Low Reynolds Number Fluid Research Articles

Rayleigh–Taylor turbulence in strongly coupled dusty plasmas

The turbulence mixing initiated by the Rayleigh–Taylor instability has been reported in a two-dimensional (2D) strongly coupled dusty plasma system using classical molecular dynamics simulation. The entire evolution cycle, including the initial equilibrium, the instability, turbulent mixing, and, finally, a new equilibrium through the thermalization process, has been demonstrated via the respective energy spectra. The fully developed spectrum follows the Bolgiano-Obukho k−11/5 scaling at smaller wavenumbers, a characteristic 2D buoyancy-driven turbulent flow feature. At higher wavenumbers, the energy spectrum E(k)∝k represents the thermalization of the system and is a characteristic feature of 2D Euler turbulence. At longer timescales, the system reflects the Kolmogorov scale of k−3. Moreover, strong coupling slows the turbulent mixing process, though the final state is a complete thermalized system. Our results also help us to understand the thermalization process in Yukawa fluids, other strongly coupled plasma families, and turbulent mixing in low Reynolds number fluids.

Influence of sedimentary structure and pore-size distribution on upscaling permeability and flow enhancement due to liquid boundary slip: A pore-scale computational study

Low-permeability sedimentary formations, such as tight sandstones, exhibit fluid flow and transport phenomena distinct from those in conventional porous systems due to the dominance of micro- to nanometer-sized pores and variable amounts of boundary slip. The widely used traditional no-slip boundary condition often fails to accurately describe fluid behavior in these formations. A knowledge gap exists in understanding how liquid slip influences fluid dynamics in complex, heterogeneous sedimentary structures, as previous studies have primarily focused on simplified, homogeneous pore geometries. In this study, we investigated the impact of boundary slip on low-Reynolds number fluid dynamics within synthetically designed two-dimensional graded and random pore networks with varying pore-size distributions to account for heterogeneity. Our results showed that velocity variance increased with increasing heterogeneity, following a power-law relationship. The power-law exponents decreased with boundary slip, quantifying how boundary slip mitigated the impact of heterogeneity on velocity variance. We developed a theoretical model to predict asymptotic flow enhancement and derived constitutive relations to estimate the coefficient C and maximum flow enhancement (ΔE) based on the pore-to-grain size ratio and porosity. Energy dissipation increased with both heterogeneity and boundary slip, which we identified as the primary mechanism contributing to asymptotic flow enhancement. This relationship was illustrated by a 1:1 linear correlation between maximum energy dissipation and maximum flow enhancement, regardless of heterogeneity, indicating that energy dissipation due to boundary slip entirely controls the emerging fluid dynamics. The presented theoretical model and constitutive equations offer practical applications for optimizing fluid dynamics in heterogeneous formations.

Transportation Efficiency of Hydrodynamically Coupled Spherical Oscillators in Low Reynolds Number Fluids

Transportation Efficiency of Hydrodynamically Coupled Spherical Oscillators in Low Reynolds Number Fluids

Generating, from scratch, the near-field asymptotic forms of scalar resistance functions for two unequal rigid spheres in low Reynolds number flow

The motion of rigid spherical particles suspended in a low Reynolds number fluid can be related to the forces, torques, and stresslets acting upon them by 22 scalar resistance functions, commonly notated X11A, X12A, Y11A, etc. Near-field asymptotic forms of these resistance functions were derived in Jeffrey and Onishi [“Calculation of the resistance and mobility functions for two unequal rigid spheres in low-Reynolds-number flow,” J. Fluid Mech. 139, 261–290 (1984)] and Jeffrey [“The calculation of the low Reynolds number resistance functions for two unequal spheres,” Phys. Fluids A 4, 16–29 (1992)]; these forms are now used in several numerical methods for suspension mechanics. However, the first of these important papers contains a number of small errors, which make it difficult for the reader to correctly evaluate the functions for parameters not explicitly tabulated. This short article comprehensively corrects these errors and adds formulas that were originally omitted from both papers, so that the reader can verify and implement the equations independently. The corrected expressions, rationalized and using contemporary nondimensionalization, are shown to match mid-field values of these scalars, which are calculated through an alternative method. A Python script to generate and evaluate these functions is provided.

