Abstract
The transport of motile microorganisms is strongly influenced by fluid flows that are ubiquitous in biological environments. Here we demonstrate the impact of fluid inertia. We analyze the dynamics of a microswimmer in pressuredriven Poiseuille flow, where fluid inertia is small but nonnegligible. Using perturbation theory and the reciprocal theorem, we show that in addition to the classical inertial lift of passive particles, the active nature generates a ‘swimming lift’, which we evaluate for neutral and pusher/pullertype swimmers. Accounting for fluid inertia engenders a rich spectrum of complex dynamics including bistable states, where tumbling coexists with stable centerline swimming or swinging. The dynamics is sensitive to the swimmer’s hydrodynamic signature and goes well beyond the findings at vanishing fluid inertia. Our work will have nontrivial implications on the transport and dispersion of active suspensions in microchannels.
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Introduction
Selfpropelling microswimmers often experience dynamic fluid environments and confinements, for example, pathogens in lung mucus^{1}, microorganisms in laminar flow through porous matrix^{2}, and sperm cells in the Fallopian tubes^{3}. Often these swimmers interact with microscale flows and boundaries^{4} to enhance survival probability^{5} and biofilm formation^{6} or cause intriguing collective patterns^{7}. Their envisioned artificial counterparts—designed to execute in vitro drug delivery—also would have to interact with the dynamic conditions of such biological flows^{8,9}. Examining the dynamics of microswimmers can help us get insights into the dispersion of active suspensions^{10}, it can also provide guidelines for the rational fabrication of microfluidic drug delivery and for minimizing biofilm formation in biomedical equipments^{11}.
Sheared flows in biological systems and microchannels impose substantial vorticity on the swimmer, which results in continuous tumbling. Past experimental and computational studies have shown that this tumbling, in conjunction with surface interactions, cause upstream swimming known as rheotaxis^{12,13,14,15}. Zöttl and Stark^{16,17} developed a theoretical model in the Stokes regime that captured swimming in Poiseuille flows. They reported upstream swinging and, for sufficiently strong flows, downstream tumbling states in planar and cylindrical channels. In two dimensions the phase portrait was found to be equivalent to that of a nonlinear oscillator such as the pendulum. Related experimental and theoretical research studied single swimmer trajectories^{18,19}, shearinduced trapping^{20}, and focusing of phototactic algae^{21} or magnetotactic bacteria^{22,23}. Most recently, Peng and Brady^{10} investigated Taylor dispersion in active suspensions.
In recent years, experimental and theoretical studies have shown how inertia affects the unsteady propulsion of ciliated^{24,25} and larger swimmers^{26,27}. The influence of particle inertia has been discussed in refs. ^{28,29}. With the recent advent of highspeed tunable microswimmers^{9,30,31}, understanding the effects of inertia can help in effective designs of biomedical devices. However, little is known how fluid inertia affects swimmer dynamics in sheared flows, which we will address in this article.
For passive particles, the Segré−Silberberg effect at finite Reynolds numbers has been known for decades^{32,33}. Inertial lift forces cause crossstream migration and eventually focus particles roughly halfway between channel center and walls. This effect has initiated major advances in cellsorting and flow cytometry techniques in the newly developing field of inertial microfluidics^{34,35}. To understand it, we note that a rigid particle resists the strain in background flow and generates a stresslet disturbance in the fluid decaying as 1/r^{2} ^{36}. The disturbance interacts with the curvature of the background flow and the channel walls, which in the presence of fluid inertia results in counteracting sheargradient and wallinduced lift forces that cause inertial focusing^{37}.
In his seminal work, Saffman^{38} considered a particle moving relative to a uniform shear flow under the influence of an external (gravitational) force. He showed that it also experiences a crossstreamline lift. Similar investigations were presented in recent works on electrophoresis^{39,40,41,42,43}. They stressed the key role of the leading hydrodynamic multipole generated by the particle.
