Number Of Vortices Research Articles

Induction of Controllable Vortical Flow in a Dual-Stenosis Aorta Model: A Replication of Disordered Eddies Flow in Aneurysms.

This paper presents a two-stenosis aorta model mimicking vortical flow in vascular aneurysms. More specifically, we propose to virtually induce two adjacent stenoses in the abdominal aorta to develop various vortical flow zones post stenoses. Computational fluid dynamics (CFD) simulations were conducted for the virtual two-stenosis model based on physiological and anatomical data (i.e., diameters, flow rate waveforms) from adult rabbits. The virtual model includes adult rabbits' infra-renal portion of the aorta and iliac arteries. 3D CFD simulations in five different dual-stenosis configurations were performed using a commercial CFD package (FLUENT). In-house software assessed the evolution of flow vortices. Notably, spatial-temporally averaged wall shear stress (STA-WSS) and oscillatory shear index (OSI), the total volume of vortex flow, the number of vortices, and the phase-to-phase overlap of vortex flow within each region were evaluated. In all models, we found consistent patterns of the vortex flow parameters, indicating that the adjacent stenoses induced three different hemodynamic zones, namely, stable vortical flow (after the first stenosis), transient vortical flow (after the second stenosis), and unstable vortical flow (further distal to the second stenosis). Also, different degrees of flow disturbance can be achieved in these three zones. It is significant to note that, although the 'dual-stenosis' geometry is completely hypothetical, it allows us to create various vortical flows in consecutive vessel segments for the first time. As a result, if implemented as a pre-clinical model, the proposed two-stenosis model offers an attractive, tunable environment to investigate the interplays between subject-specific hemodynamics and vascular remodeling. This aspect remains in our future directions.

Rapidly rotating quantum droplets confined in a harmonic potential

We consider a “symmetric” quantum droplet in two spatial dimensions, which rotates in a harmonic potential, focusing mostly on the limit of “rapid” rotation. We examine this problem using a purely numerical approach, as well as a semianalytic Wigner-Seitz approximation (first developed by Baym, Pethick, and their co-workers) for the description of the state with a vortex lattice. Within this approximation we assume that each vortex occupies a cylindrical cell, with the vortex-core size treated as a variational parameter. Working with a fixed angular momentum, as the angular momentum increases and depending on the atom number, the droplet accommodates none, few, or many vortices, before it turns to center-of-mass excitation. For the case of a “large” droplet, working with a fixed rotational frequency of the trap Ω, as Ω approaches the trap frequency ω, a vortex lattice forms, the number of vortices increases, the mean spacing between them decreases, while the “size” of each vortex increases as compared to the size of each cell. In contrast to the well-known problem of contact interactions, where we have melting of the vortex lattice and highly correlated many-body states, here no melting of the vortex lattice is present, even when Ω=ω. This difference is due to the fact that the droplet is self-bound. For Ω=ω, the “smoothed” density distribution becomes a flat top, very much like the static unconfined droplet. When Ω exceeds ω, the droplet maintains its shape and escapes to infinity, via center-of-mass motion. Published by the American Physical Society 2024

Analytic approach for computation of topological number of integrable vortex on torus

Detailed structures of vortices on a torus are discovered by performing an analytic method to calculate the vortex number. We focus on analytic vortex solutions to the Chern-Simons-Higgs theory, whose governing equation is the so-called Jackiw-Pi equation. The equation is one of the integrable vortex equations and is reduced to Liouville’s equation. The requirement of continuity of the Higgs field strongly restricts the characteristics and the fundamental domain of the vortices. Also considered are the decompactification limits of the vortices on a torus, in which “flux loss” phenomena occasionally occur.

The Similarity of Quasi-geostrophic Vortices Against the Background of Large-Scale Barotropic Currents

The paper proposes a theory of similarity of quasi-geostrophic vortices against the background of large-scale flows. This information is useful when planning laboratory and numerical experiments to study mesoscale and submesoscale vortex dynamics of vortices interacting with currents. Special attention is paid to the study of geometric similarity of phenomena. It is revealed that the complete set of dimensionless similarity numbers of baroclinic vortices includes four dimensionless parameters: the dimensionless intensity of the vortex, the geometric similarity of the background flow (the ratio of relative vorticity to the deformation coefficient of the background flow), the coefficient of horizontal elongation of the vortex core and the coefficient of vertical oblateness of the vortex core coinciding with the Burger number. To describe the similarity of barotropic vortices against the background of barotropic flows, the number of necessary dimensionless parameters is reduced by one number — the coefficient of vertical oblateness of the vortex core is eliminated from consideration. When studying axisymmetric vortices or vortex structures close to axisymmetric, another geometric parameter of the vortex is eliminated from consideration — the coefficient of horizontal elongation of the vortex core. As a result, the maximum possible set of similarity parameters includes four dimensionless numbers, and the minimum is two.

