3D Fluid Research Articles

Significance of gyrotactic microorganism's analysis for magnetized convectively heat 3D Sisko fluid flow with bioconvection phenomenon

In this research article, the bioconvection of oxytactic microorganisms with magneto-hydrodynamic (MHD) 3D steady flow of Sisko nanomaterial over a stretching sheet through convective conditions is deliberated and analyzed. Possessions of thermophoresis, Brownian motion as well as mixed convection in the Sisko fluid are also deliberated. Sisko fluid is supposed to be electrically conducted over and done with a constant pragmatic magnetic field. The flow problem is communicated using governing relations and problem is transmuted into dimensionless form among non-similar procedure. To obtain convergent solutions we must solve nonlinear differential systems. The consequences of several physical parameters have been deliberate and discussed. Our results displayed that the higher microorganism difference parameter condensed the microorganism profile, while an opposite influence is established for magnetic parameter. Concentration proises of the Sisko fluid flow are improved by growing Biot number whereas contracted beside the Lewis number. In recent years, the scientists have shown great attention in the field of bioconvection owing to its countless applications such as amassing the cells, bioreactors, gas-bearing sedimentary, modeling oil, exclusion of living, and nonliving cells, and lots of additional.

Boundary vorticity of incompressible 2D flows

For a homogeneous incompressible 2D fluid confined within a bounded Lipschitz simply connected domain, homogeneous Neumann pressure boundary conditions are equivalent to a constant boundary vorticity. We investigate the rigidity of such conditions.

Simulating 2D Fluid Motion with the Smooth Particle Hydrodynamic Approach

Abstract Virtual two-dimensional (2D) fluid simulation is useful for directly simulating fluids in various situations, including geological simulations for landslides and fluid simulations for teaching. This research aims to simulate the behaviour of three different types of fluids (water, coconut oil, and glycerine) in a 2D container and analyse how these three types of fluids behave under various conditions, including interactions with boundaries. The research used Python programming to simulate fluids and the Wondershare Filmora X application to combine images. The Smooth Particle Hydrodynamics method simulated fluid in a 2D container with 10,000 particles by deriving the force density field directly from the Navier-Stokes equations. The Navier-Stokes equations were utilized to find the acceleration and velocity of fluid particles by considering external forces, internal forces, and gravity through this method. Acceleration and velocity were validated due to wall collisions, collisions with boundaries, and collisions between particles, which caused changes in particle position and velocity. During visualization, the fluid velocity decreased over time due to attenuation caused by interactions between neighbour particles, particles with boundaries, and particles with walls. From the simulation, it was observed that the fluid flowed from a higher place to a lower place, with fluid particles taking the shape of the container and the surface of the fluid forming waves. On the other hand, simulations with boundaries indicated that smaller gap sizes and higher viscosities led to increased difficulty in fluid penetration into the gap within the container.

Estimating pulmonary arterial remodeling via an animal-specific computational model of pulmonary artery stenosis.

Pulmonary artery stenosis (PAS) often presents in children with congenital heart disease, altering blood flow and pressure during critical periods of growth and development. Variability in stenosis onset, duration, and severity result in variable growth and remodeling of the pulmonary vasculature. Computational fluid dynamics (CFD) models enable investigation into the hemodynamic impact and altered mechanics associated with PAS. In this study, a one-dimensional (1D) fluid dynamics model was used to simulate hemodynamics throughout the pulmonary arteries of individual animals. The geometry of the large pulmonary arteries was prescribed by animal-specific imaging, whereas the distal vasculature was simulated by a three-element Windkessel model at each terminal vessel outlet. Remodeling of the pulmonary vasculature, which cannot be measured in vivo, was estimated via model-fitted parameters. The large artery stiffness was significantly higher on the left side of the vasculature in the left pulmonary artery (LPA) stenosis group, but neither side differed from the sham group. The sham group exhibited a balanced distribution of total distal vascular resistance, whereas the left side was generally larger in the LPA stenosis group, with no significant differences between groups. In contrast, the peripheral compliance on the right side of the LPA stenosis group was significantly greater than the corresponding side of the sham group. Further analysis indicated the underperfused distal vasculature likely moderately decreased in radius with little change in stiffness given the increase in thickness observed with histology. Ultimately, our model enables greater understanding of pulmonary arterial adaptation due to LPA stenosis and has potential for use as a tool to noninvasively estimate remodeling of the pulmonary vasculature.

