Marco
De Corato
and
Gaetano
D'Avino
*

Dipartimento di Ingegneria Chimica, dei Materiali e della Produzione Industriale, Università di Napoli Federico II, P.le Tecchio 80, 80125 Napoli, Italy. E-mail: marco.decorato@unina.it; gadavino@unina.it

Received
22nd March 2016
, Accepted 30th June 2016

First published on 4th July 2016

In this paper, we investigate the dynamics of a model spherical microorganism, called squirmer, suspended in a viscoelastic fluid undergoing unconfined shear flow. The effect of the interplay of shear flow, fluid viscoelasticity, and self-propulsion on the orientational dynamics is addressed. In the limit of weak viscoelasticity, quantified by the Deborah number, an analytical expression for the squirmer angular velocity is derived by means of the generalized reciprocity theorem. Direct finite element simulations are carried out to study the squirmer dynamics at larger Deborah numbers. Our results show that the orientational dynamics of active microorganisms in a sheared viscoelastic fluid greatly differs from that observed in Newtonian suspensions. Fluid viscoelasticity leads to a drift of the particle orientation vector towards the vorticity axis or the flow-gradient plane depending on the Deborah number, the relative weight between the self-propulsion velocity and the flow characteristic velocity, and the type of swimming. Generally, pullers and pushers show an opposite equilibrium orientation. The results reported in the present paper could be helpful in designing devices where separation of microorganisms, based on their self-propulsion mechanism, is obtained.

Much effort, indeed, has been devoted to understanding the propulsion mechanism of single organisms or the collective dynamics of self-propelling particles through a Newtonian medium under quiescent conditions.^{7–11} Interesting phenomena, however, have been observed when active suspensions are subjected to external flows. For instance, experiments^{12,13} and simulations^{14} have shown that spermatozoa experience the so-called rheotaxis, that is, the tendency of swimming upstream when subjected to external flows. Active stresses generated by swimming microorganisms also significantly impact the rheology of the suspension.^{15,16}

In many situations of interest, small organisms swim through biological fluids such as mucus which display highly viscoelastic properties.^{17} The effect of a quiescent viscoelastic fluid on the locomotion of microorganisms has recently received much attention in the literature and a comprehensive review on the topic can be found in a recent book chapter.^{18} Most of the works have investigated the effect of viscoelasticity on the swimming velocity and efficiency of active particles^{19–23} or the dynamics of ciliated organisms close to boundaries.^{24–26} Much less is, instead, known when an external flow is applied. In this regard, we are aware of only two recent works where the dynamics of active microorganisms suspended in a viscoelastic fluid undergoing vortical^{27} and Poiseuille flow^{28} is investigated. In both works, the suspending medium is modelled by the so-called second-order fluid (SOF) constitutive equation which is valid for slow and slowly varying flow conditions. Nevertheless, interesting phenomena such as the emergence of well-defined patterns in vortical flow^{27} and an increasing tendency of swimming upstream in Poiseuille flow^{28} have been reported.

As the characteristic flow rates increase, viscoelastic effects become relevant and more complex microorganism dynamics are expected. The dramatic effect of fluid viscoelasticity is already evident in passive suspensions where peculiar phenomena occur as compared to the Newtonian case.^{29} Of course, the addition of a self-propulsion velocity would even more complicate the dynamics of the suspended particles.

The aim of the present work is to investigate the effect of the complex interplay between the swimming mechanism, viscoelasticity and an external flow field on the dynamics of microorganisms. Specifically, we present a detailed study on the dynamics of a spherical ciliated microorganism suspended in a viscoelastic fluid undergoing unbounded shear flow. We first perform a perturbative analysis at small Deborah numbers (defined as the product between the fluid relaxation time and the applied shear rate), for which the rotational velocity of the microorganism can be analytically computed. Numerical simulations are, then, carried out to investigate the particle dynamics at high Deborah numbers.