Density Waves of Vortex Fluids on a Sphere

Abstract We aim to find one highly nontrivial example of the solutions to the vortex fluid dynamical equation on the unit sphere (S^2) and compare it with the numerical simulation. Since the rigid rotating steady solution for vortex fluids on S^2 is
already known to us, we consider the perturbations above it. After decomposing the perturbation of the vortex number density and vortex charge density into spherical harmonics, we find that the perturbations are propagating waves. To
be precise, the velocities for different single-mode vortex number density waves are all the same, while the velocities for single-mode vortex charge density waves depend on the degree of the spherical harmonics l, which is a signal of the existence of dispersion. Meanwhile, we find that there is a beat phenomenon for the positive (or negative) vortex density wave. Numerical simulation based on the canonical equations for point vortex model agrees perfectly with our theoretical calculations.

Simulating the Rising Air Bubbles Stream through Heavy Fuel Oil Residue Bubble Column Reactor using COMSOL

Several industrial operations rely on the rise of air bubbles through heavy fuel oil residue (HFOr), including oil recovery and wastewater treatment. In order to increase efficiency and decrease expenses, it is necessary to understand the peculiarities of this process. With the help of COMSOL Multiphysics, we numerically simulate the ascent of a single air bubble via HFOr in this article. The dimensions of the container used in the simulations were (100) mm in diameter and (200) mm in height, with an air bubble diameter of 4.5 mm. For an air bubble ascending through an HFOr column, the predicted numerical outcomes are contrasted with nine experimental versions. As the bubble rose higher, the region with the greatest number of vortices was located on its side. The remaining vortex is contained inside the bubble. In addition, since drag decreases the flow, a lighter fluid usually exhibits a region of strong vortex and low pressure. Because of the oscillation effect, the results reveal that the rising velocity initially rises and then gradually falls between 0.6 and 0.65 s. The air bubbles' wavy motion is caused by the high fluid density, which becomes worse as the air velocity gets higher. The values of the drag coefficient rise sharply with decreasing air extrusion speed. It turns out that the height of the fluid has an inverse relationship with the pressure.

Anti-matching effect in a two dimensional driven vortex lattice in the presence of periodic pinning

The dynamics of a driven superconducting vortex lattice in a two-dimensional (2D) periodic potential of square symmetry is studied using Brownian dynamics simulations. The range and strength of the vortex-substrate interaction are taken to be of the same order as that of the vortex-vortex interaction. The matching effect in a driven vortex lattice in the presence of a periodic array of pinning centers refers to the enhanced resistance to the vortex lattice motion when the ratio of the number of vortices to the number of pinning centers (called the filling fraction) takes simple fractional values. In particular, one expects a pronounced matching effect when the filling fraction is one. Contrary to this expectation, a drop in the vortex lattice mobility is observed as the filling fraction is increased from value one. This anti-matching effect can be understood in terms of the structural change in the vortex lattice as the filling fraction is varied. The dip observed in vortex mobility as a function of temperature when the filling fraction equals one (Joseph T 2020PhysicaA556124737), is studied for other values of filling above and below one. The behavior is found to persist for other fillings as well and is associated with the melting of the vortex lattice. The temperature at which the lattice melts is found to increase with drive and explains the shift in the temperature at which mobility is a minimum, locally.

