From the bacterial and alga blooms of the biological world to the engineered nanoparticles of liquid crystals, the motion and collective dynamics of microswimmers have been subjects of much recent research. A focus has been on characterising the complex phenomenology that emerges when dense agglomerations of microswimmers or active particles form in a viscous fluid, in what is characterised as an active fluid (Marchetti et al. 2013;Saintillan 2018). These suspensions of active particles have been found experimentally to impart a variety of fluid transport properties, such as enhanced diffusion, clustering and complex vortex structures (Copeland & Weibel 2009;Wensink et al. 2012;Elgeti et al. 2015). These findings are further mediated by the stability associated with the swimming behaviour of individual particles and their interaction with many other particles, giving rise to phenomena such as attraction, repulsion and entrainment (Ishikawa et al. 2006;Alexander et al. 2008;Götze & Gompper 2010). The physics pertaining to the dynamics of microswimmers themselves have proven to be of interest for the development of artificial microswimmers (Kay et al. 2007;Lavergne et al. 2019), with potential applications for drug delivery (Malmsten 2006;Davis et al. 2008;Nelson et al. 2010;Yang et al. 2012).
Many of these microorganisms swim in a fluid environment that is mediated by a low-Reynolds-number regime (Re 1), where the dominance of viscous forces over inertial forces is due to the micrometre scale of the organism. This allows models to neglect the inertial component of the (nonlinear) Navier-Stokes equations when modelling the swimming motion of these microorganisms and instead use the (linear) Stokes equations, allowing models to take advantage of the superposition of point force solutions, i.e. Stokelets and their regularised counterparts, to characterise the forces acting on the microswimmer (Cortez et al. 2005). High-fidelity models of individual microorganisms use collections of point forces numbering of the order of O( 103 ) and O(10 4 ) to allow for highly detailed models of the motion of flagella and bodies (Cortez et al. 2005;Lee et al. 2018;LaGrone et al. 2019;Park et al. 2019Park et al. , 2022)). On the other hand, the interaction of high densities of microorganisms motivate a desire and necessity to reduce the computational costs associated with high-fidelity models. This has led to much work dedicated toward developing reduced models that impart the relevant details in a reduced description to allow one to characterise the many-swimmer interactions (Lushi & Peskin 2013;Delmotte et al. 2015;Wioland et al. 2016;Zöttl & Stark 2016;Stenhammar et al. 2017).
Reduced models that motivate the present work include the two-bead model (Hernandez-Ortiz et al. 2005, 2007, 2009), in which the propulsion is provided by a force on one of the beads. This model was designed to produce a force-dipole farfield flow, matching the velocity generated by swimming particles to leading order. A more recent popular modelling framework is to reduce the description of the organism's hydrodynamics to a pair, or collection, of point forces (Lauga & Powers 2009) in such a manner that results in a net-zero balance. When two point forces are placed in sequence, a force dipole is formed and can be used to model the swimmers' far-field hydrodynamics, with common categorisations of different swimmers as pullers and pushers that depend on the position of the flagella relative to the body and swimming direction. This forcedipole framework can be further extended with the inclusion of torques as rotlet dipoles to account for the torque-balanced, rotational components of the swimmers' hydrodynamics. This work was furthered in Spagnolie & Lauga (2012) by applying a multipole description of the microswimmer, with a linear combination of the fundamental solutions to describe a Stokeslet dipole, a source dipole, a Stokeslet quadrupole and a rotlet dipole. Another approach using the method of regularised Stokeslets allows for the Stokeslets and rotlets to be regularised in such a manner that the regularisation parameter, and the strengths and locations of the elements, can be tuned to fit an organism's cycle-averaged flow field (Ishimoto et al. 2020).
