Hydrodynamic interaction of swimming organisms in an inertial regime

Numerical study of microorganisms swimming near a convex wall in a Giesekus fluid

The motion of microorganisms in complex fluids stands out as a prominent subject within fluid mechanics. In our study, we utilize the fictitious domain method to investigate the locomotion of squirmers along a convex wall in Giesekus viscoelastic fluids. This study examines the influence of fluid elasticity and wall curvature on squirmer particles, analyzing their movement patterns in detail. Near the convex wall, three distinct behavioral characteristics emerge: scattering, orbiting forward, and orbiting backward. The findings reveal that, compared with Newtonian fluids, squirmers exhibit a stronger tendency to be attracted toward the wall in viscoelastic fluids. This behavior is attributed to the elastic stress of the fluid, which generates a reverse torque on microbial particles, altering their movement direction and hindering their escape from the wall. Notably, as the wall curvature decreases, the likelihood of particles escaping diminishes.

Densitaxis: Active particle motion in density gradients

Organisms often swim through density-stratified fluids. Here, we investigate the dynamics of active particles swimming in fluid density gradients and report theoretical evidence of taxis as a result of these gradients (densitaxis). Specifically, we calculate the effect of density stratification on the dynamics of a force- and torque-free spherical squirmer and show that density gradients induce reorientation that tends to align swimming either parallel or normal to the gradient depending on the swimming gait. In particular, swimmers that propel by generating thrust in the front (pullers) rotate to swim parallel to gradients and hence display (positive or negative) densitaxis, while swimmers that propel by generating thrust in the back (pushers) rotate to swim normal to the gradients. This work could be useful to understand the motion of marine organisms in ocean or be leveraged to sort or organize a suspension of active particles by modulating density gradients.

Swimming Mode of Two Interacting Squirmers under Gravity in a Narrow Vertical Channel

The swimming mode of two interacting squirmers under gravity in a narrow vertical channel is simulated numerically using the lattice Boltzmann method (LBM) in the range of self-propelling strength 0.1 ≤ α ≤ 1.1 and swimming type −5 ≤ β ≤ 5. The results showed that there exist five typical swimming patterns for individual squirmers, i.e., steady upward rising (SUR), oscillation across the channel (OAC), oscillation near the wall (ONW), steady upward rising with small-amplitude oscillation (SURO), and vertical motion along the sidewall (VMS). The parametric space (α, β) illustrated the interactions on each pattern. In particular, the range of oscillation angle for ONW is from 19.8° to 32.4° as α varies from 0.3 to 0.7. Moreover, the swimming modes of two interacting squirmers combine the two squirmers’ independent swimming patterns. On the other hand, the pullers (β < 0) attract with each other at the initial stage, resulting in a low-pressure region between them and making the two pullers gradually move closer and finally make contact, while the result for the pushers (β > 0) is the opposite. After the squirmers’ interaction, the squirmer orientation and pressure distribution determine subsequent squirmer swimming patterns. Two pushers separate quickly, while there will be a more extended interaction period before the two pullers are entirely separated.

Inertial particle under active fluctuations: Diffusion and work distributions

We study the underdamped motion of a passive particle in an active environment. Using the phase space path integral method we find the probability distribution function of position and velocity for a free and a harmonically bound particle. The environment is characterized by an active noise which is described as the Ornstein-Uhlenbeck process (OUP). Taking two similar, yet slightly different OUP models, it is shown how inertia along with other relevant parameters affect the dynamics of the particle. Further we investigate the work fluctuations of a harmonically trapped particle by considering the trap center being pulled at a constant speed. Finally, the fluctuation theorem of work is validated with an effective temperature in the steady-state limit.

Hydrodynamic interactions between swimming microorganisms in a linearly density stratified fluid

Oceans and lakes sustain intense biological activity due to the motion of marine organisms, which has significant ecological and environmental impacts. The motion of individual organisms and their interactions with each other play a significant role in the collective motion of swimming organisms. However, ubiquitous vertical density stratification in these aquatic environments significantly alters the swimmer interactions as compared to in a homogeneous fluid. Furthermore, organisms have sizes varying over a wide range which results in finite inertia. To this end, we numerically investigate the interactions between a pair of model swimming organisms in two configurations: (1) approaching each other and (2) moving side by side with finite inertia in a linearly density stratified fluid. We use the archetypal reduced-order squirmer model to numerically model the swimming organisms. We present trajectories and the contact times of interacting squirmer (puller & pusher) pairs for different Re in the range 1-50 and Ri in the range 0-10. Depending on the squirmer Re and Ri we observe that the squirmer interactions can be categorized as (i) pullers getting trapped in circular loops at high Re and low Ri, (ii) pullers escaping each other with separating angle decreasing with increasing stratification at low Re and high Ri, (iii) pushers sticking to each other after the collision and deflecting away from the collision plane for either low Re or high Ri, (iv) pushers escaping otherwise with an angle of separation increasing with stratification. Stratification also increases the contact time for squirmer pairs. The presented results can be useful to understand the mechanisms behind the accumulation of planktonic organisms in horizontal layers in a stratified environment such as oceans and lakes.