Physics simulation capabilities of LLMs

Large Language Models (LLMs) can solve some undergraduate-level to graduate-level physics textbook problems and are proficient at coding. Combining these two capabilities could one day enable AI systems to simulate and predict the physical world. We present an evaluation of state-of-the-art (SOTA) LLMs on PhD-level to research-level computational physics problems. We condition LLM generation on the use of well-documented and widely-used packages to elicit coding capabilities in the physics and astrophysics domains. We contribute ∼50 original and challenging problems in celestial mechanics (with REBOUND), stellar physics (with MESA), 1D fluid dynamics (with Dedalus) and non-linear dynamics (with SciPy). Since our problems do not admit unique solutions, we evaluate LLM performance on several soft metrics: counts of lines that contain different types of errors (coding, physics, necessity and sufficiency) as well as a more educational’ Pass-Fail metric focused on capturing the salient physical ingredients of the problem at hand. As expected, today's SOTA LLM (GPT4) zero-shot fails most of our problems, although about 40% of the solutions could plausibly get a passing grade. About 70%–90% of the code lines produced are necessary, sufficient and correct (coding & physics). Physics and coding errors are the most common, with some unnecessary or insufficient lines. We observe significant variations across problem class and difficulty. We identify several failure modes of GPT4 in the computational physics domain, such as poor physical units handling, poor code versioning, tendency to hallucinate plausible sub-modules, lack of physical justification for global run parameters (e.g., simulation time, or upper-lower bounds for parametric exploration) and inability to define steady-state or stopping conditions reliably. Our reconnaissance work provides a snapshot of current computational capabilities in classical physics and points to obvious improvement targets if AI systems are ever to reach a basic level of autonomy in physics simulation capabilities.

Research on the Collaborative Regeneration of Cohousing Under Spatial Densification Mode

Urban renewal is a necessary process for the transformation from incremental space to stock space, and old communities, as an important part of the urban space system, play a non-negligible role in the reconstruction of community spatial order. From the perspective of community remodeling, From the perspective of community remodeling, this paper summarizes the latest research progress through the analysis tool CiteSpace, explores and researches the design of old residential areas in Hangzhou based on the urban air index oriented residential renewal design based on the planning concepts of "co-living community", "urban spatial morphology" and "community microclimate system". On the basis of combining 3D wind field simulation and fluid mechanics software calculation, this paper explores the collaborative regeneration path of old residential areas under spatial densification mode, this paper studies how to combine scientific planning with resident sharing in a new mode, and on this basis, through the self-regulation function of the environment, let the residential area spontaneously form a complete shared living system, and carry out detailed exploration in the protection and reconstruction of the living environment of the old community, and how to promote the spiritual development of modern neighborhoods through transformation. In order to improve the quality of life of urban residents in Hangzhou, the renewal and regeneration design of old residential areas and the transformation and upgrading of future communities have brought new thinking.

3d fluid–structure interaction simulation with an Arbitrary–Lagrangian–Eulerian approach with applications to flying objects

3d fluid–structure interaction simulation with an Arbitrary–Lagrangian–Eulerian approach with applications to flying objects

Comparing the imaging characteristics of middle-aged patients with multiple sclerosis and CADASIL: A retrospective cross-sectional study