EWOD Chip with Micro-Barrier Electrode for Simultaneous Enhanced Mixing during Transportation.

Digital microfluidic platforms have been extensively studied in biology. However, achieving efficient mixing of macromolecules in microscale, low Reynolds number fluids remains a major challenge. To address this challenge, this study presents a novel design solution based on dielectric electro-wetting (EWOD) by optimizing the geometry of the transport electrode. The new design integrates micro-barriers on the electrodes to generate vortex currents that promote mixing during droplet transport. This design solution requires only two activation signals, minimizing the number of pins required. The mixing performance of the new design was evaluated by analyzing the degree of mixing inside the droplet and quantifying the motion of the internal particles. In addition, the rapid mixing capability of the new platform was demonstrated by successfully mixing the sorbitol solution with the detection solution and detecting the resulting reaction products. The experimental results show that the transfer electrode with a micro-barrier enables rapid mixing of liquids with a six-fold increase in mixing efficiency, making it ideal for the development of EWOD devices.

The N-Link Swimmer in Three Dimensions: Controllability and Optimality Results

The controllability of a fully three-dimensional N-link swimmer is studied. After deriving the equations of motion in a low Reynolds number fluid by means of Resistive Force Theory, the controllability of the minimal 2-link swimmer is tackled using techniques from Geometric Control Theory. The shape of the 2-link swimmer is described by two angle parameters. It is shown that the associated vector fields that govern the dynamics generate, via taking their Lie brackets, all eight linearly independent directions in the combined configuration and shape space, leading to controllability; the swimmer can move from any starting configuration and shape to any target configuration and shape by operating on the two shape variables. The result is subsequently extended to the N-link swimmer. Finally, the minimal time optimal control problem and the minimization of the power expended are addressed and a qualitative description of the optimal strategies is provided.

Light-propelled self-sustained swimming of a liquid crystal elastomer torus at low Reynolds number

The well-known Purcell's toroidal swimmer can translate in low Reynolds number fluid, by periodically changing its own shape in a nonreciprocal way to overcome the scallop theorem originating from the time independence of the Stokes equation. Based on a liquid crystal elastomer (LCE) torus motor with zero-elastic-energy mode, this paper proposes a mechanism for light-powered self-sustained swimming of a torus at low Reynolds number. By adopting the well-established dynamic LCE model and viscous hydrodynamics for the motion of a thin torus in Stokes flow, the self-sustained swimming of the LCE torus under steady and homogeneous illumination is theoretically investigated. The light-driven moment propelling the self-sustained swimming is derived, and the translation velocity and rotation angular velocity of the LCE torus swimmer in Stokes fluid are analytically formulated by force balance and moment equilibrium. The results show that the translation velocity and rotation angular velocity of the self-sustained swimming only depend on several simple dimensionless parameters, including light intensity, penetration depth, elastic modulus, curvature and payload. Especially, there exists a critical payload for reverse swimming in the opposite direction. The results provide a theoretical basis for the LCE-based self-propelled motor with zero-elastic-energy mode, and the self-propulsion mechanism may also inspire the design of self-sustained swimmers based on other stimuli-responsive toruses.

Metachronal patterns by magnetically-programmable artificial cilia surfaces for low Reynolds number fluid transport and mixing.

Motile cilia can produce net fluid flows at low Reynolds number because of their asymmetric motion and metachrony of collective beating. Mimicking this with artificial cilia can find application in microfluidic devices for fluid transport and mixing. Here, we study the metachronal beating of nonidentical, magnetically-programmed artificial cilia whose individual non-reciprocal motion and collective metachronal beating pattern can be independently controlled. We use a finite element method that accounts for magnetic forces, cilia deformation and fluid flow in a fully coupled manner. Mimicking biological cilia, we study magnetic cilia subject to a full range of metachronal driving patterns, including antiplectic, symplectic, laeoplectic and diaplectic waves. We analyse the induced primary flow, secondary flow and mixing rate as a function of the phase lag between cilia and explore the underlying physical mechanism. Our results show that shielding effects between neighboring cilia lead to a primary flow that is larger for antiplectic than for symplectic metachronal waves. The secondary flow can be fully explained by the propagation direction of the metachronal wave. Finally, we show that the mixing rate can be strongly enhanced by laeoplectic and diaplectic metachrony resulting in large velocity gradients and vortex-like flow patterns.