Multi‐Concentric Generalized Perfect Vortex Beams by Hybrid Phase Metasurfaces

AbstractMetasurface‐based vortex beams are expected to overcome the current communication challenges of massive data and limited bandwidth by their theoretical infinite and mutually orthogonal states. However, limited by the spatial extension of array vortices, only finite orbital angular momentum (OAM) eigenstates are available, of which the expansion will boost the bandwidth toward an inexhaustible stage. In this manuscript, multi‐concentric generalized perfect vortex beams (G‐PVBs) are experimentally generated by the fabricated hybrid phase metasurfaces integrating generalized focusing phases and circular domain function into vortex phases. Diverse vortex shapes and helical lobes within beam patterns and spiral interferograms are both arranged in a non‐overlapping concentric manner instead of spatial arrays, respectively, breaking the limitation of symmetric shape while avoiding inter‐modal cross‐talk. The vortex shape, topological charge and vortex number are decoupled mutually in orthogonal polarization states, realizing both space and polarization multiplexing. Furthermore, shaped into anomalous profiles with asymmetric states of polarization, multi‐concentric vector G‐PVBs are experimentally generated to transmit parallelly stable order information with high robustness in turbulence environments. The proposed multi‐concentric G‐PVBs simultaneously expanded the novel vortex dimension by anomalous shapes and information channels of multi‐concentric OAM modes, providing potential application in high‐capacity vortex optical communication and information encryption.

Interaction of inertia and magnetic force in the liquid metal flow past a magnetic obstacle

In this paper, we present a numerical study of the three-dimensional behavior of a liquid metal flow in an insulating rectangular duct of narrow cross section past a localized magnetic field (i.e., a magnetic obstacle) produced by two parallel square magnets arranged externally on the walls of the duct. A series of simulations are conducted focused mainly on describing the interplay between inertial and magnetic forces in a wide range of interaction parameters (1.8<N<48) by varying the Reynolds number while the Hartmann number is kept fixed (Ha = 75). The analyzed configuration coincides with that studied experimentally by Domínguez et al. [“Experimental and theoretical study of the dynamics of wakes generated by magnetic obstacles,” Magnetohydrodynamics 51(2), 215–224 (2015)] and, as a first step, experimental data from local variables (streamwise velocity component) and global parameters (oscillation frequency and kinetic energy of the wake) are consistently replicated by the numerical model. Furthermore, to complement the flow phenomenology, the transition to different flow structures as the interaction parameter varies is explored. It is found that when the magnetic forces predominate over inertia, stationary vortex patterns with two, four, and six vortices appear while, unlike the hydrodynamic flow past a bluff body, the increase in inertial effects leads to a reduction in the number of vortices and eventually to their disappearance, reaching a state in which the magnetic obstacle becomes imperceptible to the flow. The existence of a critical value of the interaction parameter that maximizes the kinetic energy of the wake is confirmed numerically and corroborated from the experimental data.

Hidden vortices and Feynman rule in Bose-Einstein condensates with density-dependent gauge potential.

In this paper, we numerically investigate the vortex nucleation in a Bose-Einstein condensate (BEC) trapped in a double-well potential and subjected to a density-dependent gauge potential. A rotating Bose-Einstein condensate, when confined in a double-well potential, not only gives rise to visible vortices but also produces hidden vortices. We have empirically developed Feynman's rule for the number of vortices versus angular momentum in Bose-Einstein condensates in the presence of density-dependent gauge potentials. The variation of the average angular momentum with the number of vortices is also sensitive to the nature of the nonlinear rotation due to the density-dependent gauge potentials. The empirical result agrees well with the numerical simulations and the connection is verified by means of curve-fitting analysis. The modified Feynman rule is further confirmed for the BECs confined in harmonic and toroidal traps. In addition, we show the nucleation of vortices in double-well and toroidally confined Bose-Einstein condensates solely through nonlinear rotations (without any trap rotation) arising through the density-dependent gauge potential.

Soliton sheets of Bose–Einstein condensates in optical lattices

Soliton sheets are observed in Bose–Einstein condensates in optical lattice which are formed by superposition of condensates occupying different single-particle states. These structures consist of one-dimensional stationary solitons distributed in the x-direction arranged continuously along the peaks of the optical lattice in the y-direction. Notably, the phase difference across the soliton sheets is periodic and varies linearly with y within each period. So, we refer to this configuration as a ‘soliton sheet’. A velocity difference in the y-component is observed between the two sides of the soliton sheets. Similar velocity distributions can be achieved by aligning an infinite number of isotropic vortices along the peaks of the optical lattice. And the soliton sheets are distinguished by their lack of dependence on phase singularities. This independence enables the formation of soliton sheets even in the absence of phase singularities, highlighting a unique aspect of this structure.