The present work is organized as follows: in Section 2, we present the model we used for the microswimmer; the equations governing the problem under investigation are reported in Section 3; the numerical method is briefly discussed in Section 4; in Sections 5 and 6, the results obtained from the perturbative analysis (low Deborah numbers) and numerical simulations (high Deborah numbers) are presented, respectively; and finally, conclusions are drawn in Section 7.

In the context of the Lighthill–Blake model, we consider the squirmer as a sphere of radius R propelling along a direction denoted by an orientation vector p through axisymmetric and time-independent surface tangential velocities. The velocity on the surface of the squirmer is given by

(1) |

We emphasize that, in view of the explicit time dependency of the viscoelastic stress constitutive equation, a steady (rather than a time-dependent) boundary condition on the squirmer surface may not capture all of the features of a real microorganism that swims in a sheared viscoelastic fluid. On the other hand, we believe that the simple steady slip velocity adopted in the present work allows one to understand the main characteristics of the microorganism dynamics when an external shear flow is coupled with self-propulsion and viscoelasticity, with a relatively small number of parameters to model the swimming velocity. (Notice that a time-dependent slip velocity would add at least one frequency for each mode and a shift angle.) Of course, the approximation of using a steady model is more and more accurate if the (long) time scale over which the orientation and position of the squirmer evolve is well-separated from the (short) time scale of the boundary actuation.

In the low-Reynolds number regime, the equations describing the fluid motion are the continuity and the momentum balance equations:

∇·u = 0 | (2) |

∇·Σ = 0 | (3) |

Σ = −pI + η_{s}(∇u + ∇u^{T}) + τ | (4) |

(5) |

(6) |

u = u_{∞} | (7) |

(8) |

To close the hydrodynamic problem, we need to consider the force and torque balances on the squirmer surface. Assuming negligible inertia, the squirmer is force- and torque-free:

(9) |

(10) |

Finally, an initial condition for τ is required. We assume a stress-free condition, i.e., that the stress is zero everywhere in the fluid at the initial time: τ_{|t=0} = 0.

The solution of the just presented equations gives the time evolution of the fields u, p, and τ as well as the time evolution of the microorganism rotational velocity ω and of the swimming velocity U. Once the translational and rotational velocities have been computed, the evolution of the swimmer centroid and orientation vector can be readily calculated from the following kinematic equations:

Ẋ(t) = U, X|_{t=0} = X_{0} | (11) |

ṗ(t) = ω × p(t), p|_{t=0} = p_{0} | (12) |

It is convenient to make the above equation dimensionless. We select the inverse of the imposed shear rate ^{−1} as the characteristic time, the radius R of the swimmer as the characteristic length, and η_{0} as the characteristic stress, where η_{0} = η_{s} + η_{p} is the zero-shear viscosity. By using these characteristic quantities, the governing equations in dimensionless form can be readily obtained. The continuity and the momentum balances are

∇·u* = 0 | (13) |

∇·Σ* = 0 | (14) |

Σ* = −p*I + (1 − η_{r})(∇u* + ∇u*^{T}) + τ* | (15) |

(16) |

u* = u_{∞}* | (17) |

(18) |

(19) |

(20) |

Ẋ*(t*) = U*, X*|_{t*=0} = X_{0}* | (21) |

ṗ(t*) = ω* × p(t*), p|_{t*=0} = p_{0} | (22) |

In what follows, all the symbols refer to dimensionless quantities. To simplify the notation, the stars are omitted.

A two-step decoupled formulation is adopted whereby the momentum and continuity equations are decoupled from the constitutive equation. At each time level, the stress field is first updated by solving the constitutive equation where the velocity field is taken from the previous step. Then, the viscoelastic stress tensor just computed is used as the known force term in the momentum balance which is solved together with the continuity equation and the force- and torque-free conditions to get the velocity and pressure fields, as well as the particle kinematic quantities. For the constitutive equation, a SUPG formulation^{43} is adopted with a log-conformation representation to improve the numerical stability.^{44,45} The force- and torque-free conditions in eqn (9) and (10) are imposed through Lagrange multipliers,^{46} and the unknown particle translational U and angular ω velocities are automatically computed by solving the augmented system of equations. Once the angular velocity is computed, the particle orientation can be updated through eqn (12). This is done by using quaternions,^{47,48} which easily allow one to track the orientation of bodies in space.^{47} The adopted numerical code has been validated in several previous works dealing with the dynamics of passive particles in viscoelastic fluids.^{46,48} More details on the adopted numerical procedure, the weak formulation of the governing equations and the solver can be found elsewhere.^{46,48}

