Inertial torque on a squirmer

Abstract A small spheroid settling in a quiescent fluid experiences an inertial torque that aligns it so that it settles with its broad side first. Here we show that an active particle experiences such a torque too, as it settles in a fluid at rest. For a spherical squirmer, the torque is $\boldsymbol {T}^\prime = -{\frac {9}{8}} m_f (\boldsymbol {v}_s^{(0)} \wedge \boldsymbol {v}_g^{(0)})$ where $\boldsymbol {v}_s^{(0)}$ is the swimming velocity, $\boldsymbol {v}_g^{(0)}$ is the settling velocity in the Stokes approximation and $m_f$ is the equivalent fluid mass. This torque aligns the swimming direction against gravity: swimming up is stable, swimming down is unstable.


Introduction
The motion of small plankton in the turbulent ocean is overdamped (Visser 2011).Accelerations play no role, and hydrodynamic forces and torques can be computed in the Stokes approximation.Turbulence rotates these small organisms, yet they manage to navigate upwards towards the ocean surface.Gyrotactic organisms make use of gravity to achieve this.These bottom-heavy swimmers experience a gravity torque that tends to align against the direction of gravity, so that they swim upwards (Kessler 1985;Durham et al. 2013;Gustavsson et al. 2016).Also density or shape asymmetries give rise to torques in the Stokes approximation that can change the swimming direction (Roberts 1970;Jonsson 1989;Roberts & Deacon 2002;Candelier & Mehlig 2016;Roy et al. 2019).
Larger organisms accelerate the surrounding fluid as they move, and this changes the hydrodynamic force the swimmer experiences (Wang & Ardekani 2012; Khair & Chisholm 2014;Chisholm et al. 2016;Redaelli et al. 2022b,a).Three different mechanisms cause such fluid-inertia effects, a non-zero slip velocity (Oseen problem with non-dimensional parameter Re p , the particle Reynolds number), velocity gradients of the disturbance flow (Saffman problem, shear Reynolds number Re s ), and unsteady fluid inertia (with parameter Re p Sl, where Sl is the Strouhal number).
In this paper, we show that a small spherical squirmer experiences an inertial torque analogous to the Khayat & Cox torque when it settles in a quiescent fluid.Using asymp-< l a t e x i t s h a 1 _ b a s e 6 4 = " S D u B w r h F Z q D 5 8 r Y 5 2 W T Y K R l 8 E S E = " > A A A C S 3 i c b V H L S s N A F J 1 U q z W + W l 2 6 G S y F C q U k U t R l 0 Y 3 L C n 1 J G 8 p k M q l D J g 9 m J o U S 8 h V u 9 Z P 8 A L / D n b h w 0 g b p w w s D h 3 M f 5 3 D G j h g V 0 q r T W V P 4 s 6 L V V g a 5 p J Z n 5 7 P p a J / G i j B c N 9 S 8 e 4 T P E n F R X + Z q b a W 6 D / n X T v G m 2 n l r V 9 n 2 e d A l c g E t Q B y a 4 B W 3 w C D q g B z D w w S t 4 A + / a h / a l f W s / y 9 G C l u + c g 7 U q F H 8 B U P q z F g = = < / l a t e x i t > (c) < l a t e x i t s h a 1 _ b a s e 6 4 = " 9 N c A / J Y s 6 z 0 1 p y l T S 0 f + f s 7 S v S c = " > A A A C + X i c b V L L j t M w F H X C a w i P 6 c C S T S A D Y o G q B C G G D W I E G x Y s B m n a G a m J q h v n p r V q O 5 H t d K i s f A x i g 9 j y A + z 4 B M T f 4 L S z I O 1 c y d b x u e c + f O 2 8 5 k y b O P 7 r + d e u 3 7 h 5 a + 9 2 c O f u v f v 7 g 4 M H Y 1 0 1 i u K I V r x S 5 z l o 5 E z i y D D D 8 b x W C C L n e J Y v P n T + s y U q z S p 5 a l Y 1 < l a t e x i t s h a 1 _ b a s e 6 4 = " T K O e E o 9 m Y j M W y T r 2 p x l X + q z + 5 5 4 < l a t e x i t s h a 1 _ b a s e 6 4 = " V 6 X a 8 t I K R d v a 8 7 N N t y 8 k w g R 9 9 r w = " > A 8 a 6 l 8 8 w m e I O a m u 8 j U 3 0 9 w G / e u m e d N s P b W q 7 f s 8 6 R K 4 A J e g D k x w C 9 r g E X R A D 2 D g g 1 f w B t 6 1 D + 1 L + 9 Z + l q M F L d 8 5 B 2 t V K P 4 C U t u z F w = = < / l a t e x i t > (d) totic matching, we calculate the torque to leading order in the particle Reynolds number where u c is a velocity scale, a is the radius of the squirmer, and ν is the kinematic viscosity of the fluid.The calculation shows that the inertial torque does not vanish for a spherical swimmer because swimming breaks rotational symmetry.We describe how the torque aligns the squirmer, and compare its effect with gyrotactic torques, and with the Khayat-Cox torque for a nonspherical passive particle.

