On the cross-streamline lift of microswimmers in viscoelastic flows

The current work studies the dynamics of a microswimmer in pressure-driven flow of a weakly viscoelastic fluid. Employing the second-order fluid model, we show that the 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 swimmers and show that they depend on the hydrodynamic signature associated with the swimming mechanism. We find that for neutral swimmers focusing towards the centerline is accelerated by two orders of magnitude, while for force-dipole swimmers no net modification in cross-streamline migration occurs.

cles, 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 characteristic of the self-propulsion mechanism. Since this component is absent in the recently proposed models [32], 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 viscoelastic flows [43,44]. Figure 1 shows a spherical swimmer of radius a at position r that self-propels with velocity v s = v s p in a twodimensional pressure-driven flow where v m is the maximum flow velocity and w is the half channel width. The flow profile of the second-order fluid is identical to Poiseuille flow but the pressure field varies in x−direction [39]. In the absence of noise, the swimmer's dynamics is governed bẏ r = p +v f + F(x, p) e x ,ṗ = 1 2 (∇ ×v f ) × p, (1) where the velocities are non-dimensionalized by swimming speed v s , lengths by w, and time by w/v s . F denotes 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 normal stresses also modify the particle 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,ỹ,z} as:∇ · V = 0,∇ · T = 0 (2) in order to calculate the viscoelastic lift velocity. Length, velocity, and pressure in (2) are non-dimensionalized by a, κv m , and µκv m /a, respectively. Here, κ is the particle to channel width ratio (a/2w) and µ is the fluid viscosity. In the above equation, T is the total stress tensor of a second-order fluid and thus has the form [44]: Here, E denotes the rate of strain tensor and Wi = (Ψ 1 + Ψ 2 )G/µ is the shear based Weissenberg number, where µ is the viscosity, G = v m /2w characterizes the shear rate in the background flow, and Ψ 1 , Ψ 2 represent the dimensional steady-shear normal stress coefficients that are measured experimentally [44].
We now focus on determining the lift velocity of a microswimmer 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 full velocity field (V = v ∞ + v) into background flow field v ∞ and disturbance field v. Substituting this in the governing equations (2) yields: where s is the polymeric stress tensor associated with the disturbance flow field (elaborated in the electronic supplementary material ESI). Assuming weak viscoelasticity, we perform a perturbation expansion in Wi and divide Eq. (3) into two problems: the Stokes equation for the zeroth order of the disturbance field and −∇p 1 +∇ 2 v 1 = −∇ · s 0 at first order. Following earlier works on viscoelastic lift [39,41], we use the reciprocal theorem to attain the lift velocity from the first-order problem Here, u t 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 s 0 corresponds to the Stokes solution v 0 of the microswimmer consisting of (i) a source-dipole field, which we attain from the squirmer model [48][49][50], v swim 0 = vsp 2r 3 · 3rr r 2 − I , and (ii) the passive disturbance field v passive 0 , which is led by the stresslet; higher order terms are obtained from Lamb's general solution [51].
Using the corresponding s 0 in Eq. (4), results in the viscoelastic lift velocity given in units of v s : (5) The first component in Eq. (5) is the passive lift F passive [39]. By fixing δ to a widely-used value of −0.5 (i.e. Ψ 2 = 0), we observe that F passive focuses the particle towards the centerline. The second component is the swimming lift F swim that arises due to the source-dipole disturbance created by the neutral swimmer. We note two striking features of F swim : the dependence on swimmer orientation through cos ψ and that its magnitude is larger by a factor κ −2 compared to the first term 1 .
Now, we substitute F into the dynamic equations (1) and examine the effect of F swim on the microswimmer dynamics. We find two fixed points in the x − ψ plane at x = 0, with the microswimmer oriented upstream (ψ = 0) or downstream (ψ = ±π). A linear stability analysis provides the following eigenvalues for these fixed points: For a typical value of δ = −0.5 and weak viscoelasticity limit (Wi 1), the downstream swimming corresponds to a saddle fixed point (λ 2 ), while the upstream swimming along x = 0 corresponds to a stable fixed point (λ 1 ). For δ = −0.5, the sign of the real part of λ 1 shows that both swimming and passive lift components stabilize the upstream swimming. However, the strong swimming lift can help the neutral microswimmer attain centerline  equilibrium more rapidly by a relaxation factor of κ −2 , as also shown in the trajectories of Fig.2 that are evaluated by substituting (5) in (1). Now, we shift our attention from neutral squirmers to flagellated microorganisms, such as E. coli and Chlamydomonas, that generate a force-dipole field at the leading order [35,52]: v 0 = Pr −1 Here P is the force-dipole strength in units of 8πµa 2 v s , which depends on the swimming mechanism [35,53,54]. Earlier studies on E. coli [54,55] and Chlamydomonas [56] suggest that |P| varies roughly between 0.04 -0.3. Following the procedure outlined for a source-dipole swimmer, we obtain the swimming lift velocity of the force-dipole swimmer as and find it to depend on the constant curvature in Poiseuille flow, as detailed in ESI. Although this lift is larger than F passive by a factor κ −1 , there is no net lateral motion because the pure angular dependence cancels out on average as the particle tumbles continuously. The trajectories in Fig.3 (a) and (b) show the dynamics of a force-dipole swimmer; the focusing along the centerline is purely due to F passive .
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 κ 2 and κ 3 to the evolution equations of position and orientation, respectively [10,57]: Upstream trajectories in Fig. 3(c) closely resemble the behavior reported previously for pure Newtonian fluids [10]. We observe that the hydrodynamic wall attraction of pushers [35] overcomes F passive 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. Pullers are repelled from walls [10] and, therefore, rapidly focus on the centerline. In contrast, for downstream drifting at large flow rates [ Fig. 3(d)], F passive dominates over the hydrodynamic wall interactions and all trajectories tend towards the centerline. We note that for source-dipole swimmers the hydrodynamic wall interactions are an order κ weaker, and therefore do not alter the trajectories qualitatively.
In conclusion, the current study analyzes microswimmers in weakly viscoelastic pressure-driven flows. For neutral and pusher/puller microswimmers, we derive an additional swimming lift velocity depending on the swimmer's hydrodynamic signature that adds to the passive viscoelastic lift [32,39,58,59]. 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 [32]. The current work shows that the swimming lift accelerates the centerline focusing. 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 follows the behavior in Newtonian fluids [10]: attraction of pushers towards the channel walls and repulsion of pullers. The downstream drifting along the centerline qualitatively remains the same as that of a passive particle.
These results suggest that normal stresses in viscoelastic fluids generated by the microswimmer's flow field can accelerate the centerline focusing. However, this strongly depends on the hydrodynamic signature of the microswimmer. 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 − 50mm and is quite realistic for microfluidic channels. Thereby, this work contributes to the understanding of swimming in more realistic biological fluids. Furthermore, the current work offers several new directions to explore. For instance, elongated microswimmers perform Jeffery orbits in sheared Newtonian fluids [60]. In viscoelastic fluids the flow disturbances from swimming will alter the orientation evolution of these orbits and hence the swimmer dynamics [33]. Impact of shear-thinning fluids is also an interesting outlook, which can be achieved by the use of more detailed rheological models [44]. Support from Alexander von Humboldt fellowship is gratefully acknowledged. Supplemental material for "On the cross-streamline lift of microswimmers in viscoelastic flows"