In this study, we propose a new modelling framework, from hereafter referred to as the force-doublet framework, as a reduction of the force-dipole framework. We derive the force-doublet model starting with two regularised Stokeslets in sequence with different size regularisation parameters. Taking the limit as the two Stokeslets approach each other and the regularisations approach, a common value leads to a closed, single-point, expression for a swimmer's motion. The asymmetry needed for propulsion is provided by the different regularisation parameters before the limit. This process reduces the twopoint force-dipole expression into a single point with an orientation vector. The evolution equation for the orientation of the force doublet is based on the local fluid dynamics using Jeffery's equation (Jeffery 1922). The parameters in the resulting force-doublet expression allows one to control for length scale, swimming speed and asymmetries in the cycleaveraged stroke. Furthermore, the framework allows for easy inclusion of other forces in our system.
The goal of this study is to introduce the force-doublet framework and demonstrate its utility in the broader context of the field. The study is structured as follows. In § 2, the force-doublet model is derived for the method of regularised Stokeslets from a force-dipole description of a cycle-averaged swimmer. For § 3, we explore how varying parameters of the force doublet affect the resulting flow field. We then examine the stability of the model for two particles swimming parallel to one another. We show the modular nature of this approach by extending the force-doublet framework to include rotlet doubles to model rotational flows associated with microswimmers. Lastly, we incorporate wall effects in the framework using the method of images. In § 4 we summarise the results and discuss connections to other related models.
In the following section, we describe the background and methods used to derive the forcedoublet framework.
In the following framework, the inertial forces associated with the microswimmer are considered negligible and so the Stokes equations are used to describe an immersed swimmer in a fluid
where u is the velocity of the fluid, p is pressure, μ is the dynamic viscosity, F(x) = fδ(xx 0 ) is a force density located at x 0 associated with the microswimmer and δ is a Dirac delta distrirution. Exact solutions to the Stokes equations, also known as Stokeslets, can be derived from a set of Green's functions, G(r ) and B(r
However, the resulting exact solution is singular and so one approach, known as the method of regularised Stokeslets, replaces δ with a smooth, regularised approximation of the Dirac delta distribution, φ , where is a small regularisation parameter (Cortez 2001;Cortez et al. 2005), from which regularised functions G and B can be derived. In this study, we elect to demonstrate this modelling framework with the commonly used approximation
although general solutions can be derived from any choice of a regularising function φ . Solutions to (2.1) and (2.2) correspond to a forcing terms F(x) = fφ
Figure 1. Schematic of the set-up for the derivation of the force-doublet model. Two regularised forces of equal magnitude in opposite directions have different size regularisation parameters. A limit as ℓ → 0 is taken in such a way that at x h , F(x h ) = f(ℓ)φ h , with f(ℓ) = (q 0 /ℓ)β 0 .
(xx 0 ) will be denoted by u(x) = S(x; x 0 , )f. It can be derived with the help of the functions G (r ) and B (r ) defined as solutions of G = φ (r ) and B = G . Taking the divergence of (2.1) and using the incompressibility condition, (2.2), we arrive at the equation p = (f • ∇)φ (r ), leading to the solution p = (f • ∇)G . Therefore the fluid velocity satisfies μ u = (f • ∇)∇G (r )fφ (r ) and its solution is μu = (f • ∇) ∇ B (r ) -fG . For the regularisation function used here, this solution is given by
where R = √ r 2 + 2 is used to simplify notation.
We are motivated by the conceptual model in Hernandez-Ortiz et al. (2005, 2009) and Underhill et al. (2008) where the cell body and the flagellum are represented by differentsized spheres that instantaneously exert equal but opposite forces along the axis connecting their centres.
