Large-scale simulation of steady and time-dependent active suspensions with the force-coupling method

We present a new development of the force-coupling method (FCM) to address the accurate simulation of a large number of interacting micro-swimmers. Our approach is based on the squirmer model, which we adapt to the FCM framework, resulting in a method that is suitable for simulating semi-dilute squirmer suspensions. Other effects, such as steric interactions, are considered with our model. We test our method by comparing the velocity field around a single squirmer and the pairwise interactions between two squirmers with exact solutions to the Stokes equations and results given by other numerical methods. We also illustrate our method's ability to describe spheroidal swimmer shapes and biologically-relevant time-dependent swimming gaits. We detail the numerical algorithm used to compute the hydrodynamic coupling between a large collection ($10^4-10 ^5$) of micro-swimmers. Using this methodology, we investigate the emergence of polar order in a suspension of squirmers and show that for large domains, both the steady-state polar order parameter and the growth rate of instability are independent of system size. These results demonstrate the effectiveness of our approach to achieve near continuum-level results, allowing for better comparison with experimental measurements while complementing and informing continuum models.Comment: 37 pages, 21 figure

Simulating Infinite Vortex Lattices in Superfluids

We present an efficient framework to numerically treat infinite periodic vortex lattices in rotating superfluids described by the Gross-Pitaevskii theory. The commonly used split-step Fourier (SSF) spectral methods are inapplicable to such systems as the standard Fourier transform does not respect the boundary conditions mandated by the magnetic translation group. We present a generalisation of the SSF method which incorporates the correct boundary conditions by employing the so-called magnetic Fourier transform. We test the method and show that it reduces to known results in the lowest-Landau-level regime. While we focus on rotating scalar superfluids for simplicity, the framework can be naturally extended to treat multicomponent systems and systems under more general `synthetic' gauge fields.Comment: 17 pages, 2 figure

Simulating Brownian suspensions with fluctuating hydrodynamics

Fluctuating hydrodynamics has been successfully combined with several computational methods to rapidly compute the correlated random velocities of Brownian particles. In the overdamped limit where both particle and fluid inertia are ignored, one must also account for a Brownian drift term in order to successfully update the particle positions. In this paper, we present an efficient computational method for the dynamic simulation of Brownian suspensions with fluctuating hydrodynamics that handles both computations and provides a similar approximation as Stokesian Dynamics for dilute and semidilute suspensions. This advancement relies on combining the fluctuating force-coupling method (FCM) with a new midpoint time-integration scheme we refer to as the drifter-corrector (DC). The DC resolves the drift term for fluctuating hydrodynamics-based methods at a minimal computational cost when constraints are imposed on the fluid flow to obtain the stresslet corrections to the particle hydrodynamic interactions. With the DC, this constraint need only be imposed once per time step, reducing the simulation cost to nearly that of a completely deterministic simulation. By performing a series of simulations, we show that the DC with fluctuating FCM is an effective and versatile approach as it reproduces both the equilibrium distribution and the evolution of particulate suspensions in periodic as well as bounded domains. In addition, we demonstrate that fluctuating FCM coupled with the DC provides an efficient and accurate method for large-scale dynamic simulation of colloidal dispersions and the study of processes such as colloidal gelation

A fluctuating boundary integral method for Brownian suspensions

We present a fluctuating boundary integral method (FBIM) for overdamped Brownian Dynamics (BD) of two-dimensional periodic suspensions of rigid particles of complex shape immersed in a Stokes fluid. We develop a novel approach for generating Brownian displacements that arise in response to the thermal fluctuations in the fluid. Our approach relies on a first-kind boundary integral formulation of a mobility problem in which a random surface velocity is prescribed on the particle surface, with zero mean and covariance proportional to the Green's function for Stokes flow (Stokeslet). This approach yields an algorithm that scales linearly in the number of particles for both deterministic and stochastic dynamics, handles particles of complex shape, achieves high order of accuracy, and can be generalized to three dimensions and other boundary conditions. We show that Brownian displacements generated by our method obey the discrete fluctuation-dissipation balance relation (DFDB). Based on a recently-developed Positively Split Ewald method [A. M. Fiore, F. Balboa Usabiaga, A. Donev and J. W. Swan, J. Chem. Phys., 146, 124116, 2017], near-field contributions to the Brownian displacements are efficiently approximated by iterative methods in real space, while far-field contributions are rapidly generated by fast Fourier-space methods based on fluctuating hydrodynamics. FBIM provides the key ingredient for time integration of the overdamped Langevin equations for Brownian suspensions of rigid particles. We demonstrate that FBIM obeys DFDB by performing equilibrium BD simulations of suspensions of starfish-shaped bodies using a random finite difference temporal integrator.Comment: Submitted to J. Comp. Phy