Model
We consider a steady spherical squirmer, an idealised model for a motile microorganism developed by Lighthill (1952) and Blake (1971).In this model, one imposes an active axisymmetric tangential surface-velocity field of the form (B 1 sin θ + B 2 sin θ cos θ)ê θ , (2.1) with parameters B 1 and B 2 , and where θ is the angle between the swimming direction (unit vector n) and the vector r from the particle centre to a point on its surface.The tangential unit vector at this point is denoted by êθ .One distinguishes two types of squirmers depending on the parameter β = B 2 /B 1 (Lauga & Powers 2009): 'pushers' (β < 0) and 'pullers' with β > 0. In the Stokes limit, a squirmer moving with velocity ẋ in a fluid at rest experiences the hydrodynamic force Here the superscript denotes the Stokes approximation, and f is the mass density of the fluid.Following Candelier et al. (2019), we use a prime to indicate that this is the hydrodynamic force on the squirmer, due to the disturbance it creates.Plankton tends to be slightly heavier than the fluid.Therefore we allow the squirmer to settle subject to the buoyancy force where s is the mass density of the squirmer, and g is the gravitational acceleration.In the overdamped limit, the steady centre-of-mass velocity of the squirmer is determined by the zero-force condition g .Again, the superscript denotes the Stokes limit.In this limit, the squirmer experiences no torque in a fluid at rest, T (0) = 0.

Inertial torque
Assume that the squirmer swims with swimming velocity v s and settles with settling velocity v g .The angle between v s and v g is denoted by α, as shown in Fig. 1(a).Symmetry dictates the form of the inertial torque T .It has the units mass × velocity 2 .Since the torque is an axial vector, it must be proportional to the vector product between the two velocities.The torque can therefore be written as where m f = 4π 3 a 3 f is the equivalent fluid mass, C is a non-dimensional constant, and the superscript indicates that this is the first inertial correction to the torque.Eq. (3.1) says that torque vanishes when the swimmer swims against gravity [α = π in Fig. 1(a)], and when it swims in the direction of gravity (α = 0).Bifurcation theory implies that one of these fixed points is stable, the other one unstable.The sign of the coefficient C determines which of the two is the stable fixed point.
Inertial torques can be understood as a consequence of advection of fluid momentum.In the frame translating with the squirmer, far-field momentum is advected by the transverse disturbance flow generated by the squirmer.At non-zero Re p , the head of the squirmer -the north pole of the axial velocity field (2.1) -experiences more drag than its rear, because some of the momentum imparted to the fluid by the head is advected to the trailing end, in the direction transverse to gravity.So when v s is not co-linear with v g there is an inertial torque which rotates the swimmer so that v s becomes closer to antiparallel with v g .Comparing with Eq. (3.1), this means that the coefficient C must be negative.Note that the mechanism described above is the same that creates Khayat-Cox torques on non-spherical passive particles sedimenting in quiescent fluid.For a fibre, for example, the far-field momentum is advected by the transverse flow along the fibre, leading to a torque that aligns the fibre perpendicular to gravity (Khayat & Cox 1989).

