On the cross-streamline lift of microswimmers in viscoelastic flows [original]

48 | Soft Matter, 2022, 18, 48–52 This journal is © The Royal Society of Chemistry 2022

Cite this: Soft Matter, 2022,

18, 48

On the cross-streamline lift of microswimmers in

viscoelastic flows†

Akash Choudhary * and Holger Stark *

The current work studies the dynamics of a microswimmer in

pressure-driven flow of a weakly viscoelastic fluid. Employing a

second-order fluid model, we show that a self-propelling swimmer

experiences a viscoelastic swimming lift in addition to the well-

known passive lift that arises from its resistance to shear flow. Using

the reciprocal theorem, we evaluate analytical expressions for the

swimming lift experienced by neutral and pusher/puller-type swim-

mers and show that they depend on the hydrodynamic signature

associated with the swimming mechanism. We find that, in com-

parison to passive particles, the focusing of neutral swimmers

towards the centerline can be significantly accelerated, while for

force-dipole swimmers no net modification in cross-streamline

migration occurs.

Biological microswimmers are ubiquitous in polymeric media

such as cervical, bronchial, and intestinal mucus films.

1,2

During the generation of biofilms most bacteria release a

mixture of proteins, DNA, and polysaccharides, endowing the

fluid with viscoelastic properties,

3,4

which significantly alter the

swimmer’s dynamics.

5,6

Pathogens like ulcer-causing Helico-

bacter pylori survive in a harsh acidic environment by altering

the rheology of the mucus lining in the stomach.

Several

theoretical and experimental studies on the dynamics of motile

microorganisms in Newtonian flows have revealed their rich

dynamics and ability to swim against fluid flows, which aids in

seeking nutrients and in reproduction.

8–15

Although recent works have provided insights into self-

propulsion in non-Newtonian environments,

15–32

few have stu-

died the impact of fluid rheology on the dynamics of micro-

swimmers in confined flows.

33–36

Mathijssen et al.

developed

a model to predict the dynamical states of microswimmers in

non-Newtonian Poiseuille flows. Employing a second-order

fluid model, they demonstrated that normal-stress differences

reorient the swimmers to cause centerline upstream migration

(rheotaxis). This suggests that non-Newtonian properties can

help microorganisms evade the boundary accumulation, pre-

valent in quiescent Newtonian fluids.

37,38

The evidence of centerline reorientation of microswimmers

can be traced back to pioneering studies on viscoelastic focus-

ing of passive particles in Poiseuille flows.

39–41

These studies

showed that normal stresses exert a lift force that focuses the

particles on the centerline. Ho and Leal

used the reciprocal

theorem and derived an analytical expression for this lift in

weakly elastic Boger fluids. The study suggested that the

hydrodynamic disturbances around the particle produce a

hoop stress that, in the presence of non-uniform shear rate,

generates a cross-streamline lift. Recent progress in

electrophoresis

42–44

has also shed light on the importance of

hydrodynamic disturbances in determining these lift forces.

Active microswimmers, as opposed to passive particles,

generate additional disturbance flow fields in the fluid due to

their self-propulsion. Therefore, the associated lift force or

velocity should also have an active component that is charac-

teristic of the self-propulsion mechanism. Since this compo-

nent is absent in the recently proposed models,

in this

communication we derive the ‘swimming lift’ in a second-

order fluid (SOF), and show how it aﬀects the dynamics of a

microswimmer in the Poiseuille flow of a viscoelastic fluid. We

choose the SOF model because it provides an asymptotic

approximation for a majority of slow and slowly varying viscoe-

lastic flows.

45,46

Fig. 1 shows a spherical swimmer of radius aat position r

that self-propels with velocity v

pin a two-dimensional

pressure-driven flow v

[1 (x/w)

, where v

is the

maximum flow velocity and wis the half channel width. The

flow profile of the second-order fluid is identical to Poiseuille

flow but the pressure field varies in the x-direction.