As above, so below, and also in between: mesoscale active matter in fluids

Living matter, such as biological tissue, can be viewed as a nonequilibrium hierarchical assembly, where at each scale self-driven components come together by consuming energy in order to form increasingly complex structures. The remarkable properties of living or "active-matter" systems, as they are generally known, such as versatility, self-healing, and self-replicating, have prompted the following questions: (1) do we understand the biology and biophysics that give rise to these properties? (2) can we achieve similar functionality with synthetic active materials? In this perspective we specifically focus on why it is important to study active matter in fluids with finite inertia. Finite inertia is relevant for mesoscale organisms that swim or fly covering at least three orders of magnitude in size (≈0.5 mm-50 cm) and their collective behavior is generally unknown. As a result, we are limited both in our understanding of the biology of mesoscale swarms and processes but also in our design of self-powered machines and robots at those scales. We expect interesting collective behavior to emerge because with finite inertia, come nonlinearities and the many-body hydrodynamic interactions between the organisms/particles can become quite complex, potentially leading to phenomena, such as novel flocking states and nonequilibrium phase transitions that have not been observed before and which could have great impact in materials applications.

Direct-forcing fictitious domain method for simulating non-Brownian active particles

We present a direct-forcing fictitious domain method for simulating non-Brownian squirmer particles with both the hydrodynamic interactions and collisions being fully resolved. In this method, we solve the particle motion by distributing collocation points inside the particle interior domain that overlay upon a fixed Eulerian mesh. The fluid motions, including those of the "fictitious fluids" being extended into the particle, are solved on the entire computation domain. Pseudo-body forces are used to enforce the fictitious fluids to follow the particle movement. A direct-forcing approach is employed to map physical variables between the overlaid meshes, which does not require additional iterations to achieve convergence. We perform a series of numerical studies at both small and finite Reynolds numbers. First, accuracy of the algorithm is examined in studying benchmark problems of a free-swimming squirmer and two side-by-side squirmers. Then we investigate statistic properties of the quasi-two-dimensional collective dynamics for a monolayer of squirmer particles that are confined on a surface immersed in a bulk flow. Finally, we explore the physical mechanisms of how a freely moving short cylinder interacts with a monolayer of active particles, and find out that the cylinder movement is dominated by collision. We demonstrate that a more directional migration of cylinder can be resultant from an inhomogeneous distribution of active particles around the cylinder that has an anisotropic shape.

Dynamics near planar walls for various model self-phoretic particles

For chemically active particles suspended in a liquid solution and moving by self-phoresis, the dynamics near chemically inert, planar walls is studied theoretically by employing various choices for the activity function, i.e., the spatial distribution of the sites where various chemical reactions take place. We focus on the case of solutions composed of electrically neutral species. This analysis extends previous studies of the case that the chemical activity can be modeled effectively as the release of a "product" molecular species from parts of the surface of the particle by accounting for annihilation of the product molecules by chemical reactions, either on the rest of the surface of the particle or in the volume of the surrounding solution. We show that, for the models considered here, the emergence of "sliding" and "hovering" wall-bound states is a generic, robust feature. However, the details of these states, such as the range of parameters within which they occur, depend on the specific model for the activity function. Additionally, in certain cases there is a reversal of the direction of the motion compared to the one observed if the particle is far away from the wall. We have also studied the changes of the dynamics induced by a direct interaction between the particle and the wall by including a short-ranged repulsive component to the interaction in addition to the steric one (a procedure often employed in numerical simulations of active colloids). Upon increasing the strength of this additional component, while keeping its range fixed, significant qualitative changes occur in the phase portraits of the dynamics near the wall: for sufficiently strong short-ranged repulsion, the sliding steady states of the dynamics are transformed into hovering states. Furthermore, our studies provide evidence for an additional "oscillatory" wall-bound steady state of motion for chemically active particles due to a strong, short-ranged, and direct repulsion. This kind of particle translates along the wall at a distance from it which oscillates between a minimum and a maximum.

Swimming sheet near a plane surfactant-laden interface

In this work we analyze the velocity of a swimming sheet near a plane surfactant-laden interface by assuming the Reynolds number and the sheet's deformation to be small. We observe a nonmonotonic dependence of the sheet's velocity on the Marangoni number (Ma) and the surface Péclet number (Pe_{s}). For a sheet passing only transverse waves, the swimming velocity increases with an increase in Ma for any fixed Pe_{s}. When Pe_{s} is increasing, on the other hand, the swimming velocity of the same sheet either increases (at large Ma) or it initially increases and then decreases (at small Ma). This dependence of the swimming velocity on Ma and Pe_{s} is altered if the sheet is passing longitudinal waves in addition to the transverse waves along its surface.