Microswimmers also move relative to an applied background flow. In this article, we consider the generic source–dipole and force–dipole microswimmers and calculate the resulting swimming lift in a planar Poiseuille flow when fluid inertia is small but nonnegligible. We demonstrate that, in combination with the passive inertial lift, this gives rise to rich complex dynamics in channel flow, which goes well beyond the findings in refs. ^{16,17}. Our work thereby opens up a new direction in the field of active matter by connecting research on microswimmers to the field of inertial microfluidics with all its biomedical applications^{34,35}.
In the following, we consider a spherical swimmer of radius a that selfpropels with velocity v_{s} = v_{s}p in a twodimensional Poiseuille flow v_{f} = v_{m}[1 − (x/w)^{2}] e_{z} where v_{m} is the maximum flow velocity and w the half channel width (see Fig. 1). The overdamped motion of a noisefree swimmer can be described by dynamic equations for swimmer position (r) and orientation (p) vector,
where we rescaled velocities by swimming speed v_{s}, lengths by w, and time by w/v_{s}. \({{{{{{{\mathcal{F}}}}}}}}\) denotes the total inertial lift velocity, which comprises the passive and swimming lift. It vanishes when fluid inertia becomes negligible and the system moves in the Stokesian regime as studied in ref. ^{16}. The passive inertial lift is wellexplored^{37,44,45} and, except in close vicinity to the channel walls, can be well approximated by \({{{{{{{{\mathcal{F}}}}}}}}}_{{{\mbox{passive}}}}(x)\approx {{{\mbox{Re}}}}_{p}\ \kappa {\bar{v}}_{m}\ x(1{x}^{2}/{x}_{{{{{{\rm{eq}}}}}}}^{2})\) (see “Methods” section). Here, x_{eq} denotes the stable equilibrium positions, κ = a/2w the ratio of swimmer radius to channel width, and \({\bar{v}}_{m}={v}_{m}/{v}_{s}\) is the scaled centerline flow velocity. The swimmer Reynolds number Re_{p} = ρv_{m}κa/μ is based on the characteristic shear around a swimmer; ρ and μ represent the fluid density and viscosity, respectively.
Results and discussion
Neutral swimmers
To evaluate the additional swimming lift \({{{{{{{{\mathcal{F}}}}}}}}}_{{{{{{\rm{swim}}}}}}}\), we find the disturbance field v created by the microswimmer using the continuity and the quasisteady Navier–Stokes equations in the comoving swimmer frame \(\{\tilde{x},\tilde{y},\tilde{z}\}\),
Here, \({{{{{{{\boldsymbol{f}}}}}}}}={{{{{{{\boldsymbol{v}}}}}}}}\cdot \tilde{\nabla }{{{{{{{{\boldsymbol{v}}}}}}}}}_{\infty }+{{{{{{{{\boldsymbol{v}}}}}}}}}_{\infty }\cdot \tilde{\nabla }{{{{{{{\boldsymbol{v}}}}}}}}+{{{{{{{\boldsymbol{v}}}}}}}}\cdot \tilde{\nabla }{{{{{{{\boldsymbol{v}}}}}}}}\) results from the convective acceleration with v_{∞} the Poiseuille flow field in the swimmer frame, and \({\mathsf{\sigma}}=p {\mathsf{I}} + 2{\mathsf{e}}\) is the Newtonian stress tensor of the disturbance field, where p and \({\mathsf{e}}\) represent pressure and the rateofstrain tensor, respectively. First, we consider a neutrally buoyant microswimmer that generates a source–dipole disturbance that, in leading order, resembles the flow field generated by some ciliated microswimmers^{46} and active droplets^{47,48}. Before evaluating the inertial swimming lift, we will perform an orderofmagnitude analysis to predict its scaling for small Re_{p}. This will provide a fundamental understanding of how weak inertia affects swimmer motion.