In the

absence of noise, the swimmer’s dynamics is governed by

r¼pþ

vfþFðx;pÞex;_

p¼1

2r

vfÞp;ð(1)

Institute of Theoretical Physics, Technische Universita

¨t Berlin, 10623 Berlin,

Germany. E-mail: [email protected]berlin.de, holger.stark@tu-berlin.de

†Electronic supplementary information (ESI) available. See DOI: 10.1039/

d1sm01339d

Received 15th September 2021,

Accepted 24th November 2021

DOI: 10.1039/d1sm01339d

rsc.li/soft-matter-journal

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where the velocities are non-dimensionalized by swimming speed

), lengths by w,andtimebyw/vs. Fdenotes the

total viscoelastic lift velocity, which comprises the passive and

swimming lift. Below, we derive the analytical expressions for the

lift velocities. We note that normalstressesalsomodifytheparticle

rotation and the drift velocity along the channel axis. We evaluated

these modifications and found that they do not play a significant

role in determining the swimmer dynamics.

The inertia-less or creeping flow hydrodynamics is governed

by the continuity equations for mass and momentum, which we

formulate here in the co-moving swimmer frame {x

˜,y

˜,z

˜} as:

rV¼0;~

rT¼0(2)

in order to calculate the viscoelastic lift velocity. The length,

velocity, and pressure in (2) are non-dimensionalized by a,kv

and mkv

/a, respectively. Here, kis the particle to channel

width ratio (a/2w) and mis the fluid viscosity. In the above

equation, Vis the total velocity field, Tis the total stress tensor

of a second-order fluid and thus has the form:

T=PI+2E+

WiS. Here, Edenotes the rate of strain tensor, Sthe total

polymeric stress tensor, and Wi = (C

)G/mis the shear

based Weissenberg number, where mis the viscosity, G=v

/2w

characterizes the shear rate in the background flow, and C

represents the dimensional steady-shear normal stress coeﬃ-

cients that are measured experimentally.

The polymeric stress

tensor S¼4EEþ2dE

Dis non-linear in Eand contains the

lower-convected time derivative of Edenoted by D. The visco-

metric parameter d=C

/2(C

) generally varies from 0.5

to 0.7 for most viscoelastic fluids.

47–49

We now focus on determining the lift velocity of a micro-

swimmer that disturbs the background flow in two ways. First

of all, the microswimmer resists straining by the flow and

second, it generates a flow field characteristic of its swimming

mechanism, for which we first take a source-dipole swimmer.

We split the total velocity field (V=v

+v) into background

flow field v

(in our case the Poiseuille flow in the co-moving

swimmer frame) and disturbance field v, and adopt the same

nomenclature for the polymeric stress tensor, S=s

+s, where

sis the polymeric stress tensor associated with the disturbance

flow field. We show its full form in the ESI.†Substituting this in

the governing eqn (2) yields the equation for the disturbance

field v:

rv¼0;~

rpþ~

r2v¼Wið~

rsÞ:(3)

The methodology of evaluating the lift velocity closely follows

the recent analysis of Choudhary et al.

Here, we only sum-

marize the basics and elaborate on details in the ESI.†Assum-

ing weak viscoelasticity (Wi {1), we perform a perturbation

expansion in Wi and only take the first two contributions of

eqn (3) in this expansion: the Stokes equation for the zeroth

order of the disturbance field, ~

rp0þ~

r2v0¼0and ~

rp1þ

r2v1¼~

rs0for the first order. Following earlier works on

viscoelastic lift,

41,43

we use the reciprocal theorem to attain the

lift velocity from the first-order problem,

F¼

6pðVf

s0:~

rutdV:(4)

Here, u

is the auxiliary or test velocity field that belongs to a

forced particle moving along the x-direction with unit velocity

in a Newtonian fluid. The polymeric stress tensor s

is asso-

ciated with the Stokes solution v

of the microswimmer. The

zeroth order of the disturbance flow field consists of (i) a

source-dipole field, which we adopt from the squirmer

model,

50–52

vswim

0¼~

vsp

r33~

r2I



;and (ii) the passive distur-

bance field v

passive

, where v

/(v

k). For the latter, the

leading contribution is the stresslet, while higher-order terms

due to the curvature in the Poiseuille flow profile are obtained

from Lamb’s general solution.

