Inertial Effects Research Articles

Inaction inertia in online shopping: impact of promotional formats and sequence

Purpose Inaction inertia occurs when people are less likely to act on a similar but inferior option after missing a superior opportunity, compared to if they had not missed out. This study aims to explore how promotional formats and their sequence affect the inaction inertia effect in online shopping, under the assumption of economic equivalence. Design/methodology/approach The authors performed two online experiments and analyzed the data by analysis of variance. Findings The findings indicate that, under the premise of economic equivalence: Monetary promotions exhibit a higher inaction inertia effect on consumers than nonmonetary promotions. When consumers miss a more favorable promotion and subsequently encounter a relatively less attractive one presented in a different promotional format, the inaction inertia effect is lower than when reencountering the same promotion format. When consumers miss a better monetary promotion and presently encounter a relatively less attractive nonmonetary promotion, the inaction inertia effect is lower than when they miss a superior nonmonetary promotion and currently encounter a relatively less attractive monetary promotion. Originality/value This study reveals the sequence effects of promotional formats, indicating that nonmonetary promotions following monetary ones effectively reduce inaction inertia. A strategically sequenced set of formats enhances consumer recommendations, mitigating inaction inertia. These findings open new research paths, providing insights into the impact of promotional format sequences on the inaction inertia effect. Consequently, this knowledge helps e-retailers in implementing effective promotional strategies and driving online purchases.

How Are You Feeling Today? Dynamic and Static Indices of Daily Affect Predict Psychological Adjustment One Year Later in a Multi-cohort, Longitudinal Investigation

Abstract Background Dominant psychotherapies target how individuals experience and understand their daily emotion. Therefore, research examining how daily emotions influence long-term mental health outcomes may help inform treatment development. Methods This investigation applied a multi-cohort (n = 378; n = 460), longitudinal design to test how reports of daily emotion predict psychological symptoms, loneliness, and wellbeing one-year later. Dynamic indices (polarity, inertia) reflecting “how” emotional experiences are conceptualized moment-to-moment and static indices (person-mean, standard deviation) of emotion were extracted from 10 daily reports. Each index was modelled individually, in concert with others, and in relation to a key dispositional factor in symptom development: trait anxiety. Results Dynamic indices predicted outcomes one-year later, but only the effect of positive emotional inertia remained significant after accounting for mean intensity of affect. Daily reports of emotion also predicted small but significant variance in outcomes beyond trait anxiety. Conclusions Results highlight the role of daily subjective experiences of emotion in long-term mental health outcomes and reinforce their importance as targets for treatment.

A simplified weak coupling dynamic analysis method for running safety evaluation of train on bridges under earthquake excitation

This study proposes a simplified analysis method that couples the bridge and train-track subsystem in a weak form to facilitate the dynamic analysis of the train-track-bridge coupled system (TTBS) under earthquakes. The method includes three main steps: (i) performing dynamic analysis of the bridge under seismic conditions to obtain the response of the bridge deck; (ii) interpolating the bridge displacement beneath each wheelset based on the train's velocity, departure time, and the initial positions of the wheelsets; and (iii) assessing the running safety of the train by incorporating the interpolated bridge response as track irregularity into the train-track coupled system. Steps (i) and (iii) can be analyzed separately using finite element and multibody dynamics software, eliminating the need to create a complex TTBS model that is challenging to achieve within a single software platform. Additionally, based on the method’s assumptions, the earthquake excitation on the train can be separated into the seismic inertial effect and seismic-induced bridge deformation, enabling a quantitative analysis of the impact of these two types of excitations on train safety. Taking a 10-span simply supported bridge and a long-span cable-stayed bridge as examples, the proposed method’s assumptions are verified by comparing its results with those obtained from a complex TTBS model. Furthermore, the impact characteristics of inertial effects and seismic-induced bridge deformation on train safety for these two types of bridges are analyzed. The proposed method offers robust support for seismic design of bridges and facilitates a comprehensive analysis of the mechanisms through which seismic excitations affect train safety.