The classical analyses of Oseen^{49} and Saffman^{38} demonstrated that the magnitude of inertial perturbations increases with distance from the swimmer until an asymptotic “crossover radius”, beyond which the perturbations become singular. For the current swimmer system, the crossover radius is \({r}_{c} \sim {\,{{\mbox{Re}}}\,}_{p}^{1/2}\)^{38} that divides the entire domain in inner (regular) and outer (singular) regions. Substitution of r_{c} in the hydrodynamic signature of a neutral swimmer (~1/r^{3}) suggests that singular lift is inferior to regular lift, i.e., \({{{{{{{{\mathcal{F}}}}}}}}}_{{{\mbox{swim}}}}\propto {{{\mbox{Re}}}}_{p}\) (see Supplementary Note 2). This is in contrast to the Saffman lift of a forced particle, where the singular contribution \(\propto {\,{{\mbox{Re}}}\,}_{p}^{1/2}\) dominates. Hence, we implement a regular perturbation expansion, which turns the Navier–Stokes equations (2) into Stokes problems of zeroth (\(\tilde{\nabla }\cdot {{{{{{{{\boldsymbol{\sigma }}}}}}}}}_{0}=0\)) and first order (\(\tilde{\nabla }\cdot {{{{{{{{\boldsymbol{\sigma }}}}}}}}}_{1}={{{{{{{{\boldsymbol{f}}}}}}}}}_{0}\)), as detailed in the “Methods” section. Using the reciprocal theorem, we are able to calculate the swimming lift velocity from the firstorder problem^{37}
Here, the auxiliary velocity field v^{t} belongs to a forced particle moving along the xdirection^{37}. The convective acceleration f_{0} corresponds to the Stokes solution v_{0} of the microswimmer consisting of a source–dipole field, which we adopt from the squirmer model^{50,51,52}, and a stresslet generated by the shearing background flow with rateofstrain tensor e_{∞},
where \({\tilde{v}}_{s}={v}_{s}/({v}_{m}\kappa )\). We use here the farfield approximation to represent the flow field around microswimmers, which also strictly implies that the microswimmers should not come too close to the bounding walls. Using the corresponding f_{0} in Eq. 3 and e_{∞} for the Poiseuille flow, results in the inertial swimming lift velocity given in units of v_{s}: \({{{{{{{{\mathcal{F}}}}}}}}}_{{{\mbox{swim}}}}=(7/6){{{\mbox{Re}}}}_{p}\ x\cos \psi\). Thus, the total inertial lift to be used in Eq. 1 becomes
where we skip the factor 7/6 for simplicity. We also calculated the modification to zdirection swimmer velocity and ydirection rotational velocity. The former is \(5/18{{{{{{\rm{Re}}}}}}}_{p}x\sin \psi\) and latter is found to be identically zero at the present order of approximation.
The inertial lift profile of Eq. 5 causes a complex dynamics of the microswimmer governed by Eq. 1, which we now explore step by step. First of all, we identify two fixed points in the x−ψ plane at x = 0, with the microswimmer either swimming upstream along the centerline (ψ = 0) or downstream (ψ = ±π). A linear stability analysis reveals the following approximate eigenvalues for these fixed points:
Downstream swimming corresponds to a saddle fixed point (λ_{2}), while upstream swimming along the centerline (λ_{1}) is stable for weak flows (\({\bar{v}}_{m} \; < \; {\kappa }^{1}\)) and unstable otherwise. The inertial lift profile plotted in Fig. 2 for a moderate flow strength and for different swimmer orientations ψ, shows the passive lift velocity at ψ = π/2 with an unstable position in the center and the two inertial focusing points at ±x_{eq}. In the presence of the swimming lift, the centerline position is stabilized at ψ = 0. For strong flows (\({\bar{v}}_{m} \; > \; {\kappa }^{1}\)) the centerline position becomes unstable. However, the swimmer cannot focus on a nonzero x position, because due to the nonzero vorticity of the Poiseuille flow, it continuously tumbles while drifting downstream. In the state diagram presented in Fig. 3a, we vary swimmer size κ versus flow strength \({\bar{v}}_{m}\) and find these two limiting cases in the lower left and upper right region, respectively. Around the dashed stability line, \(\kappa ={\bar{v}}_{m}^{1}\), we observe that fluid inertia engenders rich dynamics, which we discuss now.