We consider two equal and opposite forces exerted at two different points separated by a distance ℓ in the direction of a unit vector β 0 . In this case, the force, F(x h ), depends on the separation between the points. Referring to figure 1, the forces are applied at x h and x t , and we denote by x 0 the midpoint between them. The model will result from the limit as the points x h and x t approach x 0 . Here, β 0 is the unit vector from x t to x h . We make two assumptions about the forces. The first one is that the regularisation parameters, h and t , are different in order to break the symmetry of the system so that the velocity at x 0 is nonzero and in the direction of β 0 . Specifically, we set t = -Cℓ/2 and h = + Cℓ/2 for some constants and C. The second assumption is that f(ℓ) is inversely proportional to ℓ, denoted by f(ℓ) = (q 0 /ℓ)β 0 for some constant q 0 . This assumption is needed to prevent the two forces from simply cancelling each other.
The fluid velocity they generate at an arbitrary point x is S(x; x h , h )q 0 β 0 /ℓ -S(x; x t , t )q 0 β 0 /ℓ.
(2.5)
We note that S(x; x h , ) depends on the difference xx h , which is given by x -
Therefore, the fluid velocity at x can also be written as
With these assumptions, we take the limit as ℓ → 0 and find that the fluid velocity is
(2.7)
For the regularisation function φ (r ) = 15 4 (8π R 7 ) -1 , this is
(2.8)
The first term on the right side is the directional derivative of the Stokeslet in the direction of β 0 , and is therefore a stresslet. It turns out that the factor in parentheses in the last term is the regularised potential dipole
which is derived from the regularising function φ (r ) = 3 2 (4π R 5 ) -1 ) (Cortez & Varela 2015). Defining b 0 = Cq 0 , the fluid velocity at x is
(2.10) This is the proposed force-doublet model. We make the following comments.
(i) The self-propulsion of the particle is proportional to b 0 and is entirely provided by the potential dipole in the last term in the expression. Since the elements are regularised, the velocity is finite everywhere; in particular, it can be evaluated at x = x 0 , yielding
(2.11)
For the particle to propel itself in the direction of β 0 , we require b 0 < 0. (ii) The first term in the velocity expression is a stresslet, and it is symmetric about the point x 0 so that replacing the factor (xx 0 ) with -(xx 0 ) results in a change in the sign of the stresslet term (see figures 2 and 3 following these comments). (iii) A particle located at x 0 with orientation vector β 0 experiences a velocity u(x 0 ) due to itself and to other particles or obstacles nearby. The orientation vector is rotated by this flow according to Jeffery (1922), Saintillan (2018) and Tseng ( 2022)
where u(x 0 ) is the fluid velocity at x 0 due to all elements, and D is the rate-ofdeformation tensor with components
The parameter ξ is a factor that depends on the shape of the particle. For a spherical particle, ξ = 0, while for a slender particle, ξ = 1. As this article is concerned with reduced models of flagellated organisms, we consider the case of ξ = 1, which simplifies the orientation vector equation to
(2.13) (iv) The velocity field along the axis of the organism,
The motion of Dipole -1.5 -1.5 -1.0 -1.0 -0.5 -0.5 0 0.5 1.0 1.5 -1.5 -1.0 -0.5 0 0.5 1.0 1.5 -1.5 -1.0 -0.5 0 0.5 1.0 1.5 0 0.5 1.0 1.5 -1.5 -1.0 -0.5 0 0.5 1.0 1.5 -1.5 -1.0 -0.5 0 0.5 1.0 1.5 + = Stresslet Swimmer Figure 3. The fluid velocity produced by the model 'pusher' particle is the sum of a potential dipole (left) and a stresslet (middle).
The arrow shows the swimming direction and the black dot in the centre of the arrow is the particle location. The green dot behind the particle is the stagnation point. The parameters used were = 0.4, b 0 = -1, q 0 = 4, so that the stagnation point is located at b 0 /q 0 = -1/4.
an organism classified as a 'pusher' will typically generate a stagnation point downstream from the cell body, which corresponds to choosing q 0 > 0 in the model. On the other hand, a puller will typically generate a flow with a stagnation point in front of the cell body. This case corresponds to q 0 < 0.