Mesh and time convergence has been checked for all the simulations presented in this work. A dimensionless time step size Δt = 0.01 and a mesh with about 1500 triangular elements on the particle surface and 30000 tetrahedral elements in the domain suffice to assure convergence for most of the simulations. As expected, the most critical cases are found for the highest values of the parameters B_{1}*, De, and (the absolute value of) γ. Furthermore, time and spatial convergence for pushers requires lower time step sizes and finer meshes on the particle surface as compared with pullers and neutrals. For these cases, the dimensionless time step size needs to be reduced to Δt = 0.005 and a mesh with about 3000 triangular elements on the particle surface and 50000 tetrahedral elements in the domain is required.

u = u_{0} + De u_{1} + (De^{2}) | (23a) |

p = p_{0} + De p_{1} + (De^{2}) | (23b) |

Σ = Σ_{0} + De Σ_{1} + (De^{2}) | (23c) |

τ = τ_{0} + De τ_{1} + (De^{2}) | (23d) |

ω = ω_{0} + De ω_{1} + (De^{2}) | (23e) |

U = U_{0} + De U_{1} + (De^{2}) | (23f) |

In Appendix B, we show that the first-order rotational velocity ω_{1} can be analytically computed without the explicit knowledge of the first-order velocity, pressure, and stress fields, by means of the generalized reciprocity theorem^{49} from Stokes flow theory. To this aim, we only need to compute the zeroth-order (i.e., Newtonian) fields. Once ω_{0} and ω_{1} are known, the evolution of the orientation vector, accurate to first-order in De, is then obtained from eqn (22) with the total rotational rate given by eqn (23e).

The derivation of the zeroth-order fields is worked out in Appendix A. We want to emphasize here that, regardless of the kind of microorganism (neutral, pusher or puller), the dimensionless rotational velocity is simply equal to the rotational velocity of a ‘passive’ sphere in a shear flow, that is

(24) |

(25) |

The total rotational velocity is thus given by

(26) |

As a first step, we assess the range of De-values for which our perturbative analysis is applicable. In this regard, we consider squirmers with initial orientation belonging to the xy-plane (i.e., the shear plane). Indeed, in this validation case, the microorganism keeps tumbling in the shear plane (as this is a plane of symmetry), displaying a single non-zero component of the rotational velocity, namely the z-component ω_{z}. We compare ω_{z} as obtained from eqn (26) with that obtained from direct numerical simulations. In Fig. 2, we report ω_{z} as a function of the (positive) angle θ between the flow direction and the orientation vector for three different values of the Deborah number De = 0.05, De = 0.1, and De = 0.15. For De = 0.05, the analytical solution in eqn (26) is nearly indistinguishable from the numerical results for both the pusher and puller. As De increases, some deviations appear although the perturbative solution quantitatively well describes the numerical predictions. The agreement between the analytical and numerical solutions depicted in Fig. 2 also rules out possible instabilities that may arise in weakly nonlinear expansions of viscoelastic fluids as discussed by Morozov and Spagnolie.^{54}

Fig. 2 Plot of the z-component of the rotational velocity versus the positive angle θ between the flow direction and the orientation vector, for three different values of De and for B_{1}* = 1. Solid lines represent the values obtained from our direct numerical simulations, and dashed lines are the values given by eqn (26). |