The bounding channel walls

modify these bulk flow fields. However, by using the method of

reflections for small particles (k{1), one can show that they

do not alter the lift velocity in leading order in k.

Therefore,

it is sufficient to perform the integration in eqn (4) in the

infinite domain. Using the s

corresponding to Stokes velocity

field in eqn (4), results in the viscoelastic lift velocity given in

units of v

Fðx;cÞ¼F

passive þFswim

¼Wi 80

9x

vmk2ð1þ3dÞxð1þdÞcos c



(5)

The first component in eqn (5) is the passive lift Fpassive.

fixing dto a widely-used value of 0.5 (i.e. C

= 0), we observe

that Fpassive focuses the swimmer towards the centerline. The

second component is the swimming lift Fswim that arises due to

the source-dipole disturbance created by the neutral swimmer.

We note two striking features of Fswim: the dependence on

swimmer orientation through cos cand that its magnitude is

larger by a factor k

compared to the first term.‡

Now, we substitute Finto the dynamic eqn (1) and examine

the eﬀect of Fswim on the microswimmer dynamics. We find

two fixed points in the x–cplane at x= 0, with the micro-

swimmer oriented upstream (c= 0) or downstream (c=p). A

linear stability analysis provides the following eigenvalues for

these fixed points:

lup Wi

18 9ð1þdÞþ80ð1þ3dÞk2



i

vm1=2;

ldown Wi

18 9ð1þdÞþ80ð1þ3dÞk2





vm1=2:

(6)

Fig. 1 A spherical microswimmer with velocity v

pmoves in a pressure-

driven flow of a second-order fluid inside a channel with half width w.The

coordinate frame {x

˜,y

˜,z

˜} co-moves with the swimmer.

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For a typical value of d=0.5 and weak viscoelasticity limit

(Wi {1), the downstream orientation corresponds to a saddle

fixed point (associated with l

down

), while the upstream orienta-

tion and swimming along x= 0 corresponds to a stable fixed

point (associated with l

). For d=0.5, the sign of the real

part of l

shows that both swimming and passive lift compo-

nents stabilize the upstream orientation. Fig. 2(a) and (b) show

the upstream and downstream trajectories of the swimmer. In

both cases the swimmer attains the upstream orientation,

which is stabilized by both Fpassive and Fswim. The strong

swimming lift helps the neutral microswimmer reach the

centerline more rapidly, as suggested by the factor of k

eqn (5). To quantify this accelerated focusing, we calculate the

focusing time tas the time required for the oscillatory ampli-

tudes to decrease to 5% of the half channel width, i.e.,x= 0.05.

For upstream swimming at low flow rate (%

= 0.9), Fig. 2(c)

shows how the focusing time changes with initial position and

orientation. We further evaluate the impact of Fswim on the

focusing time in Fig. 2(d). At low flow rates (%

B1), the

centerline focusing is much faster when Fswim is taken into

account, however, this difference withers as the flow rate

increases; this can also be observed from the analytical expres-

sion (5). We further analyze the focusing time over a wider

range of parameters in the ESI.†

Now, we shift our attention from neutral squirmers to

flagellated microorganisms, such as E. coli and Chlamydomo-

nas, that generate a force-dipole field at the leading order:

37,54

v0¼P

vsr1

r3þ3rpðÞ

. Here Pis the force-dipole strength in

units of 8pma

, which depends on the swimming

mechanism.

37,55,56

Earlier studies on E. coli

56,57

and Chlamydo-

monas

suggest that P

varies roughly between 0.04 and 0.3.