Regionally-triggered geomagnetic reversals

Systematic studies of numerical dynamo simulations reveal that the transition from dipole-dominated non-reversing fields to models that exhibit reversals occurs when inertial effects become strong enough. However, the inertial force is expected to play a secondary role in the force balance in Earth’s outer core. Here we show that reversals in numerical dynamo models with heterogeneous outer boundary heat flux inferred from lower mantle seismic anomalies appear when the amplitude of heat flux heterogeneity is increased. The reversals are triggered at regions of large heat flux in which strong small-scale inertial forces are produced, while elsewhere inertial forces are substantially smaller. When the amplitude of heat flux heterogeneity is further increased so that in some regions sub-adiabatic conditions are reached, regional skin effects suppress small-scale magnetic fields and the tendency to reverse decreases. Our results reconcile the need for inertia for reversals with the theoretical expectation that the inertial force remains secondary in the force balance. Moreover, our results highlight a non-trivial non-monotonic behavior of the geodynamo in response to changes in the amplitude of the core-mantle boundary heat flux heterogeneity.

Mass-Based Separation of Active Brownian Particles in an Asymmetric Channel

Inertial effects should be considered for micro and nanoswimmers moving in a low-density medium confined by irregular structures that create entropic barriers, where viscous effects are no longer paramount. Here, we present a separation mechanism of self-propelled particles in a two-dimensional asymmetric channel, which leads to the drift of particles of different masses in opposite directions. In particular, this mechanism is based on the combined action of the spatial asymmetry of the channel structure, the temporal asymmetry inherent in particles dynamics and an external static force. This work is relevant for potential applications that can be found in the development of lab-on-a-chip devices and artificial channels for separating particles of different masses.

A High-Performance piezoelectric micropump designed for precision delivery

A piezoelectric micropump (PE pump) was proposed featuring a multi-plate cantilever valve (MPCV) and a ramp channel (RC) to deliver high performance in a compact design. Both the MPCV and RC underwent thorough theoretical, simulation-based, and experimental evaluations. A specialized driver plate was then developed to precisely control the PE pump. Key parameters of the PE pump were optimized based on test results. The final phase involved assessing the PE pump's output performance through tests, including a simulated insulin infusion scenario. The findings revealed that the MPCV exhibited a pronounced rectification capability, significantly enhancing the pump's operational efficiency over traditional designs. The RC, on the other hand, effectively minimized fluid resistance, capitalizing on the fluid inertia effect for improved performance. Integrating the MPCV and RC aligned the fluid path from inlet to outlet closely parallel to the PE vibrator. This alignment reduced backflow, thereby improving the micropump's output efficiency. Operating at 240 voltage peak-to-peak (Vpp) and 230 Hz, the PE pump demonstrated a flow rate of 6.8 ml/min and an output pressure of 68.6 kPa while maintaining a compact size of merely 10 × 10 × 1.5 mm3. The precision in flow rate adjustment was refined to 0.15 μl, with the deviation in insulin delivery being approximately ± 3 %. The study presents a prototype for a continuous subcutaneous insulin infusion protocol, leveraging the PE pump's advanced features.

Elasto-inertial instability in torsional flows of shear-thinning viscoelastic fluids

It is well known that inertia-free shearing flows of a viscoelastic fluid with curved streamlines, such as the torsional flow between a rotating cone and plate or the flow in a Taylor–Couette geometry, can become unstable to a three-dimensional time-dependent instability at conditions exceeding a critical Weissenberg ( $Wi$ ) number. However, the combined effects of fluid elasticity, shear thinning and finite inertia (as quantified by the Reynolds number $Re$ ) on the onset of elasto-inertial instabilities are not fully understood. Using a set of cone–plate geometries, we experimentally explore the entire $Wi$ – $Re$ phase space for a series of nonlinear viscoelastic fluids (with the dependence on shear rate $\dot{\gamma}$ quantified using a shear-thinning parameter $\beta _P(\dot {\gamma })$ ). We tune $\beta _P(\dot {\gamma })$ by varying the dissolved polymer concentration in solution. This progressively reduces shear thinning but leads to finite inertial effects before the onset of elastic instability, and thus naturally results in elasto-inertial coupling. Time-resolved rheometric measurements and flow visualization experiments allow us to investigate the effects of flow geometry, and document the combined effects of varying $Wi, Re$ and $\beta _P(\dot {\gamma })$ on the emergence of secondary motions at the onset of instability. The resulting critical state diagram quantitatively depicts the competition between the stabilizing effects of shear thinning and the destabilizing effects of inertia. We extend the curved streamline instability criterion of Pakdel & McKinley (Phys. Rev. Lett., vol. 77, no. 12, 1996, p. 2459) for the onset of purely elastic instability in curvilinear geometries by using scaling arguments to incorporate shear thinning and finite inertial effects. The augmented condition facilitates predictions of the onset of instability over a broader range of flow conditions, and thus bridges the gap between purely elastic and elasto-inertial curved streamline instabilities.