The evolution of the orientation of the microorganism in a weakly viscoelastic shear flow can be obtained from eqn (12) with ω given by eqn (26). An investigation of the equilibrium points (ṗ(t) = 0) reveals that the only physically admissible solution is p_{z} = 1, that is the vorticity axis. To determine the stability of this equilibrium point (i.e., the microorganism aligns with or is repelled from the vorticity axis), we solved eqn (12) for different initial orientations out of the shear flow plane and for several choices of the rheological parameters. Typical trajectories obtained for pullers and pushers are shown in Fig. 3a and b, respectively, for a selected set of parameters. In these figures, the three components of the orientation vector are reported as a function of time for a squirmer suspended in a Newtonian (dashed lines) and viscoelastic (solid lines) fluid. First of all, it can be readily observed that viscoelasticity slightly reduces the frequency at which the microorganism tumbles around the vorticity. Nevertheless, similar to the Newtonian case, the trends for the viscoelastic suspending fluid are periodic meaning that the first-order perturbative solution does not predict any drift of the microorganism orientation towards a preferential orbit. To better visualize the squirmer orientational dynamics, in Fig. 4 the phase diagram p_{y}vs. p_{x} for the same set of parameters and initial conditions used in Fig. 3 is shown. The curves for a puller and a pusher in a viscoelastic fluid are shown with blue and red lines, respectively. In the same figure, we also show the case of a swimmer in a Newtonian fluid (green line). We clearly see that weak viscoelastic effects change the shape of the closed orbit described by the squirmer orientation vector, without, however, qualitatively changing the behavior of its orientational dynamics. Hence, we conclude that an orientational drift (if any) towards some stable equilibrium solution would require expansions to higher-orders in De, involving lengthy and cumbersome calculations. For this reason, the analysis at high Deborah numbers, with specific interest in the orientational drift, is performed through direct numerical simulations as discussed in the following section.

Fig. 4 Phase diagram of p_{y}vs. p_{x} for a puller (blue line) and a pusher (red line) suspended in a viscoelastic fluid with De = 0.1 and B_{1}* = 1 obtained from our perturbative solution. The same initial conditions as in Fig. 3a for the puller and in Fig. 3b for the pusher are used. The trend for a Newtonian suspending fluid (green curve) is also reported. |

Further simulations have been carried out to investigate the influence of the parameters De and B_{1}*. We found that, by decreasing these two parameters, the drift dynamics slows down. Indeed, as De is decreased, the system tends to the Newtonian case where no drift occurs. This is also confirmed by the perturbative analysis presented in the previous section where no drift is found at first-order in De. As B_{1}* goes down, the case of a passive spherical particle in a viscoelastic fluid is recovered where, again, there is no drift (the only non-zero component of the angular velocity is that around the vorticity axis). Despite variations in De or B_{1}* affecting the transients, the long-time regime remains the same, i.e., the vorticity axis, for any set of parameters in the investigated range.

Our numerical results for a neutral squirmer are summarized in Fig. 6a, where a map of the equilibrium solutions is reported as a function of De and B_{1}*. The black circle in this figure denotes vorticity alignment that, as just discussed, is found for any combination of De and B_{1}* investigated in this work. It is worth emphasizing that, for De < 0.15, the drift towards or away from the vorticity axis is found to be negligible and, then, the corresponding data are not reported in Fig. 6a.

In Fig. 6b and c, we show the trends of the particle kinematic quantities when the microorganism attains the equilibrium orientation as a function of the Deborah number and for three different values of the parameter B_{1}*. Specifically, Fig. 6b reports the magnitude of the relative translational velocity of the microorganism |U_{rel}| = |U − u_{∞}| (i.e., the particle translational velocity without the background shear flow field). Fig. 6c shows the magnitude of the microorganism rotational velocity |ω|. Both velocities are normalized by the corresponding translational and rotational velocities of a squirmer in an unbounded sheared Newtonian fluid, denoted by U_{N} = 2/3B_{1} and ω_{N} = /2, respectively. Since a neutral squirmer always aligns along the vorticity direction, the magnitudes of the translational and angular velocities are |U_{z}| and |ω_{z}|, respectively. The angular velocity of a passive spherical particle (B_{1}* = 0) in a sheared viscoelastic fluid is also shown in Fig. 6c (black symbols and lines). In agreement with previous calculations for active particles in a quiescent viscoelastic fluid,^{22} fluid viscoelasticity slows down the squirmer translational velocity as compared to the Newtonian case. The trends also show a decreasing slope as De increases, again in agreement with the results of Zhu et al.,^{22} where the existence of a minimum in the normalized translational velocity as a function of the Weissenberg number (defined as Wi = λB_{1}/(2R) = De B_{1}*/2) was reported. Similar to a passive sphere, also the squirmer rotation rate slows down as fluid viscoelasticity increases. However, higher angular velocities are found for larger values of the parameter B_{1}*, i.e., the squirmer rotates faster around the vorticity axis as the relative weight between the self-propulsion velocity and the flow characteristic velocity is higher.