We first look at smaller microswimmers with κ ≲ 0.1 and move along the white dashed line in the state diagram with increasing \({\bar{v}}_{m}\). At \({\bar{v}}_{m} \; < \; 1\) the swimmer quickly reaches the centerline and moves upstream, while at moderate flow velocities \({\bar{v}}_{m} \; > \; 1\), it is drifted downstream by the Poiseuille flow and slowly relaxes towards the centerline (see Fig. 3b for a trajectory in x–z plane). On further increasing \({\bar{v}}_{m}\), a subcritical Hopf bifurcation occurs^{53}, where the stable centerline state and tumbling motion around x_{eq} coexist (see Fig. 3c). The schematic phase portrait in Fig. 3f shows how the stable fixed point and tumbling, a type of stable limit cycle, are separated by an unstable limit cycle. According to the bifurcation schematic next to the state diagram, the unstable limit cycle shrinks to zero and the fixed point becomes unstable. Hence, one observes a pure tumbling state (see Fig. 3d) with an amplitude that shrinks with increasing \({\bar{v}}_{m}\).
For larger microswimmers, we first concentrate on the white dashed line at κ > 0.23. When the fixed point becomes unstable at \(\kappa ={\bar{v}}_{m}^{1}\), a supercritical Hopf bifurcation occurs; the stable limit cycle, where the microswimmer performs a swinging motion about the centerline, gradually expands and then splits into two stable tumbling limit cycles. However, in the range 0.1 < κ < 0.23 the swinging limit cycle first enters a small region where it coexists with the tumbling state (multiple limit cycles)^{54} (see Fig. 3e). They are separated by an unstable limit cycle as the schematic phase portrait in Fig. 3g shows. As the flow rate further increases, the two inner limit cycles annihilate each other and the pure tumbling state remains.
In experiments, the time period of the swinging and tumbling states, as well as the drift velocity of the microswimmer along the channel axis, are measurable quantities. Figure 4a shows the time period T of the oscillatory states exhibited by the source–dipole swimmer for different rescaled swimmer sizes κ. At κ = 0.15, the two branches have an overlapping \({\bar{v}}_{m}\) region. Here, swinging and tumbling states coexist as indicated in Fig. 3a and the swimmer state depends on the initial condition. Dashed lines indicate sharp transitions between the two states. As already observed in Fig. 3a, larger swimmers enter the oscillatory states at lower flow rates. In the swinging state, we obtain a weak dependence of the time period on \({\bar{v}}_{m}\). Only close to the transition T rises with \({\bar{v}}_{m}\) and then, in the tumbling state, it decreases slowly. Figure 4b shows that the drift speed along the channel axis rises linearly with \({\bar{v}}_{m}\) with a slope one in the state of centerline swimming as expected. Also, in the swinging state (κ = 0.15 and 0.25) the slope is close to one. After the sharp drop to the tumbling state indicated by the dashed line, all three curves fall again nearly on top of each other. The slope of these straight lines is around 0.5, indicating that tumbling occurs outside of the centerline.
Pusher/pullertype swimmers
So far we have concentrated on microswimmers that generate a source–dipole flow field. Since the swimming lift crucially depends on the swimmer’s hydrodynamic signature and thus on its propulsion mechanism, we also expect a fundamentally distinct dynamics. Microswimmers that selfpropel by rotating or beating flagella, such as E. coli and Chlamydomonas, generate a force–dipole flow field at the leading order^{55,56}: \({{{{{{{{\boldsymbol{v}}}}}}}}}_{0}={{{{{{{\mathcal{P}}}}}}}}{\tilde{v}}_{s}{{{{{{{\boldsymbol{r}}}}}}}}\left[\frac{1}{{r}^{3}}+3\frac{{\left({{{{{{{\boldsymbol{r}}}}}}}}\cdot {{{{{{{\boldsymbol{p}}}}}}}}\right)}^{2}}{{r}^{5}}\right]\). Here \({{{{{{{\mathcal{P}}}}}}}}\) is the dimensionless force–dipole strength normalized by 8πμa^{2}v_{s}, which depends on the swimming mechanism^{56,57,58}. Earlier studies on E. coli^{56,58,59} and Chlamydomonas^{60} suggest that \( {{{{{{{\mathcal{P}}}}}}}}\) varies roughly between 0.04 and 0.3.