Accelerating the force-coupling method for hydrodynamic interactions in periodic domains

The efficient simulation of fluid-structure interactions at zero Reynolds number requires the use of fast summation techniques in order to rapidly compute the long-ranged hydrodynamic interactions between the structures. One approach for periodic domains involves utilising a compact or exponentially decaying kernel function to spread the force on the structure to a regular grid where the resulting flow and interactions can be computed efficiently using an FFT-based solver. A limitation to this approach is that the grid spacing must be chosen to resolve the kernel and thus, these methods can become inefficient when the separation between the structures is large compared to the kernel width. In this paper, we address this issue for the force-coupling method (FCM) by introducing a modified kernel that can be resolved on a much coarser grid, and subsequently correcting the resulting interactions in a pairwise fashion. The modified kernel is constructed to ensure rapid convergence to the exact hydrodynamic interactions and a positive-splitting of the associated mobility matrix. We provide a detailed computational study of the methodology and establish the optimal choice of the modified kernel width, which we show plays a similar role to the splitting parameter in Ewald summation. Finally, we perform example simulations of rod sedimentation and active filament coordination to demonstrate the performance of fast FCM in application

Large-scale simulation of steady and time-dependent active suspensions with the force-coupling method

We present a new development of the force-coupling method (FCM) to address the accurate simulation of a large number of interacting micro-swimmers. Our approach is based on the squirmer model, which we adapt to the FCM framework, resulting in a method that is suitable for simulating semi-dilute squirmer suspensions. Other effects, such as steric interactions, are considered with our model. We test our method by comparing the velocity field around a single squirmer and the pairwise interactions between two squirmers with exact solutions to the Stokes equations and results given by other numerical methods. We also illustrate our method’s ability to describe spheroidal swimmer shapes and biologically-relevant time-dependent swimming gaits. We detail the numerical algorithm used to compute the hydrodynamic coupling between a large collection (10^4–10^5) of micro-swimmers. Using this methodology, we investigate the emergence of polar order in a suspension of squirmers and show that for large domains, both the steady-state polar order parameter and the growth rate of instability are independent of system size. These results demonstrate the effectiveness of our approach to achieve near continuum-level results, allowing for better comparison with experimental measurements while complementing and informing continuum models

Synchronized states of hydrodynamically coupled filaments and their stability

Cilia and flagella are organelles that play central roles in unicellular locomotion, embryonic development, and fluid transport around tissues. In these examples, multiple cilia are often found in close proximity and exhibit coordinated motion. Inspired by the flagellar motion of biflagellate cells, we examine the synchrony exhibited by a filament pair surrounded by a viscous fluid and tethered to a rigid planar surface. A geometrically-switching base moment drives filament motion, and we characterize how the stability of synchonized states depends of the base torque magnitude. In particular, we study the emergence of bistability that occurs when the anti-phase, breast-stroke branch becomes unstable. Using a bisection algorithm, we find the unstable edge-state that exists between the two basins of attraction when the system exhibits bistability. We establish a bifurcation diagram, study the nature of the bifurcation points, and find that the observed dynamical system can be captured by a modified version of Adler’s equation. The bifurcation diagram and presence of bistability reveal a simple mechanism by which the anti-phase breast stroke can be modulated, or switched entirely to in-phase undulations through the variation of a single bifurcation parameter