Effect of Shear Displacement and Stress Changes on Fracture Hydraulic Aperture and Flow Anisotropy

Fluid flow in fractures has the potential to drastically change the economic, environmental, and safety risks associated with a subsurface operation. This work focuses on the analysis of experimental shear fracturing and subsequent shear displacement of a shale specimen that contains multiple pre-existing natural fractures. A triaxial direct shear apparatus with concurrent µCT imaging allowed continuous monitoring of sample permeability and changes in fracture geometry during fracture creation, displacement, and changes in stress. Steady-state, low Reynolds number fluid flow simulations were used to determine the changes of each individual fracture’s hydraulic aperture due to displacement and effective stress. The development of anisotropy in the hydraulic aperture of a generated shear fracture was determined for shear displacements of 0.38, 0.74, and 2.04 mm. At high confining stress (30 MPa), shear displacement significantly reduced the hydraulic aperture of the shear fracture to below flow detectable limits and X-ray imaging resolution (23 × 23 × 23 µm3). The shear generated fracture was found to be more sensitive to changes in effective stress compared to the pre-existing natural fractures. Existing models derived from fluid flow equations accounting for stochastic fracture roughness were able to partially fit the simulated flow properties of the pre-existing fractures, but no model was found to fit the flow properties of the newly formed shear fracture. These results are the first report of simultaneous changes in the geometry and flow properties of surrounding, parallel natural fractures as a shear fracture is generated with varying stress and shear displacement conditions.

Regularized Stokeslets Lines Suitable for Slender Bodies in Viscous Flow

Slender-body approximations have been successfully used to explain many phenomena in low-Reynolds number fluid mechanics. These approximations typically use a line of singularity solutions to represent flow. These singularities can be difficult to implement numerically because they diverge at their origin. Hence, people have regularized these singularities to overcome this issue. This regularization blurs the force over a small blob and thereby removing divergent behaviour. However, it is unclear how best to regularize the singularities to minimize errors. In this paper, we investigate if a line of regularized Stokeslets can describe the flow around a slender body. This is achieved by comparing the asymptotic behaviour of the flow from the line of regularized Stokeslets with the results from slender-body theory. We find that the flow far from the body can be captured if the regularization parameter is proportional to the radius of the slender body. This is consistent with what is assumed in numerical simulations and provides a choice for the proportionality constant. However, more stringent requirements must be placed on the regularization blob to capture the near field flow outside a slender body. This inability to replicate the local behaviour indicates that many regularizations cannot satisfy the no-slip boundary conditions on the body’s surface to leading order, with one of the most commonly used blobs showing an angular dependency of velocity along any cross section. This problem can be overcome with compactly supported blobs, and we construct one such example blob, which can be effectively used to simulate the flow around a slender body.

Velocity of suspended fluid particles in a low Reynolds number converging flow

We studied a pressure-driven, low Reynolds number fluid flow through a planar channel whose spanwise width along the flow varied inversely as the streamwise coordinate such that the extensional rate on the centerline was near constant. The effect of the near constant extensional rate on an immiscible droplet of silicone oil was studied by tracking its deformation. The droplet rapidly deformed into an ellipsoid and displayed a consistent lag velocity compared to the single phase background flow at the same point. The observations were attributed to the flow induced deformation of the immiscible droplet, which was a function of the magnitude of the initial capillary number. The streamwise component of the single phase velocity along the centerline of the converging flow was also estimated as leading order using lubrication theory. The estimated velocity is compared favorably with numerical simulations; validation with experimental measurement of the flow of castor oil through the channel by tracking tracer particles is performed. The accuracy of the determination of the velocity field by the lubrication theory allowed for the careful measurement of the velocity difference between the drop and suspended fluid velocities. This research validated lubrication theory predictions of the flow velocity through a converging channel and provided an experimental insight into the behavior of a suspended phase.

Metachronal patterns in artificial cilia for low Reynolds number fluid propulsion.