Following the procedure outlined for a source-dipole swimmer,

we obtain the swimming lift velocity of the force-dipole swim-

mer in the units of v

Fswim ¼ð8=3ÞWikPð1þ3dÞsin 2c;(7)

and find it to depend on the constant curvature in Poiseuille

flow, as detailed in the ESI.†Although this lift is O(k

) larger

than Fpassive, it does not result in net cross-stream migration

because it is independent of lateral position. The trajectories in

Fig. 3(a) and (b) show the dynamics of a force-dipole swimmer.

In the steady state, the swimmer shows a stable upstream

orientation and swims along the centerline since a deviation

from c= 0 is counteracted by flow vorticity. For c=0,Fswim

vanishes and the focusing along the centerline is purely due to

Fpassive.

So far we have neglected the hydrodynamic interactions of

microswimmers with the bounding channel walls. For force-

dipole swimmers, the hydrodynamic wall interactions add a

modification of order k

and k

to the evolution equations of

position and orientation, respectively:

11,59

x¼sin cþF3Pð3 sin2c1Þ

8k21

ð1xÞ21

ð1þxÞ2



;

c¼x

vm3Psin 2c

16 k31

ð1xÞ3þ1

ð1þxÞ3



(8)

Upstream trajectories in Fig. 3(c) closely resemble the behavior

reported previously for pure Newtonian fluids.

We observe

that the hydrodynamic wall attraction of pushers

overcomes

Fpassive and results in swinging across the whole channel cross

section, where the strong vorticity near the walls always re-

orients the swimmer away from it. In contrast, for downstream

Fig. 2 Trajectories showing the centerline focusing of a neutral (source-

dipole) swimmer oriented upstream while (a) swimming upstream

= 0.9, ’) and (b) swimming downstream (%

=8,-). The blue and

green trajectories only diﬀer in the initial condition x

; for both trajectories

we choose c

= 0. Other parameters: Wi = 0.1, k= 0.1, d=0.5.

and c

Parameters: %

= 0.9, Wi = 0.1, k= 0.1. Trajectories are considered focused

when the amplitude of oscillations reaches x= 0.05. (d) Focusing time for

various flow rates and two diﬀerent Fswim swimmer sizes. Dashed lines

depict the focusing time in the absence of Fswim. We plot the maximum

focusing time that occurs for the initial condition x

= 0.9, c

=0.

Fig. 3 Trajectories showing the centerline focusing of pusher P¼0:3

ðÞ

(a) upstream swimming (%

= 0.9, ’) and (b) downstream swimming

=3,-). The trajectories for a puller are similar. The blue and green

trajectories in (a) and (b) only diﬀer in the initial condition x

; for both

trajectories we choose c

= 0. (c) Upstream (%

= 0.9) and (d) downstream

trajectories (%

= 3) with hydrodynamic wall interactions (8) incorporated.

For the initial condition, x

= 0.9, the blue trajectory in the background of

(d) shows downstream swinging across the whole channel cross

section. We employ hard-core repulsion at the walls. Other parameters:

Wi = 0.1, k= 0.1, d=0.5.

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swimming at larger flow rates [Fig. 3(d)], Fpassive competes with

the hydrodynamic wall interactions, which results in some

trajectories focusing at the centerline and some following

channel-wide swinging depending on the initial condition.

The state diagram in Fig. 4(a) illustrates this behavior of

pushers for a wide range of parameters. At low flow rates

(region I) hydrodynamic wall interactions dominate and thus

channel-wide swinging occurs. As the flow rate increases the

system enters region II. Here, the realized trajectories depend

on the initial condition as depicted in Fig. 3(d): swimmers

starting near the walls follow downstream swinging across the

channel cross-section, whereas swimmers near the center focus

at the centerline. A further increase in flow rate or swimmer

size increases the magnitude of Fpassive relative to wall eﬀects,

which results in centerline focusing (region III). For the same

reason, as the force-dipole strength decreases [see Fig. 4(b)], the

regions of swinging state (region I) and coexisting states (region

II) shrink. Pullers, on the other hand, are hydrodynamically

repelled from the walls

and therefore always complement

Fpassive in centerline focusing [Fig. 3(c) and (d)]. We note that

for source-dipole swimmers the hydrodynamic wall interac-

tions are weaker compared to force dipoles since they scale

with k

, and therefore hardly influence the trajectories. In

Table 1, we list expressions for swimming lift and wall interac-

tions that alter the cross-stream migration.