The slow decay of the force–dipole field (∼1/r^{2}) suggests that the swimming lift obtained from singular perturbation now also is linear in \({{{{{{{{\rm{Re}}}}}}}}}_{p}\) as the lift evaluated within regular perturbation theory (see Supplementary Note 2). Thus, similar to the case of passive inertial lift^{37,45}, one can use either regular perturbation theory or matched asymptotic expansions to calculate the swimming lift in leading order of Re_{p}. A comparison of results from singular perturbation approach of Asmolov^{45} and results using regular perturbation theory^{37}, which strictly requires a channel Reynolds number Re_{c} ≪ 1, shows a close match of the liftforce profiles at Re_{c} = 15 (see Fig. 8 in^{45}). This suggests a smooth transition between the two approaches. Therefore, we continue with the approach used for neutral microswimmers and employ regular perturbation theory in combination with the reciprocal theorem in Eq. 3, as detailed in the Supplementary Note 3. The slower decay of the force–dipole field poses an additional challenge: one has to account for the finite integration domain of the microchannel, otherwise the lift would diverge logarithmically. Thus, we correct the zerothorder flow field v_{0} by including wall terms, which we obtain from the method of reflections. Our investigation shows that the angular dependence of the force–dipole swimming lift, \({{{{{{{{\mathcal{F}}}}}}}}}_{{{{\rm swim}}}}\propto \sin 2\psi\), differs from that of the source dipole. Fitting the numerical results for \({{{{{{{{\mathcal{F}}}}}}}}}_{{{{\rm swim}}}}\), we can approximate the total inertial lift velocity in units of v_{s} by
In Fig. 5a, the liftvelocity profile for a force dipole shows a clear difference to the profile in Fig. 2. Compared to the passive lift (ψ = 0, π/2), the profile either shifts up or down for varying ψ. Thus, depending on \({{{{{{{\mathcal{P}}}}}}}}\) and \({\bar{v}}_{m}\kappa\), the fixed point (\({{{{{{{\mathcal{F}}}}}}}}=0\)) in one channel half can vanish completely. We note that the profiles of force dipoles with the same strength but opposite signs follow from each other by adding π/2 to ψ.
Although the fixed points are identical to the previous case, the stability analysis with the eigenvalues
reveals that upstream swimming (ψ = 0) is always unstable, as suggested by the lift velocity. Through an unstable spiral, the trajectories enter a stable limit cycle, which for lower flow rates corresponds to a swinging motion about the centerline. The swimmer effectively swims upstream for \({\bar{v}}_{m} \; < \; 1\) as depicted in Fig. 5b, while it moves downstream for \({\bar{v}}_{m} \; > \; 1\), similar to the black trajectory in Fig. 3e.
Hydrodynamic wall interactions of the force–dipole field add weak modifications of the order of κ^{2} and κ^{3} to the evolution equations of position and orientation, respectively^{16,61,62,63}. Therefore, they mainly influence the dynamics when the flow rates are weak, i.e., for upstream swinging motion. Figure 5c shows a pusher approaching the wall as the hydrodynamic interactions are attractive^{56}. Since the strong vorticity near the walls reorients the swimmer, it will ultimately oscillate between both walls. In contrast, pullers are hydrodynamically repelled from walls^{16} and hence swim in a swinging limit cycle with an amplitude smaller compared to Fig. 5b. Finally, Fig. 5d shows that downstream swinging in stronger flows is hardly affected. For neutral swimmers, the wall effects are weaker by an additional factor of κ^{62,63,64,65} and we verified that they do not have a significant effect on the dynamics.