Perfect Off-Axis Optical Vortex Lattice

Optical vortex lattices (OVLs) with diverse modes show potential for a wide range of applications, such as high-capacity optical communications, optical tweezers, and optical measurements. However, vortices in typical regulated OVLs often exhibit irregular shapes, such as being narrow and elongated. The resulting increase in asymmetry negatively impacts the efficiency of particle trapping. Additionally, the vortex radii expand with an increase in topological charge (TC), limiting the TC value of the vortices and hindering their ability to fully utilize orbital angular momentum (OAM). Herein, we propose an alternative approach to custom OVLs using off-axis techniques combined with amplitude modulation. Amplitude modulation enables the precise generation of an OVL with perfect vortex properties, known as a perfect off-axis OVL. Further, the number of vortices in the perfect off-axis OVL, the off-axis distances, and the TC can be freely modulated while maintaining a circular mode. This unique OVL will promote new applications, such as the complex manipulation of multi-particle systems and optical communication based on OAM.

Numerical Analysis of Flow Structure Evolution during Scour Hole Development: A Case Study of a Pile-Supported Pier with Partially Buried Pile Cap

This study numerically investigates a pile-supported pier, which comprises a column with a partially buried pile cap and a group of piles, recognizing that partially buried pile caps lead to the highest scour depth. Most research has focused on equilibrium scour conditions in laboratory settings, overlooking the detailed dynamics of horseshoe vortices around pile groups. This study aims to clarify the flow structure and vortex dynamics at a pile-supported pier during local scour hole development stages using an in-house developed numerical model. The model’s accuracy is validated against flat-channel and compound pier reference cases. For the pile-supported pier, fixed bed geometry was used in flow simulations at selected scouring stages. Results show significant changes in flow structure and vortex formation with scour hole time development, particularly as the bed surface moves away from the pile cap. The study reveals variations in vortex size, number, and positioning, alongside turbulent kinetic energy and Reynolds shear stress distributions over time. High positive Reynolds shear stress near the bed during intermediate scouring stages highlights the complex interactions within the flow field. This research provides the first detailed visualization of flow structure evolution within a scour hole at a pile-supported pier.

Performance analysis and optimization of the Gyroid-type triply periodic minimal surface heat sink incorporated with fin structures

The Triply Periodic Minimal Surface (TPMS) structures demonstrate substantial value and development potential in applications and research concerning heat sinks and heat exchangers. To further enhance the convective heat transfer performance of TPMS, this study proposes a novel method of incorporating fin structures into TPMS. This method is applicable to various types of TPMS, and the new structure constructed using this method is termed “TPMS with Fin Structures (TFS).” Subsequently, a TFS-Gyroid heat sink based on the Gyroid-type TPMS was constructed. This study presents a calculation method for the fin height in TFS-Gyroid (HTFS-Gyroid). Numerical simulation methods were employed to investigate the influence and mechanism of HTFS-Gyroid on the convective heat transfer performance and flow resistance of the TFS-Gyroid heat sink model, with air as the working fluid and under constant wall temperature (100 °C) conditions. The results indicate that within the flow range of 24–120 L/min, as HTFS-Gyroid increases from 0 to 1.6 mm, the average surface convective heat transfer coefficient (h̅) of the TFS-Gyroid heat sink model increases by 27.4 %-34.6 %, and the inlet–outlet pressure drop (ΔP) increases by 42.5–632.8 Pa. As HTFS-Gyroid increases, the PEC of the TFS-Gyroid heat sink model gradually decreases, whereas j/f gradually increases. The primary mechanisms by which the fin structures enhance the convective heat transfer performance of the TFS-Gyroid heat sink are: 1) increased heat exchange area, 2) the fin structures induce a large number of longitudinal vortices and secondary flows within the flow channels, and 3) the formation of “Fins Occupying the Perforated Structure (FOPS)” within the flow channels.

Tailored micromixing in chemically patterned microchannels undergoing electromagnetohydrodynamic flow.