Fig. 7 reports the dynamics of the orientation vector of a puller swimmer with the same parameters (De = 1 and B_{1}* = 1) and initial condition as in Fig. 5. As for the neutral squirmer, the orientation vector drifts towards the vorticity axis following a transient characterized by damped oscillations. The phase diagram shown in the right panel of Fig. 7, however, reveals that the spiraling orbit deforms from a circular to an elliptical shape, with the major axis crossing the first and third quadrants.

By changing the parameters De and B_{1}*, however, a qualitatively different dynamics is found. For instance, for De = 1 and B_{1}* = 0.2, both p_{x} and p_{y} oscillate with increasing amplitude as time goes on, up to spanning the whole range [−1, 1]. As a consequence, the trend of p_{z} tends to zero at long times. Therefore, the equilibrium solution is a ‘tumbling’ of the orientation vector on the flow-gradient plane, around the vorticity axis. In conclusion, by decreasing the parameter B_{1}*, the vorticity axis changes stability. We also found a much slower dynamics for B_{1}* = 0.2 as compared to the case B_{1}* = 1 (for B_{1}* = 0.2, in 1000 dimensionless time units the equilibrium orientation is not reached whereas 500 time units are sufficient for the case depicted in Fig. 7).

The combined effect of De and B_{1}* on the long-time regime is depicted in the solution map of Fig. 8a, where open symbols denote that the flow-gradient plane is a stable equilibrium solution. From the diagram we observe that, for De ≥ 1.5, the orientation vector of a puller always drifts towards the vorticity axis, whatever is the value of B_{1}*. On the other hand, in the case of De = 0.15, for all the values of B_{1}* investigated, the orientation vector of a puller moves towards the flow-gradient plane and the microorganism tumbles around the vorticity axis. Notice that this result is at variance with the perturbative solution derived in the previous section meaning that, already for De = 0.15, higher-order terms in De are required to predict the drift. For Deborah numbers ranging from 0.15 to 1.5, tumbling in the shear plane or alignment of the puller orientation vector along the vorticity axis occurs at low and high B_{1}*-values, respectively.

In Fig. 8b and c, the equilibrium translational and angular velocities of the puller squirmer are shown as a function of the Deborah number and for three different values of the parameter B_{1}*. As for the neutral squirmer, the filled symbols represent the velocities attained when the squirmer aligns along the vorticity direction, corresponding to the closed circles in Fig. 8a. Now, however, for low values of De and B_{1}*, the squirmer tumbles around the z-axis. In this case, the magnitude of the particle relative translational velocity in Fig. 8b is given by , whereas the magnitude of the angular velocity is again |ω| = |ω_{z}|. Since both velocities are periodic, we take the average over a period. Similar to the neutral squirmer, both De and B_{1}* slow down the translational and angular velocities as compared to the Newtonian case. A comparison between Fig. 6b and 8b also shows that a puller swims slower than a neutral squirmer, in agreement with previous numerical calculations.^{21}

The overall picture of the long-time orientational dynamics of the pusher is depicted in the solution diagram of Fig. 10a. Here, the same notation as before is adopted. At large Deborah numbers (De ≥ 1.5), for all the values of B_{1}* considered, the orientation vector drifts towards the shear plane and tumbles around the vorticity axis (open symbols in Fig. 10a); on the other hand, if De = 0.15, a pusher always aligns with the vorticity (closed symbols in Fig. 10a). As shown in Fig. 10a, in an intermediate range of the Deborah number (0.15 < De < 1.5), the stable equilibrium regime depends on B_{1}*, with tumbling and vorticity alignment promoted at low and high values of this parameter.