In Fig. 6a, b we show the resulting state diagrams for a puller and pusher, respectively. The diagrams are clearly disparate to that of a neutral swimmer (Fig. 3a). For flow rates \({\bar{v}}_{m}\) below one, larger pullers swim upstream along the centerline (region I) since hydrodynamic wall interactions dominate the inertial lift and push pullers to the center. Otherwise, pushers and pullers show upstream swinging (region II) and for \({\bar{v}}_{m} \; > \; 1\) downstream swinging (region III). At even larger \({\bar{v}}_{m}\) they transition into the tumbling state (region IV). For pushers, this transition occurs at larger \({\bar{v}}_{m}\) due to the hydrodynamic wall interactions. Finally, in Supplementary Note 4, we provide the time period of the oscillatory states and the axial drift speed as a function of \({\bar{v}}_{m}\) for pusher and puller with \({{{{{{{\mathcal{P}}}}}}}}=\pm 0.3\). Note that at zero \({{{{{{{{\rm{Re}}}}}}}}}_{p}\) with hydrodynamic wall interactions included, the swimmer states realized at a specific flow speed \({\bar{v}}_{m}\) differ from that in Fig. 6. For sufficiently large \({\bar{v}}_{m}\) pushers always show stable swinging around the center line, while pullers either move along the centerline or tumble close to the wall depending on their initial conditions^{16}.
Conclusions
In summary, we have studied how swimming at low fluid inertia in Poiseuille flow adds a swimming lift to the known passive inertial lift velocity. We have concentrated on the generic source–dipole and force–dipole microswimmers and showed that their swimming lift velocities depend differently on the lateral swimmer position and orientation. This gives rise to the emergence of complex dynamics including bistable states, where tumbling coexists with stable centerline swimming or swinging. The Reynolds number determines the overall dynamics relative to the flow speed. Deriving a nonlinear oscillator equation for ψ in full analogy to ref. ^{16}, reveals a reduced relaxation time \(\propto {\,{{\mbox{Re}}}\,}_{p}^{1}\) towards the stationary states.
Recent experimental studies^{14,20,57,66} operate within the parameter ranges of microswimmer size, 10−200 μm, and channel width, 100−500 μm. Thus for the maximum flow speed v_{m} ~ 1 mm/s, Re_{p} ranges from 0.001 to 0.1 and the time taken to attain steady states, w/(v_{s}Re_{p}), roughly varies from 10 to 10^{3} s for narrow microchannels. These estimates suggest that effects of fluid inertia are observable for large microswimmers (≳50 μm) and moderate to strong flows. For instance, Volvox carteri will be of interest as it has a radius of ~200 μm and swims with ~200 μm/s^{57}. Additionally, artificial microswimmers with tunable high speeds larger than 200 μm/s exist^{9,30,31}. All this should offer the possibility to experimentally observe the dynamic features reported here at small but nonnegligible fluid inertia depending on the hydrodynamic signature of a microswimmer. Furthermore, the current insights may encourage investigations in marine ecosystem, where recent literature^{67,68} suggests that inertial lift can drive planktons out of the turbulent eddies and induce plankton blooms.
Our work extends the research on microswimmers by bringing the role of fluid inertia into focus, which has not been looked at so far. For passive particles, this has spawned the field of inertial microfluidics^{34,35}. We envisage a similar development for microswimmers, which offers numerous aspects to look at. For example, elongated microswimmers perform Jeffery orbits^{69}, which also influence their dynamics in a Poiseuille flow^{17}. Adding them to the current work is not straightforward since fluid inertia induces an orientational drift^{70}. The hydrodynamics of the swimming motion might also add an active component to the Jeffery orbits. We also stress that thermal or biological noise acting on the swimmer orientation will disturb the motion in the limit cycles and also induce transitions between coexisting states but not influence the principal behavior outlined in this article. Finally, we note that incorporation of higherorder multipoles can provide rich dynamical behavior near the walls in the presence of inertia^{71}.