Perturbation theory for the coefficient C
The inertial torque is computed from where σ (1) mn = −p (1) δ mn + 2µS (1) mn are the elements of the stress tensor σ (1) with pressure p (1) , S (1) mn are the elements of the strain-rate tensor of the disturbance flow, and µ = f ν is the dynamic viscosity.The integral goes over the particle surface S , r is the vector from the particle centre to a point on the particle surface, and ds is the outward surface normal at this point.In the Stokes approximation the torque vanishes, T (0) = 0, as mentioned above.
The disturbance stress tensor is determined by solving the steady Navier-Stokes equations for the incompressible disturbance flow w, with boundary conditions w = ẋ + (B 1 sin θ + B 2 sin θ cos θ)ê θ for |r| = 1, and w → 0 as |r| → ∞.Here we assumed that the squirmer has no angular velocity.We nondimensionalised Eq. (4.2) using the radius a of the squirmer as a length scale, and with the velocity scale u c = v

Direct numerical simulations
We solved the three-dimensional Navier-Stokes equations for the incompressible flow using an immersed-boundary method (Peskin 2002).The interaction between squirmer and fluid was implemented by the direct-force method (Uhlmann 2005): in order to satisfy the boundary condition (2.1), the algorithm calculates the predicted fluid velocity on the surface of the squirmer.Based on the mismatch between the predicted velocity and Eq.(2.1), an appropriate immersed-boundary force is applied to the fluid phase to maintain the boundary conditions (2.1) on the surface of the squirmer.We implemented the improved algorithm described in (Kempe & Fröhlich 2012;Breugem 2012;Lambert et al. 2013), because it is more precise for nearly-neutrally buoyant particles.We used a cubic computational domain of side length L = 20a with periodic boundary conditions.This is large enough to account for convective inertia for Re p > 1.The computational domain was discretised using a cubic mesh with resolution ∆x.The Navier-Stokes equations were integrated using a second-order Crank-Nicholson scheme for the time-integration (Kim et al. 2002) with time step ∆t, while the motion of the squirmer was integrated using a second-order Adams-Bashforth method (Hairer et al. 2000).
The numerical simulation of solid-body motion in a fluid is challenging at small Re p .The mesh resolution ∆x must be fine enough to resolve the shape of the body, so that the viscous stresses near its surface are accurately represented (Andersson & Jiang 2019).In addition, the time step ∆t must be small enough to resolve the viscous diffusion of the disturbance, ∆t < ∆x 2 /ν.In Appendix A we briefly describe our convergence checks.We found that our algorithm fails to converge for Re p smaller than unity.In the following we discuss our numerical results for Re p 1.
To determine the torque, we froze the orientation of the squirmer at a given angle α, but allowed the squirmer to translate.It was initially at rest.We measured the centreof-mass velocity and the torque after the transient, when the disturbance flow was fully established.Figure 2(a) shows the numerical results for the inertial torque on a spherical squirmer for Re p = 1, in comparison with the theory (4.7).The remaining parameter values used in the simulations are quoted in the Figure caption.Although the theory is valid for Re p 1, it nevertheless agrees qualitatively with the numerical results for Re p = 1.This is encouraging, because it allows as to draw qualitative conclusions about the effects of the torque on small plankton (Section 6).Panel (a) shows that the theory overestimates the amplitude of the torque by a factor of two, but that its angular dependence is roughly the same.We note, however, that the numerical results exhibit an asymmetry in their dependence on α.Since the small-Re p theory yields a symmetric angular dependence of the torque, we attribute the asymmetry to higher-order Re p -corrections.Panel (b) confirms that the difference between theory and numerical simulation increases for larger Re p , as expected.