In this study, we used the regularisation function that is most commonly used with the method of regularised Stokeslets (Cortez et al. 2005). The derivation of the reduced model presented here can be done with a different regularisation function, if desired. In general, the flow at an arbitrary location x may involve contributions from N d force doublets with parameters (q j , b j , β j ) located at x j , and possibly from N f forces f k located at y k generated by solid boundaries, steric forces or other mechanisms. The total fluid velocity at x due to these elements is the superposition of all of the elements.
It is convenient to write the model equations in dimensionless form in order to identify dimensionless groups and reduce the number of parameters. A characteristic length and a characteristic speed for the problem are needed. Here, we use as the normalising length scale and the swimming speed v 0 as the characteristic speed. Time will be scaled by /v 0 . We use the notation x = x/ , ỹ = y/ , R = R/ , t = tv 0 / and ũ = u/v 0 to write the equation for the velocity due to forces, force doublets and rotlet doubles as
where the following dimensionless coefficients are identified
(2.15)
The change in the orientation β n of the force doublet at x n is computed with Jeffery's equation (Jeffery 1922)
(2.16)
The expressions for the velocity gradients are found by differentiating the velocity expression and are given in Appendix A.
This section presents studies performed using the force-doublet model. First, we explore the choice of model parameters discussed in § 2.2. Then, we examine the stability of two particles swimming parallel to each other. Next, we demonstrate the potential of the framework to incorporate other flow characteristics by incorporating rotlet doubles into the force-doublet framework. Lastly, we employ the method of images for a force doublet interacting with an infinite plane wall.
The choice of parameters in the force-doublet model of (2.10) is dependent on the organismal system. Importantly, the model parameters can be estimated experimentally based on the organism's swimming velocity, its geometry and features of the flow it generates.
Using the self-velocity shown in (2.11), in order to model a swimmer with a cycleaveraged swimming speed of v 0 , we choose
from which the stagnation point, s 0 = b 0 /q 0 , could be used as a constraint for describing flow features.
Plots of the velocity streamlines along the centre plane of the force doublet (v 0 = 1.0, δ = 1.0, b 0 = -4π ) over the scalar field of the velocity magnitude for (a) q 0 = -2π, (b) q 0 = -4π and (c) q 0 = -8π , along with (d) the speed u plot plotted along the centre axis for varying q 0 . Note that, while the velocity is held fixed, the resulting stagnation point approaches the origin as q 0 increases.
In figure 4(a-c), we plot the resulting velocity streamlines along the central plane over the scalar field of the velocity magnitude of puller for varying q 0 . Here, b 0 is fixed for a velocity v 0 = 1.0 and = 1.0. Initially setting q 0 = -4π yields a stagnation point at x = 1.0 (figure 4a), increasing (figure 4b,c) q 0 leads to a shift in the stagnation point towards x 0 , following the reciprocal relation of 1/q 0 for a fixed b 0 . Plotting the velocity along the centre line in figure 4(d), we note the change in the resulting flow profile as |q 0 | increases, where the flow anti-symmetry associated with the stresslet becomes more pronounced.
Similarly, varying b 0 , in figure 5, we note the role the dipole plays in determining the resulting flow profile. Given the initial flow profile of a puller in figure 5(a) for b 0 = -2π , we note that as b 0 increases (figure 5b,c) the strength of the dipole contribution increases as well. This corresponds to an increase in the self-swimming velocity at x 0 , as seen when plotting the velocity on the centre line (figure 5d). Increasing b 0 also increases the resulting stagnation point, due to its linear relationship with b 0 for a fixed q 0 .
Varying the regularisation parameter also affects the resulting flow field. Plotting the resulting flow fields for varying , given a fixed b 0 = -4π and q 0 = -4π , in figure 6(a), we note how increasing spreads the resulting flow field, as is natural to associate with a regularisation parameter. Plotting the centre line velocity in figure 6(d), we note that the choice of the regularisation parameter does affect the resulting self-swimming speed, given that it is proportional to 1/ 3 (3.1). However, given the fixed choice of b 0 and q 0 , the resulting stagnation point remains fixed.