BackgroundFew studies have quantitatively analyzed the imaging disparities between multiple sclerosis (MS) and cerebral autosomal dominant arteriopathy with subcortical infarcts and leukoencephalopathy (CADASIL). We aimed to compare the imaging characteristics of MS and CADASIL in middle-aged patients. Materials and MethodsThis retrospective study used a single-center database and included patients aged 40–60 years with MS and CADASIL who underwent the designated imaging protocol including 3D T1-weighted imaging and fluid attenuated inversion recovery (FLAIR), diffusion tensor imaging and susceptibility-weighted imaging between January 2018 and March 2023. Patients with MRI-detected macrobleeds were excluded. ResultsA total of 27 patients with MS (mean age, 46.7 years ± 4.4, 8 men) and 30 patients with CADASIL (mean age, 51.6 years ± 5.8, 14 men) were included. No significant differences were observed in the Fazekas grades of white matter lesions (WMLs). Patients with CADASIL exhibited greater external capsule involvement (56.7% vs.18.5 %; p = 0.006), whereas the MS group had more lesions in the corpus callosum (81.5% vs. 53.3 %, p = 0.02) and brainstem (74.1% vs. 46.7 %, p = 0.04). The CADASIL group exhibited a higher incidence of microbleeds (12.07 vs. 0.11, p = 0.001). The WMLs in the MS group exhibited a lower T1 lesion/cerebrospinal fluid signal index (2.206 vs. 2.882, p < 0.001). A value of ≤2.57 demonstrated a sensitivity of 92.6 % and a specificity of 90.0 % in differentiating MS. Patients with MS had a thinner corpus callosum (7.18 mm vs 7.86 mm, p = 0.04), while patients with CADASIL showed significantly higher mean diffusivity (0.8776 × 10–3 vs. 0.7637 × 10–3 mm2/s, p = 0.03) and lower fractional anisotropy (0.7581 vs. 0.8389, p = 0.04) in the splenium of the corpus callosum. ConclusionMiddle-aged patients with MS and CADASIL showed comparable Fazekas grades for WMLs. However, lesion distribution, T1 signal characteristics, and splenic diffusivity changes can help differentiate between MS and CADASIL.

Horseshoes and spiral waves: capturing the 3D flow induced by a low-mass planet analytically

ABSTRACT The key difficulty faced by 2D models for planet–disc interaction is in appropriately accounting for the impact of the disc’s vertical structure on the dynamics. 3D effects are often mimicked via softening of the planet’s potential; however, the planet-induced flow and torques often depend strongly on the choice of softening length. We show that for a linear adiabatic flow perturbing a vertically isothermal disc, there is a particular vertical average of the 3D equations of motion that exactly reproduces 2D fluid equations for arbitrary adiabatic index. There is a strong connection here with the Lubow–Pringle 2D mode of the disc. Correspondingly, we find a simple, general prescription for the consistent treatment of planetary potentials embedded within ‘2D’ discs. The flow induced by a low-mass planet involves large-scale excited spiral density waves that transport angular momentum radially away from the planet and ‘horseshoe streamlines’ within the coorbital region. We derive simple linear equations governing the flow that locally capture both effects faithfully simultaneously. We present an accurate coorbital flow solution allowing for inexpensive future study of corotation torques, and predict the vertical structure of the coorbital flow and horseshoe region width for different values of adiabatic index, as well as the vertical dependence of the initial shock location. We find strong agreement with the flow computed in 3D numerical simulations, and with 3D one-sided Lindblad torque estimates, which are a factor of 2–3 lower than values from previous 2D simulations.

A mixed-dimensional formulation for the simulation of slender structures immersed in an incompressible flow

We consider the simulation of slender structures immersed in a three-dimensional (3D) flow. By exploiting the special geometric configuration of the slender structures, this particular problem can be modeled by mixed-dimensional coupled equations. Taking advantage of the slenderness of the structure and thus considering 3D/1D coupled problems raise several challenges and difficulties. From a mathematical point of view, these include defining well-posed trace operators of co-dimension two. On the computational standpoint, the non-standard mathematical formulation makes it difficult to ensure the accuracy of the solutions obtained with the mixed-dimensional discrete formulation as compared to a fully resolved one. Here we proposed to circumvent theses issues by imposing the fluid–structure coupling conditions on the 2D fluid–structure interface but in a reduced way still taking advantage of the 1D dynamic of the slender structure. We consider the Navier–Stokes equations for the fluid and a Timoshenko beam model for the structure. We complement these models with a mixed-dimensional version of the fluid–structure interface conditions, based on the projection of kinematic coupling conditions on a finite-dimensional Fourier space on each beam cross section. Furthermore, we develop a discrete fictitious domain formulation within the framework of the finite element method, establish the energy stability of the scheme, provide extensive numerical evidence of the accuracy of the discrete formulation, notably with respect to a fully resolved (ALE based) model and a standard reduced modeling approach.