The small-Re p theory (4.7) says that the torque is independent of β.Panel (c) shows that this is not the case for the numerical results at Re p = 1.This suggests, again, that higher-Re p corrections matter at Re p = 1, and that they exhibit a β-dependence.Another indication that higher-order Re p -corrections may be important comes from measuring settling and swimming speeds in the numerical simulations.We extracted the swimming speed using ẋ = v s n − v g ê2 .Solving for v s gives v s = ẋ • ê1 /(n • ê1 ).Fig. 2(d) compares the measured swimming and settling speeds.The settling speed is substantially smaller than the Stokes estimate, consistent with a significant Re p -correction.The swimming speed is much closer to the Stokes estimate.This is because the data shown is for β = 0, and the known Re p -corrections to the swimming speed (Khair & Chisholm 2014), vanish for β = 0.

Conclusions
We showed that a spherical squirmer settling in a fluid at rest experiences an inertial torque, and computed the torque using matched asymptotic expansions.The calculation is similar to that of Cox (1965) for the inertial torque on a nearly spherical, passive particle settling in a quiescent fluid.This torque vanishes for a passive sphere, a consequence of spherical symmetry.A spherical swimmer experiences an inertial torque because swimming breaks this symmetry.The torque causes the squirmer to align with gravity so that it swims upwards.In other words, this torque acts just like Kessler's gyrotactic torque for bottom-heavy organisms.
For plankton, the effect of the inertial torque is much smaller than the gyrotactic torque, at least for spherical shapes.We can see this by comparing the corresponding reorientation times.This time scale is defined as τ I = 1 2 (8πµa 3 )/T max , where 8πµa 3 is the rotational resistance coefficient for a sphere (Kim & Karrila 2013), and T max is the maximal magnitude of the torque.For the inertial torque, one obtains τ I = 8ν/(3v g ) (this and all following expressions are quoted in dimensional units).The reorientation time for the gyrotactic torque is τ G = 3 s ν/( f gh) (Pedley & Kessler 1987), where h is the offset between the centre-of-mass and the geometrical centre of the squirmer, and g = |g|.The ratio of these time scales is (Kessler 1986)], we see that swimming and settling speeds need to be of the order mm/s for the reorientation times to be comparable.For small plankton, typical speeds tend to be much smaller (Kessler 1986).For larger organisms, however, the inertial torque can be significant.With typical values for a small copepod (Titelman & Kiørboe 2003), v s = 1 mm/s, v g = 0.2 mm/s, as well as ν = 10 −6 m 2 /s, one finds an inertial reorientation time of the order of τ I ∼ 10 s.Kolmogorov times for ocean turbulence range from τ K = ν/E = 100 s for dissipation rate per unit mass E = 10 −6 cm 2 /s 3 to τ K = 1 s for E = 10 −2 cm 2 /s 3 .So the non-dimensional reorientation parameter Ψ = τ I /τ K (Durham et al. 2013) ranges from 0.1 for weak turbulence to 10 for strong turbulence.The Reynolds number is of order Re p ∼ 1 for speeds of the order of 1 mm.This means that the Re p -perturbation theory does not strictly apply, but we can nevertheless conclude that for weak turbulence, the inertial torque can have a significant effect on the angular dynamics of the organism.
Some motile microorganisms are non-spherical (Berland et al. 1995;Faust & Gulledge 2002;Smayda 2010).It has been suggested that this can give additional contributions to the torque (Qiu et al. 2022).Since the boundary conditions are different, and since swimming breaks fore-aft symmetry, these additional contributions may be different from the Khayat-Cox torque for passive particles.However, we expect that the torque is still determined by the same physical mechanism, advection of fluid-momentum transverse to gravity.This may give rise to terms proportional to sin(2α), whereas the torque is proportional to sin(α) for the spherical squirmer.To make these speculations definite, one could compute the inertial torque for a nearly spherical squirmer in perturbation theory.A second open question is to determine the inertial torque for bottom-heavy, non-spherical organisms, the analogue of the inertial torque on passive particles with mass-density asymmetries (Roy et al. 2019).