Finally, the normalized equilibrium translational and angular velocities of the pusher squirmer are shown in Fig. 10b and c as a function of the Deborah number and for three different values of the parameter B_{1}*. The same notation as in Fig. 8 is adopted. The translational velocity follows a trend similar to that observed for neutral and puller particles, decreasing with both De and B_{1}*. However, for any value of these two parameters, the pusher swimming velocity is always faster than the corresponding velocity of neutral and puller squirmers. The angular velocity shows a different trend as compared to neutral and puller particles. Indeed, up to De = 1.5, higher B_{1}*-values slow down the rotation rate. Only at De = 2, a faster rotation rate is found for increasing values of B_{1}*. However, the maximum variation of the angular velocity from B_{1}* = 0 to B_{1}* = 2 is lower than 5–6%. Hence, the effect of B_{1}* on the angular velocity for a pusher microorganism in a sheared viscoelastic fluid is relatively weak.

The data reported in these figures show that negative (positive) values of γ tend to slow down (speed up) the drift of the orientation vector towards the vorticity axis or speed up (slow down) the drift of p towards the flow-gradient plane. This is clearly visible for the neutral and puller squirmers with (De, B_{1}*) = (1, 1) (Fig. 11b and 12b) where vorticity alignment is reached in times that significantly depend on γ. A similar effect of γ is also found for a pusher (Fig. 13b) where the tumbling on the flow-gradient plane is reached at a higher or lower rate whether γ is negative or positive, respectively. Notably, for the highest γ-value investigated (black line), the equilibrium dynamics changes from tumbling to vorticity alignment.

The influence of the third mode on the equilibrium orientation is more relevant for the case (De, B_{1}*) = (0.5, 0.6) reported in panels (a) of the previous figures: the equilibrium orientation for the neutral and pusher squirmers switches from vorticity alignment to tumbling by decreasing γ from 0 to −0.5; the opposite is observed for a puller by increasing γ from 0 to 0.5. Furthermore, as before, higher and lower γ-values speed up vorticity alignment and tumbling, respectively. In conclusion, the third mode of Blake's expansion can change the equilibrium orientation attained by the particle, especially for low values of the parameters De and B_{1}*. It is important to point out, however, that the effect of the parameter γ is decoupled from β in the sense that, regardless of the kind of microorganism, positive or negative γ-values promote vorticity alignment or tumbling on the flow-gradient plane.

Panels (c) of the previous figures show the normalized translational velocity of the microorganism for different values of the parameters γ and for the two sets of (De, B_{1}*) corresponding to the data in panels (a) and (b). For all the three kinds of squirmer, an increasing translational velocity is found as γ increases. This effect is more evident for higher values of De and B_{1}*, and for a neutral and a puller squirmer. Finally, we mention that the equilibrium particle angular velocity is essentially unaffected by the third mode, with the maximum deviations in the investigated range of γ being lower than 1%.

The rheological properties of the suspending fluid might also have an important role in the microorganism dynamics. We have already mentioned that the first normal stresses are the main factors responsible for the observed drift. It is worthwhile to point out, however, that the Giesekus model employed in this work also predicts shear-thinning and a non-zero second normal stress difference in shear flow. Both these features might contribute in some way to the observed particle dynamics (as happens, for instance, for spheroidal passive particles^{48}). To clarify the detailed effect of the fluid rheology, the constitutive parameters of the viscoelastic model should be varied and simulations by changing the constitutive model are needed. This investigation will be part of future work.