The study unveils a simple, non-invasive method to perform micromixing with the help of spatiotemporal variation in the Lorentz force inside a microchannel decorated with chemically heterogeneous walls. Computational fluid dynamics simulations have been utilized to investigate micromixing under the coupled influence of electric and magnetic fields, namely, electromagnetohydrodynamics, to alter the direction of the Lorentz force at the specific locations by creating the reverse flow zones where the pressure gradient, . The study explores the impact of periodicity, distribution, and size of electrodes alongside the magnitude of applied field intensity, the flow rate of the fluid, and the nature of the electric field on the generation of the mixing vortices and their strength inside the microchannels. The results illustrate that the wall heterogeneities can indeed enforce the formation of localized on-demand vortices when the strength of the localized reverse flow overcomes the inertia of the mainstream flow. In such a scenario, while the vortex size and strength are found to increase with the size of the heterogeneous electrodes and field intensities, the number of vortices increases with the number of heterogeneous electrodes decorated on the channel wall. The presence of a non-zero pressure-driven inflow velocity is found to subdue the strength of the vortices to restrict the mixing facilitated by the localized variation of the Lorentz force. Interestingly, the usage of an alternating current (AC) electric field is found to provide an additional non-invasive control on the mixing vortices by enabling periodic changes in their direction of rotation. A case study in this regard discloses the possibility of rapid mixing with the usage of an AC electric field for a pair of miscible fluids inside a microchannel.

Numerical investigation on characteristics of vortex dissipation in multi-horizontal submerged jets stilling basin.

Multi-horizontal submerged jets stilling basins have been utilized in large-scale water conservancy and hydropower projects due to its stable flow pattern, high energy dissipation rate and less atomization. This study employs vorticity criterion, Q criterion, λ2 criterion and Ω criterion to investigate the characteristics of vortex formation and turbulent dissipation in multi-horizontal submerged jets stilling basins with various configurations, including crest overflowing orifice alone (COO), combination of crest overflowing orifice and mid-discharge orifice (COO-MO) and mid-discharge orifice alone (MO). The results indicate that the Q criterion and λ2 criterion are effective in identifying vortex structure within multi-horizontal submerged jets stilling basin. Specifically, the stronger intensity of vortex structure and vortex dissipation are mainly distributed in the vicinity of the vertical drop, which gradually weakens for the increasing distance to the vertical drop. Furthermore, the intensity and number of vortexes with COO-MO are the largest. This conclusion can provide guidance for energy dissipation and bottom protection of stilling pool.

Induction of Controllable Vortex Flow in a New Aorta Stenosis Model for Potential Aorta Mechanobiology Study: A Flow Simulation Platform

Background: Abdominal aortic aneurysms (AAA) are common, particularly among individuals over age 65, with a prevalence of 9%. To better understand the natural history of AAAs, one critical knowledge gap must be filled to elucidate the role of vortical flow in vascular degradation and thrombosis formation. We aimed to induce various types of vortical flow with aneurysm-like hemodynamic environments using a novel aorta stenosis model. Such a model will enable us to investigate the interplays between disturbed hemodynamics and vascular pathological remodeling. Methods: In this study, the hemodynamics and geometry of the aorta in rabbits were used to simulate the physiological flow conditions after two adjacent stenoses were created virtually with varying degrees of area reduction (30-70%) and spacing (5-6.8 mm). A physiological blood flow (rate) waveform measured in rabbits using ultrasound Doppler (normalized to 3.0 mL/s) was prescribed to the inlet (i.e., descending aorta), and zero pressure boundary conditions were applied at two outlets (i.e., iliac arteries). 3D computational fluid dynamics (CFD) simulation in five different stenosis configurations was performed using commercial software (Fluent, Version 20, Ansys Inc., PA). In-house software assessed the evolution of flow vortices over three distinct regions post stenosis using an established method. The total volume of vortex flow, the number of vortices, and the phase-to-phase overlap of vortex flows within different post-stenosis regions were derived to characterize the vortex flow. Results: Quantitative and qualitative flow assessments were performed for all five CFD models. Firstly, in all models, we found consistent patterns of the three vortex flow parameters, indicating that the adjacent stenoses induced three different hemodynamic zones in the post-stenosis regions, namely, stable vortical flow (after first stenosis), transient flow (after second stenosis), unstable vortical flow (further distal to second stenosis). Moreover, increased stenosis degree or reduced spacing led to reduced vortex flow number but increased total volume of vortex flow. The distinctions of the three regions were most significant in the model with 50% and then 30% stenoses, and with a 5 mm (center-to-center) spacing between two stenoses. Conclusions: CFD simulations can offer a powerful tool to predict complex flows using subject-specific vasculature. Such a flow simulation platform is valuable for guiding experimental design or therapy in various diseases. We demonstrated that it is feasible to adjust and achieve multiple types of vortical flows with the novel stenosis model. The three zones with distinct hemodynamic environments allow us to investigate the vascular remodeling regulated by the specifics of the eddies. Our ongoing work is implementing optimized aorta stenosis models to preclinical or in vitro studies to reveal the interplays between varying disturbed flow and aneurysm remodeling. We acknowledge funding support from NIH/NIBIB (R01-EB029570A1) and the Health Research Institute of Michigan Technological University. This is the full abstract presented at the American Physiology Summit 2024 meeting and is only available in HTML format. There are no additional versions or additional content available for this abstract. Physiology was not involved in the peer review process.

