This report focuses specifically on the motion at high Reynolds number, where inertial forces dominate the fluid dynamics. Current models of emergent behavior in the inertial regime are mainly phenomenological and do not account for the fluid-mediated interactions between bodies. We are particularly interested in the fluid mediated interactions between self-propelled bodies and how this can lead to collective behavior.
An individual body, such as a fish or a bird, causes a fluid disturbance and affects the fluid forces on all other bodies in the system. The Reynolds number quantifies the relative importance of inertial and viscous forces in the liquid medium, [13–15] where the density ρ and the viscosity µ of the liquid are material parameters. The fluid inertia cannot be neglected at moderate and large Reynolds numbers, which is a major problem.
Of course, a finite element scheme for the fluid can be constructed with deformable solid bodies. Historically, the modeling of budding phenomena at high Reynolds number sidestepped the Navier-Stokes equations by completely neglecting the fluid medium. The added mass physically captures the effective increase in inertia that a particle of the fluid experiences when it translates into potential flow.
We have made two fundamental fluid mechanical assumptions: the fluid flow is inviscid and irrotational.
![Figure 1.1: Macroscopic examples [7] of emergent phenomena: () fish school- school-ing and (a) starlschool-ings flockschool-ing](https://thumb-ap.123doks.com/thumbv2/123dok/10401316.0/9.918.209.711.551.939/figure-macroscopic-examples-emergent-phenomena-school-starlschool-flockschool.webp)
POTENTIAL FLOW THEORY
For a constant mass density (incompressible fluid), the kinetic energy of the fluid is given by the integral of the local kinetic energy density over the area of the fluid. Further manipulation and application of the divergence theorem to relate the potential gradient to the boundary conditions of all N particles results. The negative sign in Eq. 2.7) is a consequence of the convention of a normal vector pointing into the fluid domain from the solid—established by the no-flow boundary conditions in Eq. 2.3)) is linear, the solution must be linear with respect to particle velocity through no-flow boundary conditions.
A translating sphere creates a leading-order dipolar potential perturbation O(r−2) [2, 3], and higher-order polar moments can be neglected, as shown in Bonnecaze & Brady simplifying the kinetic energy into a quadratic form of the particle velocities. The source of the interesting phenomena we study is the configuration dependence of the added mass tensor. The added mass tensor is calculated via a Taylor expansion of the potential Φ around the center positions of the particles.
Therefore, the added mass is proportional to the gradient of the leading-order dipolar potential, which is O(r−3). The Lagrangian of the fluid is therefore entirely determined by the velocities of the particles and their configuration. Now we need to add the real mass and potential energy of the particles to solve the Lagrangian of the whole system.
The total Lagrangian for the system is the sum of the Lagrangians from the fluid and the particles. If the added mass were independent of the particle configuration, the equations of motion immediately reduce to Newton's equations for particles of constant mass. For simplicity, we denote the relationship between mass densities as Γ = ρ(p)/ρ and the fluid mass displaced by a single sphere as ν=ρ V, so that the mass tensor is written more compactly. 2.20) The importance of the configuration-dependent added mass can also be seen directly from the hydrodynamic forces.
They can either accelerate their surrounding fluid (U˙k 6= 0), or change their relative positions to take advantage of the added mass tensor (cf. This implies that the Lagrangian must be invariant for a rigid spatial translation of the whole system (i.e. A method for determining the effective conductance of dispersions of particles. Proceedings of the Royal Society of London.
COLLINEAR SWIMMER
With no external forcing and momentum-free initial conditions, the total linear momentum of the body mass center must forever be zero (Pz = 0). Integrating the leading-order Taylor expansion shows that only the cross terms (with both X and Y coordinates in the integrand) are non-zero due to simple trigonometric identities. Importantly, the dimensionless displacement is scaled as O(R0) to leading order, which is expected from the derivative of the added mass tensor.
If the phase difference is an integer multiple of π, no translation is possible in leading order, regardless of the internal mechanics. To establish the validity of the leading-order expansion, we integrate Z˙ numerically (similar initial accelerations were set by the time derivatives of the velocity constraints and closed with a force-free condition on the overall swimmer.
As mentioned before, the swimmer is able to self-promote due to the configuration dependence of the elements of the total mass tensor. The direct coupling of the added mass Mxz and Myz through internal degrees of freedom is clearly a key source of self-propulsion for the swimmer shown. The hydrodynamic resistance tensor Rˆ is only a function of the particle configuration, analogous to the mass tensor added to the potential flow.
Note that the Stokes equation of motion is analogous to the corresponding impure equation (cf. 3.2)) with the drag tensor taking the place of the total mass tensor. Since the drag tensor is only a function of the particle configuration, the center of mass translation derivative is the same as in the inviscid regime, resulting in . We therefore expect an inverse quadratic decay of the center of mass translation for the separation of large particles, which is what we observe in Fig.3.5a.
The swimmer model similarly predicted a quadratic scaling of self-propulsion ∆Z (cf. 3.6)) with the relative oscillation amplitude. They find that a general three-sphere swimmer must have a phase difference between the prescribed kinematics of the particle pairs, otherwise self-propulsion will not occur. Conversely, when the phase difference reaches an integer multiple of π, the ellipse collapses into a line that does not cover any area.
Therefore, by taking the ratio of the square to the circular area, we predict the relative translation after one period of π/4 (continuously deforming: Golestansk [10]), which is the same ratio that we recover numerically. Since the Stokes regime possesses a much larger leading-order interaction strength, the self-propulsion of the Stokes swimmer is therefore larger in magnitude for well-separated particle pairs.
FURTHER DETAILS ON MASS TENSORS
The gradient of the mass tensor with respect to all particle centers Rα is required for the particles' equations of motion. We denote the gradient of the added mass tensor first component with respect to the tek-th dimension of particle centers as Mij,k(1). The gradient particle introduces an antisymmetry into the mathematics, since the tensor element depends on Ra−Rβ.
Exploiting these symmetries yields the following simplifications in the calculation of the added mass tensor. The calculation of the full-mass tensor gradient is complicated because a tensor, A, must be computed numerically via tensor inversion A = B−1. We can avoid this problem by using the identity tensor to define ∇(B−1) in terms of ∇B and B−1 itself.
RAYLEIGH DISSIPATION FUNCTION
The leading-order expansions yield different analytic forms for each set of {η, ν, λ} particles, depending on their relationship, and are summarized in tensor form as. The semicolon indicates that the subtensor Cνη depends parametrically on the integration surface of particleλ. According to Rayleigh's dissipation theorem, the viscous dissipative force can then be calculated via the velocity derivative axis.
GENERALIZED WORK-ENERGY THEOREM
The work done on the particles by the general force is a state function of the kinetic energy evaluated on both sides of the interval of interest.
OVERVIEW OF STOKESIAN DYNAMICS
![Figure 1.2: Reynolds numbers of swimming and flying animals from Ref. [23]. Organisms from left to right: algae, bacterium, paramecium, nematode, fairyfly, brine shrimp, larval squid, wasp, pteropod, dragonfly, jellyfish, whale, swallow.](https://thumb-ap.123doks.com/thumbv2/123dok/10401316.0/10.918.229.697.111.403/reynolds-swimming-organisms-bacterium-paramecium-nematode-dragonfly-jellyfish.webp)