Density-contrast induced inertial forces on particles in oscillatory flows

Oscillatory flows have become an indispensable tool in microfluidics, inducing inertial effects for displacing and manipulating fluid-borne objects in a reliable, controllable and label-free fashion. However, the quantitative description of such effects has been confined to limit cases and specialized scenarios. Here we develop an analytical formalism yielding the equation of motion of density-mismatched spherical particles in oscillatory background flows, generalizing previous work. Inertial force terms are systematically derived from the geometry of the flow field together with analytically known Stokes number dependences. Supported by independent, first-principles direct numerical simulations, we find that these forces are important even for nearly density-matched objects such as cells or bacteria, enabling their fast displacement and separation. Our formalism thus consistently incorporates particle inertia into the Maxey–Riley equation, and in doing so provides a generalization of Auton's modification to added mass, as well as recovering the description of acoustic radiation forces on particles as a limiting case.

Non-inertial interpretation of the Dirac oscillator

Non-inertial physics is seldom considered in quantum mechanics and this contrasts with the ubiquity of non-inertial reference frames. Here, we show an application to the Dirac oscillator which provides a fundamental model of relativistic quantum mechanics. The model emerges from a term linearly dependent on spatial coordinates added to the momentum of the free-particle Dirac Hamiltonian. The definition generates peculiar features (mutating vacuum energy, non-Hermitian momentum, accidental degeneracies of the spectrum, etc). We interpret these anomalies in terms of inertial effects. The demonstration is based on the decoupling of the Dirac equation from the stereographic projection that maps the 3D geometry of the dynamical problem to the complex plane. The decoupling shows that the fundamental mechanical model underpinning the Dirac oscillator reduces to the representation of the oscillator in the rotating reference frame attached to the orbital angular momentum. The resulting Coriolis-like contribution to the Hamiltonian accounts for the peculiarities of the model (mutating vacuum energy, form of the non-minimal correction to the momentum, classical intrinsic spin and gain of its quantum value, accidental degeneracies of the energy spectrum, supersymmetric potential). The suggested interpretation has an interdisciplinary character where stereographic geometry, classical physics of the Coriolis effect and quantum physics of Dirac particles contribute to the definition of one of the few exactly soluble models of relativistic quantum mechanics.

A two-phase flow model for sedimentation and consolidation

Sedimentation and consolidation, a multi-physical phenomenon of great significance in aquatic environments, usually involves dynamic pore pressure, inertial effects, fluid-particle interphase interaction and solid stress. However, simplified models for sedimentation and consolidation typically assume hydrostatic mixture pressure and neglect inertial effects without proper justifications. Here, a one-dimensional three-equation two-phase flow model (TTP) is proposed for sedimentation and consolidation, which directly resolves dynamic fluid pressure and inertial terms. The present TTP model is benchmarked against a series of experimental cases and two existing four-equation two-phase flow (FTP) models. It features encouraging performance as compared to measured data and computed results of the existing FTP model. Furthermore, the present TTP model shows superior computing efficiency over the FTP models. To investigate the influences of inertial effects and non-hydrostatic mixture pressure, two simplified versions of the TTP model are constructed and compared with the TTP model. It is shown that incorporating inertial effects and non-hydrostatic fluid pressure are important for accurately predicting the sedimentation-consolidation process. The present study facilitates a promising framework for modelling sedimentation and consolidation, thereby supporting effective sediment management.

