), or F j (s) ∝ 1 in Laplace space.The velocity of the probe after this initial perturbation is thus given by: where m i (k, s) depends on the eigenvalues of M ij and on the form of the initial perturbation.We will not focus on its precise expression in the following discussion, but the crucial point is that m i (k, s) is a rational fraction in s.This implies that the inverse Laplace transform can be computed as: where the s * ℓ (k) are the poles in s of the rational fraction m i and where the m i,ℓ (k) are the residues of m i at s = s * ℓ (k).As a consequence of this simple form, a sufficient condition for the long-time stability of the probe is directly given by the sign of the real part of the poles of m i : which is the condition given in the main text.

Incompressible limit
To continue our analysis while keeping the technicalities to a minimum, we now restrict to the incompressible case.In this simpler case, M ij is proportional to the identity tensor, such that the probe velocity can be written as: where we have defined with τ 0 = ρ 0 a 2 /η s , and we only consider instantaneous perturbations (F i (s) constant).As discussed in the previous section, the long-time stability is then given by the poles in s of r.They correspond to the roots of the third-order polynomial P (s) = s 3 + a 2 s 2 + a 1 s + a 0 with where we have considered the case ν ′ s,o = ν s,o .The roots s * ℓ (k) of P (s) all have negative real parts iff a 0,1,2 > 0 and a 2 a 1 − a 0 > 0 according to the Routh-Hurwitz stability criterion.We note that a 2 is always positive.The conditions a 0,1 > 0 are equivalent to: A similar analysis can be conducted for ν ′ s,o ̸ = ν s,o . (a) FIG. 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.In order to compute the integrals Gi along the contours Ci, the radius of the large circle R and the width 2ε between the segments above and below the real axis are sent to ∞ and 0, respectively.
This inverse Laplace transform can be computed using residues.We first note that the function g(s) has branch cuts for Re [s(s − s 2 )/(s − s 1 )] < 0, that is for Re(s) < s 2 and 0 < Re(s) < s 1 , and is analytic otherwise.We thus define a closed integration contour (see Fig. 6) that avoids the branch cuts and such that i G i = 0 where G i = Ci ds g(s)e st /(2iπ).In the following, we detail the computation of the G i>1 .
Contours C 2 and C 14 .We set s = Re iθ with θ ∈ [π/2, π] such that: and where we have used the identities K 0 (ix) − K 0 (−ix) = −iπJ 0 (x) for x > 0. Importantly, we verify that in the long-time limit the behavior of M ∥ (t) ∼ t→∞ e s1t is set by the real part of s 1 (since (s 2 − u)t is always negative), and thus diverges, as expected for our choice s 1 > 0.

Introduction 1 I 8 IV. Discussion 9 A. Constitutive equations for active odd materials 10 1 .
. probe particle in an active odd viscoelastic fluid 2 A. Constitutive equations for the fluid 2 B. Jeffreys model for the odd viscoelastic fluid 3 C. Probe particle 4 II.Linear stability of a probe particle in an odd active viscoelastic fluid 5 A. General case 5 B. Stability of probe motion in the incompressible limit 5 III.Drag and lift response 7 A. Drag response for an incompressible fluid 7 B. Drag and lift response for a compressible fluid Passive chiral viscoelastic fluid 11

FIG. 1 .
FIG. 1. Schematic picture of a probe particle moving with velocity U in an odd viscoelastic fluid, experiencing a drag force F ∥ and a lift force F ⊥ .
2 J 8 X M = " > A A A B / 3 i c b V D N S 8 M w H E 3 n 1 5 x f V c G L l + A Q P I 1 W h n o c e v E 4 w X 3 A W k q a p l t Y m p Q k F U b d w X / F i w d F v P p v e P O / M d 1 6 0 M 0 f h D z e + z 3 y 8 s K U U a U d 5 9 u q r K y u r W 9 U N 2 t b 2 z u 7 e / b + Q V e J T G L S w Y I J 2 Q + R I o x y 0 t F U M 9 J P J U F J y E g v H N 8 U e u + B S E U F v 9 e T l P g J G n I a U 4 y 0

1 <
y 8 u g e 9 5 w L x r N u 2 a 9 d V 3 W U Q X H 4 A S c A R d c g h a 4 B W 3 Q A R g 8 g m f w C t 6 s J + v F e r c + 5 q s V q / Q c g j 9 j f f 4 A e 3 O W a A = = < / l a t e x i t > l a t e x i t s h a 1 _ b a s e 6 4 = " s l 4 l z V k n m U m e 0 i x D n A o Z n z B r X J s = " > A A A B / 3 i c b V D N S 8 M w H E 3 9 n P O r K n j x E h y C p 9 G O o R 6 H X j x O c B + w l p K m 6 R a W J i V J h V F 3 8 F / x 4 k E R r / 4 b 3 v x v T L c e d P M H I Y / 3 f o

2 <
l a t e x i t s h a 1 _ b a s e 6 4 = " C t + P l 6 U 1 N + f 1 5M 1 t u 2 f u i j O I H S w = " > A A A B + X i c b V D L S s N A F L 2 p r 1 p f U Z d u g k V w V R I R d V k U w W U F + 4 A 2 h M l k 0 g 6 d z I S Z S a G E / o k b F 4 q 4 9 U / c + T d O 2 i y 0 9 c A w h 3 P u Z c 6 c M G V U a d f 9 t i p r 6 x u b W 9 X t 2 s 7 u 3 v 6 B f X j U U S K T m L S x Y E L 2 Q q Q I o5 y 0 N d W M 9 F J J U B I y 0 g 3 H d 4 X f n R C p q O B P e p o S P 0 F D T m O K k T Z S Y N u D U L B I T R N z 5 f e B N w v s u t t w 5 3 B W i V e S O p R o B f b X I B I 4 S w j X m C G l + p 6 b a j 9 H U l P M y K w 2 y B R J E R 6 j I e k b y l F C l J / P k 8 + c M 6 N E T i y k O V w 7 c / X 3 R o 4 S V Y Q z k w n S I 7 X s F e J / X j / T 8 Y 2 f U 5 5 m m n C 8 e C j O m

FIG. 4 .
FIG. 4. Drag response ηsM ∥ of an odd incompressible viscoelastic fluid as a function of the dimensionless time t = t/τ0 with τ0 = a 2 ρ0/ηs.(a) In the absence of the odd elastic stress relaxation (γo = 0) and for different values of νo.(b) In the presence of an odd elastic stress relaxation (γoηs = 8) and for different values of νo.(c) Drag response for a varied odd elastic stress relaxation γo and a vanishing νo.(d) Drag response for a varied odd elastic stress relaxation γo and νo = −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: νs = 1, γsηs = 0.3, Gsτ0/ηs = 0.25.