Heat transfer mechanism of different vortex generators arrangements embedded in heat exchange tubes

Heat exchange is widely used in many fields to improve heat transfer efficiency and save energy, while longitudinal vortex generator (VG) is widely used in many heat transfer heat exchangers. In this work, the effect of VGs with different numbers and different pitch ratios (PR) on the heat transfer characteristics in the tube was studied numerically. The results show that when the fluid flows through the VG, the reflux region will be generated, and the fluid velocity in the reflux region will be inhibited, resulting in uneven temperature distribution and increased Nu. When multiple VGs are arranged in a row, the number of vortices is increased, and the area affected is increased. This improves the overall heat transfer performance. With the decrease of PR, the number of VG rows increases, and the heat transfer performance increases first and then decreases with the decrease of PR.

Numerical study on influence of protrusion heights on Reynolds stress and viscous stress variations in turbulent vortical structures

Analysing the influence mechanism of the riblet protrusion height on turbulent drag components is more beneficial in organising the vortical structure over the riblet surface. Therefore, the Large Eddy Simulation (LES) is used to investigate the vortex structure over the riblet surface with different protrusion heights. Then, the variations of Reynolds stress and viscous shear stress in a turbulent channel are analysed. As a result, the drag reduction rate increases from 3.4% when the riblets are completely submerged in the turbulent boundary layer to 7.9% when the protrusion height is 11.2. Further analysis shows that the protrusion height affects the streamwise vortices and the normal diffusivity of spanwise and normal vortices, thus driving the variation of Reynolds stress. Compared with the smooth surface, the vorticity strength and the number of streamwise vortices are weakened near the wall but increase in the logarithmic layer with increased protrusion height. Meanwhile, the normal diffusivity of spanwise vorticity decreases with the increase of protrusion height, and the normal diffusivity of normal vorticity is the smallest when the protrusion height is 11.2. Moreover, the protrusion height affects the velocity gradient of the riblet tip and riblet valley, thus driving the variation of viscous shear stress. With the increase of protrusion height, the velocity gradient of the riblet tip increases dramatically but decreases in the riblet valley.

Turbulent spot formation in three-dimensional yukawa liquids using large-scale molecular dynamics simulation—effect of system size

We have performed classical ‘first principles’ 3D Molecular Dynamics (MD) simulation of plane-Couette flow (PCF) in a 3D Yukawa liquid. The main aim of the present investigation is to study the effect of stream-wise (flow-direction) and span-wise (transverse direction to the flow) width on the subcritical transition to turbulence in plane Couette flow (PCF), separately. In the past, taking very large-aspect ratio systems, subcritical transition to turbulence in PCF has been studied in great detail. However, the effect of stream-wise and span-wise width on the turbulent dynamics separately, have not been studied in detail. In the present work, we have investigated the effect of stream-wise and span-wise width or length and found that the stream-wise length enhances the large-scale energy, which gives rise to strong large-scale flow, and span-wise width enhances the small-scale energy, which gives rise to strong small-scale structures. In other words, topology of the turbulent spot is observed to change with the change in system size. The spectral separation between the modes governing the large and small-scale dynamics improves with the increase in system sizes. The connection between the stream-wise vortices and the stream-wise velocity streaks is investigated in details. We have found that the number stream-wise velocity streaks is one unit larger than the number of stream-wise vortices. A qualitative comparison between the results obtained from the MD simulation and the hydrodynamics experiment is presented in this article. The behavior of the system was observed to vary with the range of interactions among the dust grains.