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

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

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

A generalised drift-correcting time integration scheme for Brownian suspensions of rigid particles with arbitrary shape

32 pages and 11 figuresThe efficient computation of the overdamped, random motion of micron and nanometre scale particles in a viscous fluid requires novel methods to obtain the hydrodynamic interactions, random displacements and Brownian drift at minimal cost. Capturing Brownian drift is done most efficiently through a judiciously constructed time-integration scheme that automatically accounts for its contribution to particle motion. In this paper, we present a generalised drift-correcting (gDC) scheme that accounts for Brownian drift for suspensions of rigid particles with arbitrary shape. The scheme seamlessly integrates with fast methods for computing the hydrodynamic interactions and random increments and requires a single full mobility solve per time-step. As a result, the gDC provides increased computational efficiency when used in conjunction with grid-based methods that employ fluctuating hydrodynamics to obtain the random increments. Further, for these methods the additional computations that the scheme requires occur at the level of individual particles, and hence lend themselves naturally to parallel computation. We perform a series of simulations that demonstrate the gDC obtains similar levels of accuracy as compared with the existing state-of-the-art. In addition, these simulations illustrate the gDC's applicability to a wide array of relevant problems involving Brownian suspensions of non-spherical particles, such as the structure of liquid crystals and the rheology of complex fluids

Large-scale motion in a sperm suspension from From flagellar undulations to collective motion: predicting the dynamics of sperm suspensions

The simulation of 1000 swimmers with stochastically-varying undulation frequencies starting from a polar configuration and interacting though fully resolved hydrodynamics. The domain size is 13.29 swimmer lengths and the effective area fraction is 1.42. There is only one frame per period in order to observe suspension evolution. An image from this simulation is shown in Fig. 3C of the main text

Flagellar undulations in a suspension synchronized sperm from From flagellar undulations to collective motion: predicting the dynamics of sperm suspensions

An excerpt from the simulation of 1000 swimmers with fixed undulation frequencies starting from a polar configuration and interacting through fully resolved hydrodynamics. The domain size is 13.29 swimmer lengths and the effective area fraction is 1.42. There are multiple frames per period to show what is occurring at the time-scale of flagellar undulations. An image from this simulation is shown in Fig. 3A of the main text