An extended continuum-mechanics standard linear solid rheology for fluid-saturated porous rock

SUMMARY In a large body of rock-physics research, seismic wave velocity dispersion and attenuation in fluid-saturated porous rock are studied by constructing analytical or numerical models for time- or frequency-dependent dynamic (effective, or viscoelastic) moduli. A key and broadly used model of such kind is the Zener's, or the standard linear solid (SLS). This model is qualitatively successful in explaining many field and laboratory observations and serves as the key element of many generalizations such as the Burgers model for plastic deformations or the generalized SLS explaining band-limited or near-constant seismic attenuation. However, as a physical model of fluid-saturated porous rock, the SLS has several major limitations: disregard of inertial effects, absence of secondary wave modes and lack of key physical parameters such as porosity and Skempton coefficients. Grainy and porous rock is an unconsolidated material in which the effective density is frequency-dependent, and its effects on wave velocities may exceed those of the dynamic modulus. To overcome these limitations of the empirical SLS, we propose a rigorous rheologic model based on classical continuum mechanics and called the extended SLS, or eSLS. This rheology explains the available attenuation and dispersion observations equally well, but it is also close to Biot's model, honours all poroelastic relations, includes inertial effects, and reveals several new physical properties of the material. Detailed comparison of the eSLS and Biot's models gives a physical-mechanism-based classification of wave-induced fluid flow (WIFF) phenomena. In this classification, the so-called ‘global-scale’ flows occur in Biot's type structures within the material, whereas the ‘local-scale’ WIFF occurs in eSLS-type structures. Combining Biot's and eSLS models gives a broad class of rheologies for linear anelastic phenomena within rock with a single type of porosity. The model can be readily generalized to multiple porosities and different types of internal variables, such as describing squirt flows, wetting or thermoelastic effects. Modelling is conducted with relatively little effort, using a single matrix equation similar to a mechanical form of the standard SLS. By combining the eSLS and Biot's models, observations of dynamic-modulus dispersion and attenuation can be inverted for macroscopic mechanical properties of porous materials.

Revisiting the mechanism responsible for the stratified-slug transition in two-phase flows

Linear stability analysis is extensively used for predicting the transition between stratified and slug flow. In the present work, a one-dimensional two-fluid flow is linearly perturbed to evaluate the behavior of the dispersion curves through parameters such as the maximum wave growth rate (ωI,max), the fastest-growing wave (kmax) and the wave that makes the problem permanently stable (kc) as a function of the gas and liquid superficial velocities. The novelty of this article relies upon coupling the analysis of the behavior of transition-related parameters to the physical effects that are responsible for stabilizing and destabilizing the flow interface. The coupling of the transition analysis with the physical parameters showed potential as a reliable way of explaining the obtained transition behavior. By doing this, the stabilizing effects of gravity and surface tension are found to be invariable to the superficial velocities of the phases. On the other hand, the destabilizing effect of inertia increased with phase superficial velocities, while the shear stress increases with the liquid superficial velocity and shows a non-monotonic behavior with the gas superficial velocity. Although the overall trend of ωI,max, kmax and kc was to increase with the superficial velocities of the phases, they were directly affected by the shear stress behavior, also showing a non-monotonic trend.

Data-Driven Path Collective Variables.

Identifying optimal collective variables to model transformations using atomic-scale simulations is a long-standing challenge. We propose a new method for the generation, optimization, and comparison of collective variables that can be thought of as a data-driven generalization of the path collective variable concept. It consists of a kernel ridge regression of the committor probability, which encodes a transformation's progress. The resulting collective variable is one-dimensional, interpretable, and differentiable, making it appropriate for enhanced sampling simulations requiring biasing. We demonstrate the validity of the method on two different applications: a precipitation model and the association of Li+ and F- in water. For the former, we show that global descriptors such as the permutation invariant vector allow reaching an accuracy far from the one achieved via simpler, more intuitive variables. For the latter, we show that information correlated with the transformation mechanism is contained in the first solvation shell only and that inertial effects prevent the derivation of optimal collective variables from the atomic positions only.

A modular computational framework for the dynamic analyses of cable-driven parallel robots with different types of actuation including the effects of inertia, elasticity and damping of cables

Abstract Dynamic simulations of the cable-driven parallel robots (CDPRs) with cable models closer to reality can predict the motions of moving platforms more accurately than those with idealisations. Hence, the present work proposes an efficient and modular computational framework for this purpose. The primary focus is on the developments required in the context of CDPRs actuated by moving the exit points of cables while the lengths are held constant. Subsequently, the framework is extended to those cases where simultaneous changes in the lengths of cables are employed. Also, the effects due to the inertia, stiffness and damping properties of the cables undergoing 3D motions are included in their dynamic models. The efficient recursive forward dynamics algorithms from the prior works are utilised to minimise the computational effort. Finally, the efficacy of the proposed framework and the need for such an inclusive dynamic model are illustrated by applying it to different application scenarios using the spatial $4$ - $4$ CDPR as an example.