SPH simulations of complex 3D fluid–solid interactions with an improved single-layer particle boundary technique preventing boundary penetration

The smoothed particle hydrodynamics (SPH) method has been widely used to simulate fluid–solid interaction (FSI) problems. The solid boundary treatment under complex geometric configurations has always been a hot topic in the SPH numerical simulation. An improved single-layer particle technique combined with an effective anti-penetration boundary improvement algorithm is proposed in this paper, which allows for multi-resolution single-layer particle distribution and effectively prevents particle penetration under complex geometric configurations. Four engineering problems are simulated with the improved technique: dam-break flows with a rectangular prism, water entry of a revolution body, oscillations of a floating body, and gearbox operation. The simulation results of four engineering problems are in good agreement with the reference results, which demonstrates that the improved single-layer particle boundary technique proposed in this paper can accurately simulate the FSI problem of complex geometric configurations.

Numerical modeling and experimental validation of fluid flow in micro- and meso-fluidic siphons

Siphons have been used for thousands of years to transfer fluids without the use of pumps or power and are present in our daily lives. Paradoxically, it is only in recent decades that the operation of siphons has been fully clarified, which is now understood to be exclusively linked to gravity and molecular cohesion. Siphons are uniquely able to offer automatic, intermittent flow, yet present the main drawback of requiring a source of energy to induce initial flow. Our research team has recently disclosed a microfluidic siphon able to self-prime and deliver a sequence of bioanalytical reagents, previously demonstrated for high-performance, multi-reagents diagnostic testing. Here we show for the first time 2D and 3D computational fluid dynamics (CFD) modeling and the experimental characterization of fluid flow in a range of miniaturized hydrophilic siphons of varying hydraulic liquid height-to-length ratios, ΔH/LT = 0–0.9, using fluids of varying viscosities. CFD simulations using velocity- and pressure-driven inlet boundary conditions were generally in good agreement with experimental fluid flow rates and pressure-balance predictions for plastic ∼0.2 mm and glass ∼0.6 mm internal diameter microfluidic siphons. CFD predictions of fluid flow in “meso-scale” siphons with 1 and 2 mm internal diameters also fully matched normalized experimental data, suggesting that miniaturized siphons are scalable. Their discharge rate and pressure drop are readily predicted and fine-tunable through the physical properties of the fluid and some design parameters of the siphon. The wide range of experimental and numerical parameters studied here provide an important framework for the design and application of gravity-driven micro- and meso-fluidic siphons in many applications, including but not limited to life sciences, clinical diagnostics, and process intensification.

Bending-driven patterning of solid inclusions in lipid membranes: Colloidal assembly and transitions in elastic 2D fluids.

Biological or biomimetic membranes are examples within the larger material class of flexible ultrathin lamellae and contoured fluid sheets that require work or energy to impose bending deformations. Bending elasticity also dictates the interactions and assembly of integrated phases or molecular clusters within fluid lamellae, for instance enabling critical cell functions in biomembranes. More broadly, lamella and other thin fluids that integrate dispersed objects, inclusions, and phases behave as contoured 2D colloidal suspensions governed by elastic interactions. To elucidate the breadth of interactions and assembled patterns accessible through elastic interactions, we consider the bending elasticity-driven assembly of 1-10 μm solid plate-shaped Brownian domains (the 2D colloids), integrated into a fluid phospholipid membrane (the 2D fluid). Here, the fluid membranes of giant unilamellar vesicles, 20-50 μm in diameter, each contain 4-100 monodisperse plate-domains at an overall solid area fraction of 17 ± 3%. Three types of reversible plate arrangements are found: persistent vesicle-encompassing quasi-hexagonal lattices, persistent closely associated chains or concentrated lattices, and a dynamic disordered state. The interdomain distances evidence combined attractive and repulsive elastic interactions up to 10 μm, far exceeding the ranges of physio-chemical interactions. Bending contributions are controlled through membrane slack (excess area) producing, for a fixed composition, a sharp cooperative multibody transition in plate arrangement, while domain size and number contribute intricacy.