Simulating squirmers with multiparticle collision dynamics

Multiparticle collision dynamics is a modern coarse-grained simulation technique to treat the hydrodynamics of Newtonian fluids by solving the Navier-Stokes equations. Naturally, it also includes thermal noise. Initially it has been applied extensively to spherical colloids or bead-spring polymers immersed in a fluid. Here, we review and discuss the use of multiparticle collision dynamics for studying the motion of spherical model microswimmers called squirmers moving in viscous fluids.

Reduced-order model for inertial locomotion of a slender swimmer

The inertial locomotion of an elongated model swimmer in a Newtonian fluid is quantified, wherein self-propulsion is achieved via steady tangential surface treadmilling. The swimmer has a length 2l and a circular cross section of longitudinal profile aR(z), where a is the characteristic width of the cross section, R(z) is a dimensionless shape function, and z is a dimensionless coordinate, normalized by l, along the centerline of the body. It is assumed that the swimmer is slender, ε=a/l≪1. Hence, we utilize slender-body theory to analyze the Navier-Stokes equations that describe the flow around the swimmer. Therefrom, we compute an asymptotic approximation to the swimming speed, U, as U/u_{s}=1-β[V(Re)-1/2∫_{-1}^{1}zlnR(z)dz]/ln(1/ε)+O[1/ln^{2}(1/ε)], where u_{s} is the characteristic speed of the surface treadmilling, Re is the Reynolds number based on the body length, and β is a dimensionless parameter that differentiates between "pusher" (propelled from the rear, β<0) and "puller" (propelled from the front, β>0) -type swimmers. The function V(Re) increases monotonically with increasing Re; hence, fluid inertia causes an increase (decrease) in the swimming speed of a pusher (puller). Next, we demonstrate that the power expenditure of the swimmer increases monotonically with increasing Re. Further, the power expenditures of a puller and pusher with the same value of |β| are equal. Therefore, pushers are superior in inertial locomotion as compared to pullers, in that they achieve a faster swimming speed for the same power expended. Finally, it is demonstrated that the flow structure predicted from our reduced-order model is consistent with that from direct numerical simulation of swimmers at intermediate Re.

Hydrodynamic Choreographies of Microswimmers

We unveil orbital topologies of two nearby swimming microorganisms using an artificial microswimmer, called Quadroar. Depending on the initial conditions of the microswimmers, we find diverse families of attractors including dynamical equilibria, bound orbits, braids, and pursuit–evasion games. We also observe a hydrodynamic slingshot effect: a system of two hydrodynamically interacting swimmers moving along braids can advance in space faster than non-interacting swimmers that have the same actuation parameters and initial conditions as the interacting ones. Our findings suggest the existence of complex collective behaviors of microswimmers, from equilibrium to rapidly streaming states.

Rotational Diffusion of Soft Vesicles Filled by Chiral Active Particles

We investigate the dynamics of two-dimensional soft vesicles filled with chiral active particles by employing the overdamped Langevin dynamics simulation. The unidirectional rotation is observed for soft vesicles, and the rotational angular velocity of vesicles depends mainly on the area fraction (ρ) and angular velocity (ω) of chiral active particles. There exists an optimal parameter for ω at which the rotational angular velocity of vesicle takes its maximal value. Meanwhile, at low concentration the continuity of curvature is destroyed seriously by chiral active particles, especially for large ω, and at high concentration the chiral active particles cover the vesicle almost uniformly. In addition, the center-of-mass mean square displacement for vesicles is accompanied by oscillations at short timescales, and the oscillation period of diffusion for vesicles is consistent with the rotation period of chiral active particles. The diffusion coefficient of vesicle decreases monotonously with increasing the angular velocity ω of chiral active particles. Our investigation can provide a few designs for nanofabricated devices that can be driven in a unidirectional rotation by chiral active particles or could be used as drug-delivery agent.

Modeling of active swimmer suspensions and their interactions with the environment

In this article, we review mathematical models used to study the behaviour of suspensions of micro-swimmers and the accompanying biophysical phenomena, with specific focus on stimulus response. The methods discussed encompass a range of interactions exhibited by the micro-swimmers; including passive hydrodynamic (gyrotaxis) and gravitational (gravitaxis) effects, and active responses to chemical cues (chemotaxis) and light intensities (phototaxis). We introduce the simplest models first, and then build towards more sophisticated recent developments, in the process, identifying the limitations of the former and the new results obtained by the latter. We comment on the accuracy/validity of the models adopted, based on the agreement between theoretical results and experimental observations. We conclude by identifying some of the open problems and associated challenges faced by researchers in the realm of active suspensions.