Direct numerical simulations of microlayer formation during heterogeneous bubble nucleation

In this article, we present direct numerical simulation results for the expansion of spherical cap bubbles attached to a rigid wall due to a sudden drop in the ambient pressure. The critical pressure drop beyond which the bubble growth becomes unstable is found to match well with the predictions from classical theory of heterogeneous nucleation imposing a quasi-static bubble evolution. When the pressure drop is significantly higher than the critical value, a liquid microlayer appears between the bubble and the wall. In this regime, the interface outside the microlayer grows at an asymptotic velocity that can be predicted from the Rayleigh–Plesset equation, while the contact line evolves with another asymptotic velocity that scales with a visco-capillary velocity that obeys the Cox–Voinov law. In general, three distinctive regions can be distinguished: the region very close to the contact line where dynamics is governed by visco-capillary effects, an intermediate region controlled by inertio-viscous effects away from the contact line yet inside the viscous boundary layer, and the region outside the boundary layer dominated by inertial effects. The microlayer forms in a regime where the capillary effects are confined in a region much smaller than the viscous boundary layer thickness. In this regime, the global capillary number takes values much larger then the critical capillary number for bubble nucleation, and the microlayer height is controlled by viscous effects and not surface tension.

Role of viscoelasticity in the appearance of low-Reynolds turbulence: considerations for modelling.

Inertial effects caused by perturbations of dynamical equilibrium during the flow of soft matter constitute a hallmark of turbulence. Such perturbations are attributable to an imbalance between energy storage and energy dissipation. During the flow of Newtonian fluids, kinetic energy can be both stored and dissipated, while the flow of viscoelastic soft matter systems, such as polymer fluids, induces the accumulation of both kinetic and elastic energies. The accumulation of elastic energy causes local stiffening of stretched polymer chains, which can destabilise the flow. Migrating multicellular systems are hugely complex and are capable of self-regulating their viscoelasticity and mechanical stress generation, as well as controlling their energy storage and energy dissipation. Since the flow perturbation of viscoelastic systems is caused by the inhomogeneous accumulation of elastic energy, rather than of kinetic energy, turbulence can occur at low Reynolds numbers.This theoretical review is focused on clarifying the role of viscoelasticity in the appearance of low-Reynolds turbulence. Three types of system are considered and compared: (1) high-Reynolds turbulent flow of Newtonian fluids, (2) low and moderate-Reynolds flow of polymer solutions, and (3) migration of epithelial collectives, discussed in terms of two model systems. The models considered involve the fusion of two epithelial aggregates, and the free expansion of epithelial monolayers on a substrate matrix.

Mechanism and evolution of the wake magnetic field generated by underwater vehicles

The motion of charged seawater in a submarine wake, which is affected by the variance in the ion mass and the presence of a background geomagnetic field, results in a consequential wake magnetic field. The magnetic field is critical for the detection and monitoring of underwater targets. Despite its importance, the intricate relationship among hydrodynamics, ion transport, and electromagnetic fields, which collectively affect the generation of magnetic fields in submarine wakes, has yet to be sufficiently and systematically investigated. Hence, we formulate a theoretical model of the submarine wake magnetic field. This innovative interdisciplinary model seamlessly integrates the effects of ion inertia and ion induction in the marine environment. By performing multiphysics coupling simulations, we meticulously replicate the evolution of the wake magnetic field induced by the movement of a submarine propeller system. We comprehensively examine the velocity and magnetic-field components to elucidate the intricate mechanism of wake magnetic field generation. Furthermore, we comprehensively analyze the effects of ion inertia and ion induction on the wake magnetic field. Our findings highlight the predominant role of the ion-induction effect in determining the wake magnetic field when a submarine maintains a constant speed. Additionally, we investigate the effect of the submarine heading on the wake magnetic field and discuss the magnetic-field characteristics and attenuation rules at various axial and radial positions in the wake for different headings. Our scientific insights contribute considerably to the understanding of wake magnetic fields and offer valuable perspectives that can potentially advance future underwater target-detection methodologies based on wake magnetic fields.