More generally, although the small-Re p perturbation theory may become quantitatively inaccurate for Reynolds numbers of order unity -where the torque begins to make a significant difference -the results tell us which non-dimensional parameters matter, and how to reason about the effect of boundary conditions, and the symmetries of the problem.The calculation illustrates the conceptual insight that the inertial torque comes from fluid motion transverse to the direction of gravity.Fluid momentum in this direction is advected along the swimmer by the transverse fluid velocity, resulting in a torque.In our case, the boundary conditions are different from those for passive particle, and so is the symmetry of the problem, because swimming breaks fore-aft symmetry.Nevertheless, the fundamental mechanism generating the torque is the same.
4 G P / 5 z k / P z x E 8 G V d p w 3 q 7 S w u L S 8 s l q 2 1 9 Y 3 N r c q 1 e 0 r F a e S Y Y / F I p Y 3 P i g U P M K e 5 l r g T < l a t e x i t s h a 1 _ b a s e 6 4 = " A G n m l 6 Q Y K J 9 x i W E D 9 D q N X K W I a T Y = " > A A A C T n i c b V H L S s N A F J 3 U V 4 2 v V p d u g q X Q R S m J F O 2 y 4 M Z l h b 6 g C W U y m b R D J p M w M y m U k N 9 w q 5 / k 1 h 9 x J z p p g / T h h Y H D u Y 9 z O O P G l A h p m p 9 a 6 e D w 6 P i k f K q f n V 9 c X l W q 1 0 M R J R z h A Y p o x M c u F J g S h g e S S I r H M c c w d C k e u c F T 3 h 8 t M B c k Y n 2 5 j L E T w h k j P k F Q K s q 2 f Q 5 R a m V p u 5 N N K z W z Z a 7 K 2 A d W A W q g q N 6 0 q j V t L 0 J J i J l E r / e J J F + x 0 k J i x O J G V o r + g k 1 Z G T k U R g e 4 R h J u l Q A I k 6 U a Q P N o Q p E q s C 2 V P 4 s 6 P V N g b 7 l p L n 5 / P p W J w 3 i n B d N 9 T U B 5 g t I v U x X + V q 7 a e 6 D 4 X 3 L e m i 1 X 9 q 1 b q N I u g x u w R 1 o A A s 8 g i 5 4 B j 0 w A A j E 4 B W 8 g X f t Q / v S v r W f 9 W h J K 3 Z u w F a V y r + 8 t L S v < / l a t e x i t > 1 48 We then checked for convergence as the step size ∆t was reduced, for fixed 2a/∆x = 36.Again, both settling and swimming speeds reached plateaus as ν∆t/∆x 2 decreased.When we halved ν∆t/∆x 2 from 0.58 to 0.29, the settling and swimming speeds varied about 1% and 2%, respectively.Therefore, we used 2a/∆x = 36 and ∆t = 0.58∆x 2 /ν for our numerical simulations at Re p = 1.0 that are shown in the main text.At these parameter values, the simulated settling speed of a passively settling particle is about 3% larger than the measurement of Maxworthy, taken from Fig. 4 of (Vesey II & Goldenfeld 2007).The swimming speed of the squirmer is about half a percent larger than the theoretical value v (0) s = 2B 1 /3.The convergence for the inertial torque was slightly worse.The torque varied about 0.5% when we changed 2a/∆x from 36 to 48 for ν∆t/∆x 2 = 0.58, and it varied about 4% when we halved ν∆t/∆x 2 from 0.58 to 0.29 for 2a/∆x = 36.We also performed convergence checks for Re p = 0.32, reducing the particle Reynolds number by increasing ν.We found that the values of ∆x and ∆t quoted above are too large for the numerical scheme to converge at this Reynolds number.Therefore we only show results for Re p 1 in the main text.

Figure 1 .
Figure 1.(a) Squirmer with swimming velocity vs and settling velocity vg, see Section 2. Gravity points in the negative ê2-direction.(b,c,d) Disturbance flow created by a squirmer with B2 = 0 (schematic).Shown are the flow lines in the frame that translates with the body.The centre-of-mass velocity ẋ is shown in green.
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