A comparison between the solution maps for a puller reported in Fig. 8a and a pusher reported in Fig. 10a shows that, for a fixed couple (De, B_{1}*) (and for γ = 0), the two kinds of microorganisms have an ‘opposite’ behavior, i.e., if the pusher orientation vector drifts towards the shear plane, a puller aligns with the vorticity and vice versa. This opposite behavior is not completely unexpected because their flow fields in a Newtonian fluid are also inverse. As previously mentioned, the normal stresses are expected to be responsible for the orientation vector drift. The inverse flow field around pushers and pullers leads to an inverse shear rate gradient profile as well that, in turn, generates an opposite distribution of normal stresses around the surface, possibly justifying the observed behavior. This interesting feature might be exploited to obtain a separation of swimming microorganisms based on their propulsion mechanism.

Finally, it is interesting to compare the relevant time scales involved in real microorganism dynamics. Several characteristic times can be identified: (i) the time t_{b} for the boundary actuation that is related to the frequency of the cilia beating on the organism surface; (ii) the characteristic time required for the viscoelastic stress development t_{v} that is related to the fluid relaxation time λ; (iii) the flow characteristic time t_{f} related to 1/; (iv) the characteristic time t_{m} of the microorganism swimming that is related to R/B_{1} (i.e. the time required to swim a distance equal to its radius); (v) the time t_{T} needed for a revolution of the orientation vector around the vorticity axis, i.e., the characteristic time of the oscillations in Fig. 5, 7 and 9, that is related to the rotation period T = 2π/ω; and (vi) the time t_{s} required to achieve the equilibrium orientation. The fluid, flow and swimming time scales can be combined as discussed in Section 3 to give De and B_{1}*. The values selected for these two parameters give (t_{v}) ≈ (t_{f}) ≈ (t_{m}). Typical values of t_{m} of real microorganisms fall in the range [0.1–1 s].^{40,41} Thus, the interval of B_{1}* investigated in the present paper corresponds to shear rates that are easily accessible through conventional rheometers.^{56} Hence, by employing dilute and semi-dilute polymer suspensions, it should be possible to experimentally observe some of the effects described in the present paper. The rotation period T depends on ω, which, in turn, depends on De, B_{1}* and β. However, in the range of parameters investigated in this work, the microorganism angular velocity is of the same order of magnitude as in the Newtonian case ω_{N} = /2 (see, e.g., Fig. 6c, 8c and 10c). Thus, t_{T} is slightly less than one order of magnitude higher than t_{f}. The time t_{s} needed to reach the equilibrium orientation depends on all the relevant dimensionless numbers as well as on the initial orientation. However, it is always found that t_{s} is much larger than t_{T} as clearly visible in Fig. 5, 7 and 9 where several oscillations are required before reaching the steady-state condition. Finally, the characteristic time of the microorganism’s beating cilia t_{b} is typically 10–20 milliseconds.^{57} Hence, in the range of parameters investigated in this work, such a time is significantly lower than the other aforementioned characteristic times justifying the use of a steady-state boundary condition on the particle surface.

The analysis was extended to high Deborah numbers by means of direct finite element simulations. The numerical results show that, at variance with the Newtonian case, spherical microorganisms in a viscoelastic fluid attain a preferential orientation that depends on the Deborah number De, the relative weight between the self-propulsion velocity and the flow characteristic velocity B_{1}*, and the type of swimming (β, γ). Specifically, for γ = 0, neutral microorganisms always align with the vorticity axis. On the other hand, tumbling around the vorticity axis is found for pullers and pushers at low and high De-values, respectively. For intermediate De-values, the stable long-time solution depends on the value of B_{1}*. We emphasize that, for a fixed set of De and B_{1}*, puller and pusher squirmers always show an opposite behavior, suggesting a possible technique to separate microorganisms based on their propulsion mechanism. Non-zero values of γ can modify the equilibrium orientation as compared to the case γ = 0 in a direction that does not depend on the kind of squirmer. For some combinations of parameters, pushers and pullers can also tend towards the same long-time regime.