Biologically relevant values of q 0 can be determined by considering the velocity given by the model along the organism's axis. One characteristic of a pusher is that Note that, while the velocity is held fixed, the resulting stagnation point approaches the origin as q 0 increases.
the fluid motion along the organism's axis shows a stagnation point located at a distance of b 0 /q 0 behind the cell body position, where the stresslet flow balances the dipole flow. For a puller, the stagnation point is also at a distance of b 0 /q 0 from the cell body position, but upstream from it. If we consider x 0 the centre of the cell body, then the distance s 0 from x 0 to the stagnation point provides a length scale for the organism and can be estimated experimentally (see e.g. Drescher et al. 2011;Mondal et al. 2021). These parameters in turn allow one to fit the model to a cycle-averaged flow field, as was done, for example, in Ishimoto et al. (2020) and Delmotte et al. (2015). For our model, experimental data can then provide an estimate for s 0 , which determines q 0 = b 0 /s 0 . In figure 7(a), we present a choice of parameters (b 0 = -40π, q 0 = 20π, = 1.0) that match many of the experimental flow features of a pusher bacteria described in Drescher et al. (2011) (figure 7b). Similarly, we can produce qualitatively similar flow fields (b 0 = -80000π, q 0 = -4000π, = 10.0) to those reported in Mondal et al. (2021), as shown in figures 7(c) and 7(d).
We consider a scenario in which two identical particles translate parallel to each other along the paths x 1 (t) and x 2 (t), with initial conditions given by x 1 (0) = (0, y 0 , 0) and x 2 (0) = (0, -y 0 , 0) with y 0 > 0. Therefore, the separation distance between them is r 0 = 2y 0 . We are interested in the motion that translates the particles at a constant speed in the x-direction. That is, the particle velocities are (U, 0, 0) where U is a constant. In addition, the orientation vectors are assumed to be fixed in time so that dβ/dt = 0. A similar situation is discussed in Lushi & Peskin (2013). Given parameters , q 0 , b 0 , the symmetry of the problem implies that the orientation vector of particle one is (β 1 , -β 2 , 0) and the orientation vector of the second particle is (β 1 , β 2 , 0). The unknowns are the separation distance r 0 , the orientation vector component β 2 , and the speed U .
The dimensionless velocity of the top particle (the fluid velocity at x 1 (t)) is given by (A2) with N = 2, f = 0 and xx j = (0, r 0 , 0). The rate of change of the orientation vector is given by (2.13). Since the motion in this example takes place in the x y-plane, the two non-zero components of the orientation vector have the same rate of change, so they lead to a single constraint equation. In component form, the unknown variables satisfy 8πμU/β 1 =q 0 3β 2 r 0 (r 2 0 + 1) 5/2
The last two equations can be solved numerically for β 2 and r 0 using Newton's method (figure 8a). Once those two variables are known, the swimming speed U is found from the first equation. In the case of a pusher, the solution is r 0 = 0.11540745284558, β 2 = 0.4730561821293 and U = 1.6928. We make the observation that changing the signs of q 0 and β 2 leaves the three equations above unchanged. Recall that q 0 > 0 corresponds to a pusher and q 0 < 0 corresponds to a puller. Thus, we conclude that two pullers also can move parallel to one another at the same separation distance as the pushers, but with the orientation vectors tilted toward one another instead of away from each other (figure 8b).
So far, the presentation of the reduced model has been based on a force dipole along the axis of the swimmer. However, a more comprehensive model would also include the effect of torques that are generated, for example, in the case of bacteria with rotating helical flagella (Spagnolie & Lauga 2012;Ishimoto et al. 2020). Here, we describe how to include this rotational flow.