Methods
Problem formulation
To evaluate the lift velocities, we work in reference frame that translates with the swimmer \((\tilde{x},\tilde{y},\tilde{z})\). Supplementary Note 1 and Supplementary Fig. 1 show the nondimensional notation, where s = d/2w and s/κ = d/a denotes the dimensionless distance from the bottom wall in units of particle radius a. For simplicity, we temporarily drop the tilde \(\tilde{}\) notation. We divide the full velocity field v_{actual} = v + v^{∞} into the background flow field v^{∞} and the disturbance field v, and then obtain the equations governing the disturbance field from the continuity and Navier–Stokes equations:
The hydrodynamic equations follow a quasisteady description as the time scale associated with swimming (a/v_{s} ~ 1s) is much larger than the characteristic vortex diffusion time (a^{2}/ν ~ 10^{−4}s). The above equations have been nondimensionalized using a, κv_{m}, μκv_{m}/a as the characteristic scales for length, velocity, and pressure, respectively. The definitions of these dimensional parameters a (particle size), κ = a/2w, and v_{m} (maximum flow velocity) are consistent with the article. In our case, v^{∞} is the undisturbed Poiseuille flow velocity in the frame of reference translating/comoving with the particle
where U_{p} is the total velocity of the swimmer, i.e., swimming velocity v_{s} plus advection due to the Poiseuille flow and the lift velocities. The constants α, β, and γ are:
where β and γ represent the shear and curvature of the background flow, respectively.
The boundary conditions of the disturbance field are:
Here, the walls are located at x = −s/κ and x = (1 − s)/κ, and v_{θ} represents the prescribed tangential surface velocity of the spherical microswimmer.
We find the inertial lift or migration velocities at O(Re_{p}) using a regular perturbation expansion. For small values of Re_{p}, the disturbance field variables are expanded as:
Here, ξ is a generic field variable that represents velocity (v), pressure (p), translational (U_{p}), and angular velocity (Ω_{p}). We substitute (11) in the equations governing the disturbance field (7), and obtain the problem at O(1) (i.e. Stokes problem) as
and at O(Re_{p}) as:
In (12), \({{{{{{{{\boldsymbol{v}}}}}}}}}_{0}^{\infty }=\left(\alpha +\beta x+\gamma {x}^{2}\right){{{{{{{{\boldsymbol{e}}}}}}}}}_{z}{{{{{{{{\boldsymbol{U}}}}}}}}}_{p0}\).
Ho and Leal^{37}, in their seminal work, used the reciprocal theorem to derive a volume integral expression for the migration velocity associated with the O(Re_{p}) Eq. 13:
The auxiliary or test field (v^{t}, p^{t}) is associated with a sphere moving in the positive xdirection (towards the upper wall) with unit velocity in a quiescent fluid:
The reciprocal theorem makes it relatively easy to find lift velocities at O(Re_{p}), as we can solve the creeping flow problem (12) using wellestablished methods^{61,72} and directly substitute its solution in (14). In other words, we do not need to solve the O(Re_{p}) problem (13) to obtain the O(Re_{p}) lift.
Lift velocity
We now use the reciprocal theorem integral (14) for evaluating the swimming lift of a source–dipole swimmer. We explicitly choose the axisymmetric neutral squirmer, which has the surface velocity field \({{{{{{{{\boldsymbol{v}}}}}}}}}_{\theta }={B}_{1}\sin \theta \, {{{{{{{{\boldsymbol{e}}}}}}}}}_{\theta }\), where θ is the polar angle and e_{θ} the corresponding base vector. The swimming velocity is directly related to this squirmer coefficient: v_{s} = 2B_{1}/3^{52,73}. The solution to the O(1) Stokes problem (12) in the Poiseuille background flow is obtained as (4). The first part in the above expression is the swimmergenerated source–dipole, and the second part is the stresslet and the higherorder octupole correction due to the local shear flow^{74}. e^{∞} is the rate of strain tensor for the background flow, which amounts to e^{∞} = (∇ v^{∞} + ∇ v^{∞†})/2 (here † represents transpose). For the case of a small neutral swimmer, we can neglect the curvature of the background flow, that would bring in a term proportional to γ ~ O(κ) in Eq. 4. Additionally, the previous work^{41} suggests that the hydrodynamic multipoles arising from the curvature have a negligible effect on the lift of a source–dipole swimmer.