Cilia are hair-like organelles, present in arrays that collectively beat to generate flow. Given their small size and consequent low Reynolds numbers, asymmetric motions are necessary to create a net flow. Here, we developed an array of six soft robotic cilia, which are individually addressable, to both mimic nature's symmetry-breaking mechanisms and control asymmetries to study their influence on fluid propulsion. Our experimental tests are corroborated with fluid dynamics simulations, where we find a good agreement between both and show how the kymographs of the flow are related to the phase shift of the metachronal waves. Compared to synchronous beating, we report a 50% increase of net flow speed when cilia move in an antiplectic wave with phase shift of -π/3 and a decrease for symplectic waves. Furthermore, we observe the formation of traveling vortices in the direction of the wave when metachrony is applied.

Controlled Noncontact Manipulation of Nonmagnetic Untethered Microbeads Orbiting Two-Tailed Soft Microrobot

A rotating two-tailed soft microrobot induces a frequency dependent flow-field in low Reynolds number fluids. We use this flow-field to achieve noncontact manipulation of nonmagnetic microbeads with average diameter of 30 μm in 2-D space. Our noncontact manipulation strategy capitalizes on exerting a rotational magnetic torque on the magnetic dipole of the microrobot. The induced flow-field enables microbeads in the surrounding fluid to orbit the microrobot along a sprocketlike trajectory due to a periodic and asymmetric flow-field caused by the two tails. A hydrodynamic model of the two-tailed microrobot and the orbiting microbeads is developed based on the method of regularized Stokeslets for computing Stokes flows. The relations between the angular velocity of the orbiting microbeads and the rotation frequency of the microrobot, their proximity (p), and tail length ratio of the microrobots are studied theoretically and experimentally. Our simulations and experimental results show that the angular velocity of the orbiting microbeads decreases nearly as | p | <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-2</sup> with the distance to the microrobot and its tail length ratio. We also demonstrate closed-loop control of the microbeads toward target positions along sprocketlike trajectories with an average position error of 23.1 ± 9.1 μm (n = 10), and show the ability to swim away without affecting the positioning accuracy after manipulation.

Emergence of metachronal waves in active microtubule arrays

The physical mechanism behind the spontaneous formation of metachronal waves in microtubule arrays in a low Reynolds number fluid has been of interest for the past several years, yet is still not well understood. We present a model implementing the hydrodynamic coupling hypothesis from first principles, and use this model to simulate kinesin-driven microtubule arrays and observe their emergent behavior. The results of simulations are compared to known experimental observations by Sanchez et al. By varying parameters, we determine regimes in which the metachronal wave phenomenon emerges, and categorize other types of possible microtubule motion outside these regimes.

Controlled propulsion of wheel-shape flaky microswimmers under rotating magnetic fields

Effective propulsion of untethered micro-/nanorobots at low Reynolds numbers can offer possibilities for promising biomedical applications. Diverse locomotion modes have been proposed for propulsion at a small scale, and rolling is an alternative method which is significantly effective. Here, we demonstrate mass produced magnetic wheel-shape flaky microswimmers fabricated via a simple and cost-effective method. Locomotion behaviors under vertical rotating magnetic fields were studied, and the propulsion mechanisms were analyzed. They exhibited two modes to swim forward as tumbling and rolling, which relied on the actuating field and the fluid. The rolling microswimmers could be propelled and steered precisely and a high velocity can be easily reached. Forward velocity and transition frequency within diverse fields and fluids were analyzed, and side slip effects when rolling at a camber angle were also observed. Such microswimmers synthesized in bulk with alternative locomotion modes and excellent swimming performances may have potential in low Reynolds number fluids.

Coupled Self-Organized Hydrodynamics and Stokes Models for Suspensions of Active Particles

We derive macroscopic dynamics for self-propelled particles in a fluid. The starting point is a coupled Vicsek–Stokes system. The Vicsek model describes self-propelled agents interacting through alignment. It provides a phenomenological description of hydrodynamic interactions between agents at high density. Stokes equations describe a low Reynolds number fluid. These two dynamics are coupled by the interaction between the agents and the fluid. The fluid contributes to rotating the particles through Jeffery’s equation. Particle self-propulsion induces a force dipole on the fluid. After coarse-graining we obtain a coupled Self-Organised Hydrodynamics–Stokes system. We perform a linear stability analysis for this system which shows that both pullers and pushers have unstable modes. We conclude by providing extensions of the Vicsek–Stokes model including short-distance repulsion, finite particle inertia and finite Reynolds number fluid regime.