In conclusion, the current study analyzes microswimmers in

weakly viscoelastic pressure-driven flows. For neutral and

pusher/puller microswimmers, we derive an additional swim-

ming lift velocity depending on the swimmer’s hydrodynamic

signature that adds to the passive viscoelastic lift.

33,41,60,61

For source-dipole (neutral) swimmers, the swimming lift is

two orders of magnitude stronger than the passive lift, which

was considered alone in a recent study.

The current work

shows that the swimming lift accelerates the centerline focus-

ing. For force-dipole swimmers (pusher/puller), the swimming

lift does not contribute to a net cross-streamline migration.

Incorporating hydrodynamic wall interactions, we show that

upstream swimming for weak flow strengths qualitatively fol-

lows the behavior in Newtonian fluids:

attraction of pushers

towards the channel walls and repulsion of pullers. The down-

stream swimming along the centerline qualitatively remains

the same as that of a passive particle. Finally, we stress that our

analysis is valid for an arbitrary swimming direction, which can

be decomposed in swimming along and perpendicular to the

shear plane.

Interestingly, the results suggest that normal stresses in

viscoelastic fluids generated by the flow field of a neutral

swimmer can accelerate the centerline focusing. However, this

strongly depends on the hydrodynamic signature of the micro-

swimmer. Even for a weakly viscoelastic fluid (Wi = 0.1), we

observe rapid focusing within a traveled distance of

10–500 times the channel width (Fig. 2), which amounts to

ca. 1–50 mm and is quite realistic for microfluidic channels.

Thereby, this work contributes to the understanding of swim-

ming in more realistic biological fluids. We note that higher

order multipoles like force quadrupole, rotlet dipole and so on,

will have their own Fswim. Therefore, we foresee further devel-

opment in understanding the trajectories of microswimmers

that exhibit higher order multipoles in their hydrodynamic

disturbance. Furthermore, the current work offers several new

directions to explore. For instance, elongated microswimmers,

like passive particles, perform Jeffery orbits in sheared New-

tonian fluids.

In viscoelastic fluids the flow disturbances from

swimming will alter the orientation evolution of these orbits

and hence the swimmer dynamics.

The impact of shear-

thinning fluids is also an interesting outlook, which can be

achieved by the use of more detailed rheological models.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

Support from the Alexander von Humboldt Foundation is

gratefully acknowledged.

Notes and references

‡The correction to the drift velocity in z-direction is Wixksin c(1 + d),

and the correction to rotation is found to be 2Wiksin c. We verified

that these modifications do not qualitatively alter the dynamics and

thus neglect their contributions for simplicity. Details are provided in

the ESI.†

1 S. S. Suarez and A. Pacey, Hum. Reprod. Update, 2006, 12,

23–37.

Fig. 4 State diagram for pushers (a) P¼0:3and (b) P¼0:1. Region I:

unstable trajectories that spiral out to result in channel-wide swinging,

Region III: stable centerline swimming, Region II: both dynamic states

coexist; they depend on the initial condition. Other parameters: Wi = 0.1,

=0.

Table 1 Summary of leading-order contributions in Wi that determine the

lift velocity of a microswimmer in the pressure-driven flow of a second-

order fluid. Here, SD and FD refer to source-dipole (neutral squirmer) and

force-dipole (pusher/puller) swimmers, respectively. In the third column 1/

represents the function 1

1x



1

1þx



Type Lift velocity pWi Wall contribution to :

Passive 80

9x

vmk2ð1þ3dÞ—

SD x(1 + d)cos c(sin c)k

FD 8

3kPð1þ3dÞsin 2cð3=8ÞPð13 sin2cÞk2=e2

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