I. PROBLEM FORMULATION
The schematic in Fig. S1 (a) shows a neutrally buoyant spherical microswimmer suspended in pressure-driven flow of a polymeric fluid between two walls. In order to derive the expressions for the lift velocities, we work in a reference frame that translates with the swimmer (x,ỹ,z). For simplicity, we temporarily drop the tildeñ otation. Fig. S1 (b) shows the non-dimensional description, where s = d/2w and s/κ = d/a is the distance from the bottom wall normalized by the particle radius a. Schematic with all lengths normalized by particle radius a. The tilde notation of the coordinates is shown to be dropped for brevity.
We split the actual velocity field into the background flow field v ∞ and the disturbance field v (i.e. v actual = v + v ∞ ), and substitute in Eq. (2) of the article. The inertia-less hydrodynamics of the disturbance field is governed by the continuity and momentum equations in the co-moving swimmer frame {x,ỹ,z} as 2 : (S1) Here, e is the rate of strain tensor for the disturbance flow (∇v + ∇v † )/2, whereas w, ∆ e and ∆ w are the different parts of the perturbation s of the polymeric tensor due to the disturbance flow field v: where e ∞ is the rate of strain tensor for the undisturbed flow, ∆ e is the lower convected derivative of e (also known 2 We follow a quasi-steady description because the time scales associated with cross-stream motions both due to swimming (a/vs ∼ 1s) and viscoelastic lift are much larger than the characteristic vorticity diffusion time (a 2 /ν ∼ 10 −4 s).
as the Rivlin-Eriksen tensor), w is the 'interaction tensor' (arising from the interaction between background flow and disturbance field), and ∆ w is its lower convected derivative.
The above equations are non-dimensionalized 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 communication article. In our case, v ∞ is the undisturbed Poiseuille flow velocity in the frame of reference translating 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 flow 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.
In (S7), v ∞ 0 = α + βx + γx 2 e z − U p 0 . Ho and Leal [39], in their seminal work, used the reciprocal theorem to derive a volume integral expression for the migration velocity associated with the O(Wi) equations (S8): The auxiliary or test field (v t , p t ) is associated with a sphere moving in the positive x-direction (towards the upper wall) with unit velocity in a quiescent fluid: v t (r) = 3 4 e x + xr r 2 The reciprocal theorem makes it relatively easy to find lift velocities at O(Wi), as we can solve the creeping flow problem (S7) using well-established methods [51,61] and directly substitute its solution in (S9). In other words, we do not need to solve the O(Wi) problem (S8) to obtain the O(Wi) lift.