The main idea used to derive the model was to take the limit as two equal and opposite regularised elements approached each other with an asymmetry in the regularisation parameter. Using Stokeslets as the regularised elements resulted in applying the operator
x 2 (0)
x 1 (0)
x 2 (T )
x to the Stokeslet, as described in (2.7). The same procedure, which has been validated experimentally (Yamamoto & Sano 2018), can be applied with a rotlet replacing the Stokeslet to include the effect of torques. Specifically, the fluid velocity due to a regularised rotlet of strength q r 0 β is given by
where we recall that G = φ . Thus, the velocity at x due to an asymmetric rotlet double located at the origin is
where C r provides the asymmetry in the blob sizes before taking the limit. The resulting flow shows rotation ahead of the particle and counterrotation behind it (figure 9). We note that the three-dimensional flow trajectories show the contributions due to the force doublet (figure 9a), the rotlet double (figure 9b) and their combined flow (figure 9c). The addition of the rotlet double to the complete formulation of § 2.3 is given in Appendix A.
Swimming microorganisms generally move in confined environments where the fluid motion is affected by the presence of surfaces and other obstacles. In the case of the model presented here, which is derived from regularised Stokeslets, the flows bounded by a plane can be computed using the method of images for regularised elements. We consider a plane wall at z = z w and the fluid domain to be all points (x, y, z) with z > z w .
The fluid velocity generated by the swimmers must vanish at the wall. To accomplish this analytically, we define the image of a point (x 0 , y 0 , z 0 ) in the fluid to be (x 0 , y 0 , 2z wz 0 ) and place regularised singularity solutions of Stokes equations at the image point of a swimmer so that they cancel the flow due to the swimmer at the wall. The formulae for the images of a Stokeslet, a force dipole and for a potential dipole have been derived in Cortez & Varela (2015). Here, we present how to use those formulae in the context of the forcedoublet swimmer model. Details of the image system used in this section are shown in Appendix B.
Given the regularising function
the components of velocity from our model are given by (2.7), (2.9) and (2.10)
where S(x; x 0 , ) is the Stokeslet velocity kernel, and D(x; x 0 , ) is the potential dipole kernel. This dipole is derived from a different regularising function, which in Cortez & Varela (2015) is called the companion blob, and is used in the image system for a Stokeslet.
Figure 10 shows the streamlines resulting from a single organism swimming parallel to a nearby (z = 0) wall (figure 10a), from a faraway (z = -1000) wall (figure 10b), as well as at an angle with respect to a nearby (z = 0) wall (figure 10c). We note how the inclusion of a nearby wall alters the resulting flow fields, with vortices formed in the region between the swimmer location and the wall. Taking apart the different components of the force-doublet flow field, we plot the contributions of the dipole (figure 10d) and the stresslet (figure 10e) of the swimmer in figure 10(a). The resulting flow shows that the symmetry of the stresslet generates two counter-rotating vortices by the edge of the domain and a single vortex due to the inclusion of the dipole, which in turn combine so as to create an asymmetry in the full force-doublet generated flow field. In figure 10(f ), we note similarities in the resulting flow profiles of the wall to previous work in Ainley et al. (2008), where two regularised Stokeslet particles are used to model a swimmer and placed near a wall, resembling the stresslet component of the force-doublet model.
Here, q 0 = -4π , b 0 = -4π , and = 1.0 for both simulations with a swimmer centred at x = (0, 0, 3). We plot (d) the dipole and (e) stresslet components of the flows generated by the swimmer in (a). The resulting flows of the stresslet near the wall are comparable to (f ) the flow fields of a two-point swimmer described in Ainley et al. (2008).