It remains to calculate e^{∞} for the Poiseuille flow of Eq. 8 in zeroth order of Re_{p}. The total velocity of the forcefree swimmer in the Stokes regime is \({{{{{{{{\boldsymbol{U}}}}}}}}}_{p0}={\tilde{{{{{{{{\boldsymbol{v}}}}}}}}}}_{s}+\alpha {{{{{{{{\boldsymbol{e}}}}}}}}}_{z}\), where the second part is obtained by the fact that the swimmer at x = 0 is advected by the flow. To complete the expression of \({{{{{{{{\boldsymbol{v}}}}}}}}}_{0}^{\infty }\), we substitute U_{p 0} in (8), and obtain:
which gives \({[{{\mathsf{e}}}^{\infty }]}_{xz}={[{{\mathsf{e}}}^{\infty }]}_{zx}=\beta /2\).
Now, we evaluate the lift integral arising from the active nature of the swimmer (in addition to the passive lift). Since the source–dipole field of the neutral swimmer decays quickly away from the swimmer (~1/r^{3}), we can neglect the wall corrections in the lift velocity integral (14). In the context of electrophoresis, Choudhary et al.^{41} showed that accounting for such wall corrections hardly affects the lift and only becomes noticeable very near the walls (see^{41}, p. 877). That work also showed that higherorder effects due to curvature are negligible in the case of a source dipole^{41}, p. 879. Hence, we evaluate the lift velocity integral by substituting (16) and (4) in (14). Integrating over the infinite space, we obtain the swimming lift velocity in units of v_{s} as
expressed in the comoving frame of the swimmer. See Supplementary Software 1 for the details of evaluation. In Eq. 17 the contribution from the disturbancedisturbance interaction term of the integral (v_{0} ⋅ ∇ v_{0}) is \((49/360){{{\mbox{Re}}}}_{p}\beta \cos \psi\) and the contribution arising from the disturbanceflow interaction term (\({{{{{{{{\boldsymbol{v}}}}}}}}}_{0}^{\infty }\cdot \nabla {{{{{{{{\boldsymbol{v}}}}}}}}}_{0}+{{{{{{{{\boldsymbol{v}}}}}}}}}_{0}\cdot \nabla {{{{{{{{\boldsymbol{v}}}}}}}}}_{0}^{\infty }\)) is \((7/45){{{\mbox{Re}}}}_{p}\beta \cos \psi\). Note, to calculate the passive inertial lift, one must account for wall interactions and the curvature γ in \({{{{{{{{\boldsymbol{v}}}}}}}}}_{0}^{\infty }\).
For a force–dipole swimmer, we obtain the lift velocity in the units of v_{s} as:
The evaluation is detailed in Supplementary Note 3. Supplementary Fig. 2 shows fits for the swimminglift profile of a pusher.
The final expressions of the swimming and passive lift velocities in the channel frame of reference can be obtained by a transformation of particlewall distance s to the channel x coordinate: s = (1 + x)/2. Using the definition of β from Eq. 9, Eq. 17 is obtained as \((7/6){{{\mbox{Re}}}}_{p}\ x\cos \psi\). Similarly, we obtain the swimming lift velocity of a force–dipole swimmer as \(0.5\ {{{\mbox{Re}}}}_{p}\ {{{{{{{\mathcal{P}}}}}}}}\ \left(12{x}^{2}\right)\sin 2\psi\).
Following the prior works on inertial migration^{37,41,75}, we reproduce the lift force profile for a passive neutrally buoyant particle suspended in Poiseuille flow, which needs the numerical evaluation of integrals. Supplementary Fig. 3 shows the comparison of our reproduced results for passive lift with Vasseur and Cox^{75} and also illustrates the fitted function used in the article:
Data availability
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
Code availability
Supplementary Software 1 files include a Mathematica code for the results.
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Support from the Alexander von Humboldt Foundation is gratefully acknowledged. We also acknowledge financial support from the Collaborative Research Center 910 funded by the Deutsche Forschungsgemeinschaft.
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A.C., F.R., and H.S. planned the research. A.C., F.R., and S.P. performed the calculations. A.C. and H.S. wrote the manuscript.
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Choudhary, A., Paul, S., Rühle, F. et al. How inertial lift affects the dynamics of a microswimmer in Poiseuille flow. Commun Phys 5, 14 (2022). https://doi.org/10.1038/s4200502100794y
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DOI: https://doi.org/10.1038/s4200502100794y
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