Strain rate and size effects on dynamic tensile behaviour of standard and high-strength concrete: An experimental study

The strain rate and specimen size are the main dominant factors affecting the splitting tensile behaviour of concrete. The static size effect is well-known for concrete behaviour, but it needs more knowledge about the dynamic size effect. The present study investigated the strain rate sensitivity and size effect dependency of dynamic tensile behaviour of two different grades of concretes (M35 and M60) using the Split Hopkinson Pressure Bar (SHPB) setup. From the outcomes of the experimental analysis, it was noted that the quasi-static tensile strength underwent a reduction of 8.96% and 11.39% in standard and high-strength concrete, respectively. This decrease occurred as the diameter increased from 29.5 mm to 45 mm, aligning notably with the principles outlined in the "Bazant Size Effect Law," as proposed by Bazant et al. [52]. Whereas, the dynamic tensile strength and DIF increased with increased specimen diameter, which is opposite to the quasi-static size effect. The maximum split tensile strength observed was 18.56 and 19.86 MPa, corresponding to 13.30 and 12.13 s−1 strain rates with 3.69 and 3.51 dynamic increase factor (DIF) for standard (M35) and high-strength concrete (M60), respectively. The dynamic results also demonstrated a size-effect dependency, where larger diameter specimens exhibited higher strain rate sensitivity, leading to increased strength and DIF due to lateral inertial and crack propagation effects. The modified DIF model of CEB-FIP proposed by the Malvar et al. [17] considering the strain rate and size effect simultaneously strongly supported the experimentally obtained DIF values with an ± 13% deviations. In addition, the strain rate significantly influenced the concrete failure pattern, but no size effect was observed. At a low strain rate, the initiated crack propagated towards the loading ends and split the specimen into two semi-cylindrical halves with penetration cracks. While at higher strain rates, additional wedge-shaped crushed regions formed at the loading end due to tensile and shear damage generated due to stress concentration and local crushing. The crack width becomes wider, and the triangular-shaped local crushing zone continuously moves toward the centre of the specimen from both loading ends and generates the crushed strap.

Instability in strongly stratified plane Couette flow, with application to supercritical fluids.

This paper addresses the stability of plane Couette flow in the presence of strong density and viscosity stratifications. It demonstrates the existence of a generalised inflection point that satisfies the generalised Fjørtoft's criterion of instability when a minimum of kinematic viscosity is present in the base flow. The characteristic scales associated with this minimum are identified as the primary controlling parameters of the associated instability, regardless of the type of stratification. To support this finding, analytical stability models are derived in the long wave approximation using piecewise linear base flows. Numerical stability calculations are carried out to validate these models and to provide further information on the production of disturbance vorticity. All instabilities are interpreted as arising from the interaction between two vorticity waves. Depending on the type of stratification, these two waves are produced by different physical mechanisms. When both strong density and viscosity stratifications are present, we show that they result from the concurrent action of shear and inertial baroclinic effects. The stability models developed for simple fluid models ultimately shed light on a recently observed unstable mode in supercritical fluids (Ren et al., J. Fluid Mech., vol. 871, 2019, pp. 831-864), providing a quantitative prediction of the stability diagram and identifying the dominant mechanisms at play. Furthermore, our study suggests that the minimum of kinematic viscosity reached at the Widom line in these fluids is the leading cause of their instability. The existence of similar instabilities in different fluids and flows (e.g., miscible fluids) is finally discussed.

Trajectory dynamics of particles in accelerated toroidal pipe: A computational study using CFD-DPM simulations

In this study, we investigate the behavior of solid particles of varying sizes within an accelerated toroidal filled with fluid using computational fluid dynamics- discrete phase model (CFD-DPM) simulations. A comprehensive grid independence study and validation were conducted to ensure the accuracy of our simulations. We focus on particles with 10-, 50-, and 100-microns diameters. Our simulations indicate that the primary flow (relative tangential velocity) of both the fluid and the particles follow the input acceleration. We also examined the secondary flow within the toroidal. Smaller particles (10 microns) closely follow the fluid flow with minimal lag, while larger particles (50 and 100 microns) exhibit a noticeable lag due to the inertia effect. Furthermore, the displacement from the initial position is also dependent on the particle size, with smaller particles traveling further distances. However, despite the ongoing acceleration of the fluid and the toroidal, the largest particles (100 microns) eventually settle and stop due to their high inertia (mass), and their velocity begins to decrease during the acceleration of the toroidal. These findings highlight the significant role of particle size in determining particle behavior within a toroidal, paving the way for future research and Particle Imaging Velocimetry Gyroscope (PIVG) application.