The results presented in this work can be extended in several ways. The effect of the fluid rheological properties as well as of a more realistic time dependent boundary condition for the beating cilia of the microorganism on the particle dynamics can be investigated. Finally, the orientational dynamics found in this study also suggests that the rheology of dilute microorganism suspensions in viscoelastic fluids could be very different from the Newtonian counterpart. The predictions of the rheology behavior of active viscoelastic suspensions will be part of future work.

∇·u_{0} = 0 | (27) |

∇·Σ_{0} = 0 | (28) |

Σ_{0} = − p_{0}I + (1 − η_{r})(∇u_{0} + ∇u^{T}_{0}) + τ_{0} | (29) |

τ_{0} = η_{r}(∇u_{0} + ∇u^{T}_{0}) | (30) |

Σ_{0} = −p_{0}I + (∇u_{0} + ∇u^{T}_{0}) | (31) |

(32) |

u_{0} = u_{∞} − U_{0} | (33) |

(34) |

(35) |

Notice that, in the low Deborah number analysis, the viscoelastic stress tensor instantaneously reaches its pseudo-steady value, and hence no initial condition is required for it.

Due to the linearity of the Stokes equations, the solution of the above system of partial differential equations is the superposition of the solution for the motion of a squirmer in a quiescent Newtonian fluid with arbitrary orientation p and that of a ‘passive’ sphere suspended in a Newtonian fluid undergoing a shear flow u_{∞} at infinity. Hence, the total zeroth-order translational velocity is given by the velocity of an unconfined squirmer in a quiescent Newtonian fluid ^{31} (as a ‘passive’ sphere in shear flow has no translational velocity in the reference frame we have chosen, see Section 3). The total rotational velocity is given by the rotational velocity of a ‘passive’ sphere in a Newtonian shear flow with k the z-axis unit vector (as the microorganism has no self-induced rotational velocity). The zeroth-order translational and rotational velocities reported above can also be derived from the reciprocal theorem as previously shown in the literature^{8} and summarized in a recent book.^{58} The total Newtonian velocity flow field u_{0} can be written as the sum of an ‘active’ contribution u_{0,a} and a ‘passive’ contribution u_{0,p}. The ‘active’ flow field is given by^{31}

(36) |

(37) |

(38) |

(39) |

u_{0} = u_{0,a} + u_{0,p} | (40) |

∇·u_{1} = 0 | (41) |

∇·Σ_{1} = 0 | (42) |

Σ_{1} = −p_{1}I + (1 − η_{r})(∇u_{1} + ∇u^{T}_{1}) + τ_{1} | (43) |

(44) |

(45) |

σ_{1} = −p_{1}I + (∇u_{1} + ∇u^{T}_{1}) | (46) |

(47) |

The first-order mass balance in eqn (41) and the first-order momentum balance in eqn (42) are equipped with the first-order boundary conditions on the particle surface, in a reference frame translating with the particle:

u_{1} = ω_{1} × r | (48) |

u_{1} = −U_{1} | (49) |

(50) |

(51) |

We proceed by defining an auxiliary Stokes problem, namely, the rotation of a sphere with a rotational velocity , suspended in an unbounded quiescent Newtonian fluid. (All the quantities related to the auxiliary problem are denoted with a hat superscript.) We emphasize that in principle any Stokes problem might be chosen as an auxiliary problem in the reciprocity theorem. Typical choices are rigid body motions like translation and rotation.^{20} The choice of the auxiliary problem made in the present work is motivated by the fact that we are interested in the rotational rate of the particle and not in the translational velocity. The dimensionless equations governing this problem are

∇·û = 0 | (52) |

∇· = 0 | (53) |

= −I + ∇û + ∇û^{T} | (54) |

û = × r | (55) |

û = 0 | (56) |

The derivation in this appendix closely follows that originally reported by Lauga,^{20,60} subsequently used in Datt et al.^{38} and De Corato et al.,^{19} and recently summarized in a book chapter.^{18} The generalized reciprocity theorem states that

(57) |

(58) |

(59) |

(60) |

(61) |

(62) |

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