An in-situ method of identifying ingredients of zeotropic refrigerants by inversion problem technique

An inversion method is applied to identify ingredients of zeotropic refrigerants in a circular duct. In the case of low Reynolds number and constant fluid pressure, the temperature distribution of direct heat transfer problem can be solved numerically. The thermophysical parameters of zeotropic refrigerants are determined by using inversion problem technique, the ingredients of refrigerants can be identified eventually. An in-situ experimental apparatus was proposed and three test samples with different composition refrigerants were conducted in this study. The experiment results show that the relative error of ingredients identification can be limited within 8.33%.

Symmetry-Breaking Cilia-Driven Flow in Embryogenesis

The systematic breaking of left–right body symmetry is a familiar feature of human physiology. In humans and many animals, this process originates with asymmetric fluid flow driven by rotating cilia, occurring in a short-lived embryonic organizing structure termed the node. The very low–Reynolds number fluid mechanics of this system is reviewed; important features include how cilia rotation combines with tilt to produce asymmetric flow, boundary effects, time dependence, and the interpretation of particle tracking experiments. The effect of perturbing cilia length and number is discussed and compared in mouse and zebrafish. Whereas understanding of this process has advanced significantly over the past two decades, there is still no consensus on how flow is converted to asymmetric gene expression, with most research focusing on resolving mechanical versus morphogen sensing. The underlying process may be more subtle, probably involving a combination of these effects, with fluid mechanics playing a central role.

Pairwise controllability and motion primitives for micro-rotors in a bounded Stokes flow

Micro-robots that can propel themselves in a low Reynolds number fluid flow by converting their rotational motion into translation have begun attracting much attention due to their ease of fabrication. The dynamics and controllability of the motion of such microswimmers are investigated in this paper. The microswimmers under consideration here are spinning spheres (or rotors) whose dynamics are approximated by rotlets, a singularity solution of the Stokes equations. While singularities of Stokes flows are commonly used as theoretical models for microswimmers and micro-robots, rotlet models of microswimmers have received less attention. While a rotlet alone cannot generate translation, a pair of rotlets can interact and execute net motion. Taking the control inputs to be the strengths of the micro rotors, the positions of a pair of rotors are not controllable in an unbounded planar fluid domain. However, in a bounded domain, which is often the case of practical interest, we show that the positions of the micro rotors are controllable. This is enabled by the interaction of the rotors with the boundaries of the domain. We show how control inputs can be constructed based on combinations of Lie brackets to move the rotors from one point to another in the domain. Another contribution of this paper is the creation of a framework for path planning and control of the motion of Stokes singularities that model the dynamics of microswimmers. This can be extended to microswimmers with other shapes moving in confined fluid domains with complex boundaries.

Independent Actuation of Two-Tailed Microrobots

A soft two-tailed microrobot in low Reynolds number fluids does not achieve forward locomotion by identical tails regardless to its wiggling frequency. If the tails are nonidentical, zero forward locomotion is also observed at specific oscillation frequencies (which we refer to as the reversal frequencies), as the propulsive forces imparted to the fluid by each tail are almost equal in magnitude and opposite in direction. We find distinct reversal frequencies for the two-tailed microrobots based on their tail length ratio. At these frequencies, the microrobot achieves negligible net displacement under the influence of a periodic magnetic field. This observation allows us to fabricate groups of microrobots with tail length ratio of 1.24 ± 0.11, 1.48 ± 0.08, and 1.71 ± 0.09. We demonstrate selective actuation of microrobots based on prior characterization of their reversal frequencies. We also implement simultaneous flagellar propulsion of two microrobots and show that they can be controlled to swim along the same direction and opposite to each other using common periodic magnetic fields. In addition, independent motion control of two microrobots is achieved toward two different reference positions with average steady-state error of 110.1 ± 91.8 μm and 146.9 ± 105.9 μm.