FSGe: A fast and strongly-coupled 3D fluid–solid-growth interaction method

Equilibrated fluid–solid-growth (FSGe) is a fast, open source, three-dimensional (3D) computational platform for simulating interactions between instantaneous hemodynamics and long-term vessel wall adaptation through mechanobiologically equilibrated growth and remodeling (G&R). Such models can capture evolving geometry, composition, and material properties in health and disease and following clinical interventions. In traditional G&R models, this feedback is modeled through highly simplified fluid solutions, neglecting local variations in blood pressure and wall shear stress (WSS). FSGe overcomes these inherent limitations by strongly coupling the 3D Navier–Stokes equations for blood flow with a 3D equilibrated constrained mixture model (CMMe) for vascular tissue G&R. CMMe allows one to predict long-term evolved mechanobiological equilibria from an original homeostatic state at a computational cost equivalent to that of a standard hyperelastic material model. In illustrative computational examples, we focus on the development of a stable aortic aneurysm in a mouse model to highlight key differences in growth patterns between FSGe and solid-only G&R models. We show that FSGe is especially important in blood vessels with asymmetric stimuli. Simulation results reveal greater local variation in fluid-derived WSS than in intramural stress (IMS). Thus, differences between FSGe and G&R models became more pronounced with the growing influence of WSS relative to pressure. Future applications in highly localized disease processes, such as for lesion formation in atherosclerosis, can now include spatial and temporal variations of WSS.

Influence of Voltage Rising Time on the Characteristics of a Pulsed Discharge in Air in Contact with Water: Experimental and 2D Fluid Simulation Study

In the context of plasma–liquid interactions, the phase of discharge ignition is of great importance as it may influence the properties of the produced plasma. Herein, we investigated the influence of voltage rising time (τrise) on discharge ignition in air as well as on discharge propagation on the surface of water. Experimentally, τrise was adjusted to 0.1, 0.4, 0.6, and 0.8 kV/ns using a nanosecond high-voltage pulser, and discharges were characterized using voltage/current probes and an ICCD camera. Faster ignition, higher breakdown voltage, and greater discharge current (peak value) were observed at higher τrise. ICCD images revealed that higher τrise also promoted the formation of more filaments, with increased radial propagation over the water surface. To further understand these discharges, a previously developed 2D fluid model was used to simulate discharge ignition and propagation under various τrise conditions. The simulation provided the spatiotemporal evolution of the E-field, electron density, and surface charge density. The trend of the simulated position of the ionization front is similar to that observed experimentally. Furthermore, rapid vertical propagation (&lt;1 ns) of the discharge towards the liquid surface was observed. As τrise increased, the velocity of discharge propagation towards the liquid increased. Higher τrise values also led to more charges in the ionization front propagating at the water surface. The discharge ceased to propagate when the charge number in the ionization front reached 0.5 × 108 charges, irrespective of the τrise value.

Computational framework for the generation of one-dimensional vascular models accounting for uncertainty in networks extracted from medical images.