In this study, we have presented a mathematical framework for modelling the fluid flows generated by microswimmers in a Stokes fluid. This mathematical framework, the forcedoublet framework, uses a reduced, cycle-averaged description of fluid flows generated by microswimmers using a single point particle to encapsulate their resulting fluid flows. This force-doublet approach is derived from taking a limit of a two-point regularised Stokeslet description of the swimmer. The two-point model takes advantage of the method of regularised Stokeslets' regularisation parameter to account for the asymmetry of the flow fields generated by the swimmer's body and flagellum, allowing us to easily recreate the asymmetries of the pusher and puller fluid dynamics. The derivation of the single point particle in turn reflects this asymmetry with the presence of a stresslet and a potential dipole in the formulation. Furthermore, we have detailed the stability of two swimmers moving in tandem and the resulting stability regions associated both with their orientation and flow field. Finally, we showed the versatility and relative ease of this modelling framework for incorporating other flow characteristics, such as rotational flows generated by the swimmer, by incorporating a rotlet double in the model framework, as well as wall effects, via the method of images.
The velocity expression in our model is a regularised version of the well-known forcedipole model (Lauga & Powers 2009;Spagnolie & Lauga 2012;Bárdfalvy et al. 2024).
To see this, we take the limit of the velocity expression (2.10) as → 0. This eliminates the last term on the right since b 0 = Cq 0 . The last term in the stresslet also vanishes and
which is the force-dipole model. In the regularised version, the additional term of the stresslet ensures that the incompressibility condition is satisfied analytically. The model we have derived is similar to the one developed in Delmotte et al. (2015), although there are some differences and our derivation is completely different. The model developed in Delmotte et al. (2015) is based on adapting the squirmer model (Lighthill 1952;Blake 1971;Ishikawa et al. 2006), where a spherical particle with prescribed axisymmetric surface velocity deforms the shape of the boundary to provide motion to the particle, using a force coupling model (Liu et al. 2009;Yeo & Maxey 2010;Maxey 2017).
Our model also shares features with the work in Ishimoto & Gaffney (2013), Ishimoto et al. (2017Ishimoto et al. ( , 2020) ) and Gaffney et al. (2021). Their model includes two equal and opposite regularised Stokeslets and two equal and opposite regularised rotlets separated by a small distance. The rotlets and Stokeslets are located at the same points, although each one of the four elements has a different regularisation parameter value. In our model, the limiting process results in a force dipole and a potential dipole at a single location (and a single regularisation parameter). As described earlier, rotlets may also be included in our model, which would introduce a rotlet double with its own regularisation parameter. The relatively low number of parameters in the model framework would allow for a similar process of fitting to experimental flow data that was detailed in Ishimoto et al. (2020).
As discussed in previous sections, the flow generated by an organism described as a pusher typically displays a stagnation point along its axis located slightly behind the cell body near the anterior end of the flagellum. For a puller, the stagnation point is generally located ahead of the cell body, in front of the organism, although this is not necessarily the case throughout all phases of the flagellar beat. Chlamydomonas reinhardtii, for example, exhibits a time-dependent instantaneous flow field in which the stagnation point moves from the back of the cell body to the front during a beat cycle. High-speed imaging measurements (Guasto et al. 2010) show fluid flow rotations that reverse direction during other phases of the cycle. See also Delmotte et al. (2015). In the method presented here, the parameters q 0 and b 0 can be time-dependent and tuned to reproduce flows where the stagnation point moves from one side of the cell body to the other.
We do note that the modelling framework does come with its own inherent limitations. The formulation is intended for capturing far-field, cycle-averaged hydrodynamics of dilute suspensions of swimmers, sacrificing the detail of high-fidelity fully descriptive models for a single-point evaluation. Some additional modification to the model could allow for a time-varying flow field, as opposed to cycle-averaged flows, by fitting the stagnation over a swimming cycle to experimental data (Delmotte et al. 2015;Ishimoto et al. 2020). Additionally, near-field flow characteristics could be further addressed by designing regularising functions that approximate some feature of the near-field flow.
As it stands, the framework derived can be implemented to understand features of collective phenomenon, with the intent of future investigations into many-swimmer interactions and correlations between swimming speed and orientation (Wu & Libchaber 2000;Hernandez-Ortiz et al. 2005;Mehandia & Nott 2008;Lushi & Peskin 2013;Schwarzendahl & Mazza 2018, 2019). Bacteria exposed to antibiotics have been shown to modify their colonial structure in order to maximise their survival (Fung et al. 2023). They seem to keep track of their past encounters and adapt as a colony, which would require some sort of chemotactic signalling (BenJacob & Levine 2006;Ryan 2019). A much more robust model would be one that incorporates the fluid dynamics with regulation of the chemoattractant emission (Eyiyurekli et al. 2008) and optimal tumbling rate (Dillon et al. 1995;Strong et al. 1998). Furthermore, the numerical experiments performed thus far were restricted to a sample of uniform size. But microorganisms are known to have variability in size, which in turn affects their swimming speeds, and in the presence of other surfaces would also affect their orientation (Ishikawa 2009). We would like to understand how the induced changes in swimming speed and orientation would also affect cell aggregation and size of the fluid rotations observed.
As demonstrated by this study, the force-doublet model lends itself easily to other extensions in the method of the regularised Stokeslets framework. The role of biofilms and polymer networks (Cogan et al. 2005;Cogan & Wolgemuth 2011) could be investigated with the inclusion of viscoelastic networks in the fluid (Wróbel et al. 2014;Schuech et al. 2022) or using a formulation of the method of regularised Stokeslets for a Brinkman fluid (Cortez et al. 2010;Ho et al. 2019;Nguyen et al. 2019). Further study into the interaction and stability of swimmers near a wall (Berke et al. 2008) can be examined using an image system to account for a boundary, as described in § 3.4 (Ainley et al. 2008;Cortez & Varela 2015;Wróbel et al. 2016). Furthermore, if differences due to the regularisation function are small, the method presented here can be extended to include the use of different blob functions.
Funding. This work was supported by National Science Foundation Grants DMS 1043626 (to R.C.) and DMS 2240770 (to A.P.H.).
The authors report no conflict of interest.
When tracking the dynamics of a force-doublet particle located at x 0 , we compute its velocity with (2.10) and use (3.5) to include the effect of torques. The particle's orientation vector β 0 evolves according to . Example of a force-doublet particle located at x 0 with orientation vector β 0 . The vector β 0 will rotate according to (A1) due to the velocity gradients produced by a different force-doublet particle located at x k with orientation vector β k , due to a force f located at x f , and due to a rotlet double of strength L = q r 0 β at x r .
The complete formulation, with rotlet doubles, is as follows:
-3(β j • (x -x j ))
xx j 8π R5 j + R2 j -3
with the additional dimensionless coefficient associated with the rotlet double qr j = q r j μv 0 3 , (A3) using the characteristic quantities of § 2.3.
The velocity gradient due to a force We consider a force f located at x f and let x = x 0x f and R 2 = x 2 + 2 . The fluid velocity that the force produces at x 0 given in (2.4) can be written as
Then, after simplification, we have that
We form the cross product [β 0 × (β 0 • ∇)v] and finally get
A.3 The velocity gradient due to a force doublet Consider a force doublet with parameters q k , b k and orientation vector β k located at x k and let x = x 0x k and R 2 = x 2 + 2 . The velocity it induces at x 0 is
The image system for each of these elements is given in Cortez & Varela (2015). The image system for (β • ∇)(H 2 (β • x)x + H 1 β) can be summarised as
), (B7)
where x = x ey I M and x e is the point where the velocity is evaluated.
The image system for the potential dipole D 2 (β • x)x + D 1 β is
(B9)
A1-3 https://doi.org/10.1017/jfm.2025.138 Published online by Cambridge University Press
https://doi.org/10.1017/jfm.2025.138 Published online by Cambridge University Press
A1-16https://doi.org/10.1017/jfm.2025.138 Published online by Cambridge University Press