One-dimensional (1D) cardiovascular models offer a non-invasive method to answer medical questions, including predictions of wave-reflection, shear stress, functional flow reserve, vascular resistance and compliance. This model type can predict patient-specific outcomes by solving 1D fluid dynamics equations in geometric networks extracted from medical images. However, the inherent uncertainty in in vivo imaging introduces variability in network size and vessel dimensions, affecting haemodynamic predictions. Understanding the influence of variation in image-derived properties is essential to assess the fidelity of model predictions. Numerous programs exist to render three-dimensional surfaces and construct vessel centrelines. Still, there is no exact way to generate vascular trees from the centrelines while accounting for uncertainty in data. This study introduces an innovative framework employing statistical change point analysis to generate labelled trees that encode vessel dimensions and their associated uncertainty from medical images. To test this framework, we explore the impact of uncertainty in 1D haemodynamic predictions in a systemic and pulmonary arterial network. Simulations explore haemodynamic variations resulting from changes in vessel dimensions and segmentation; the latter is achieved by analysing multiple segmentations of the same images. Results demonstrate the importance of accurately defining vessel radii and lengths when generating high-fidelity patient-specific haemodynamics models. KEY POINTS: This study introduces novel algorithms for generating labelled directed trees from medical images, focusing on accurate junction node placement and radius extraction using change points to provide haemodynamic predictions with uncertainty within expected measurement error. Geometric features, such as vessel dimension (length and radius) and network size, significantly impact pressure and flow predictions in both pulmonary and aortic arterial networks. Standardizing networks to a consistent number of vessels is crucial for meaningful comparisons and decreases haemodynamic uncertainty. Change points are valuable to understanding structural transitions in vascular data, providing an automated and efficient way to detect shifts in vessel characteristics and ensure reliable extraction of representative vessel radii.

Numerical Study of the Influence of Regenerator Structure on the Performance of Hot Blast Stoves

The properties and arrangement of checker bricks in regenerators are crucial for the heat exchange process of hot blast stoves. In this study, a 3D fluid flow heat transfer model is established to analyze the influence of three regenerator layered structures on the combustion and air supply performance of hot blast stoves. The results show that a “silica bricks–high-alumina bricks–clay bricks” three-layer arrangement in regenerators produces a “thermal conduction hindrance effect” at the interface between silica and high-alumina bricks during the 2 h combustion period, which raises the local temperature and improves air supply performance. Compared to the “silica bricks–clay bricks” and “high-alumina bricks–silica bricks–clay bricks” structures, this setup increases the maximum air supply temperature difference to 23 and 64 K, respectively, and extends the effective air supply time by 80 and 320 s, respectively. However, the “thermal conduction hindrance effect” diminishes over longer combustion periods, and by 4 h, the performance across all structures becomes increasingly consistent. Additionally, the study suggests that the temperature level and distribution in the upper part of the regenerator are the key factors determining the air supply performance of hot blast stoves.

The quantum perfect fluid in 2D

We consider the field theory that defines a perfect incompressible 2D fluid. One distinctive property of this system is that the quadratic action for fluctuations around the ground state features neither mass nor gradient term. Quantum mechanically this poses a technical puzzle, as it implies the Hilbert space of fluctuations is not a Fock space and perturbation theory is useless. As we show, the proper treatment must instead use that the configuration space is the area preserving Lie group {S\mathrm{Diff}}SDiff. Quantum mechanics on Lie groups is basically a group theory problem, but a harder one in our case, since {S\mathrm{Diff}}SDiff is infinite dimensional. Focusing on a fluid on the 2-torus T^2T2, we could however exploit the well known result {S\mathrm{Diff}}(T^2)\sim SU(N)SDiff(T2)∼SU(N) for N\to ∞N→∞, reducing for finite NN to a tractable case. SU(N)SU(N) offers a UV-regulation, but physical quantities can be robustly defined in the continuum limit N\to∞N→∞. The main result of our study is the existence of ungapped localized excitations, the vortons, satisfying a dispersion \omega \propto k^2ω∝k2 and carrying a vorticity dipole. The vortons are also characterized by very distinctive derivative interactions whose structure is fixed by symmetry. Departing from the original incompressible fluid, we constructed a class of field theories where the vortons appear, right from the start, as the quanta of either bosonic or fermionic local fields.

Thermal Performance of Cattaneo-Christov Heat Flux Model on 3D Micropolar Fluid with Motile Microbes Across a Permeable Surface

Thermal Performance of Cattaneo-Christov Heat Flux Model on 3D Micropolar Fluid with Motile Microbes Across a Permeable Surface

Hemodynamic Characteristics of a Tortuous Microvessel Using High-Fidelity Red Blood Cell Resolved Simulations.

Tortuous microvessels are characteristic of microvascular remodeling associated with numerous physiological and pathological scenarios. Three-dimensional (3D) hemodynamics in tortuous microvessels influenced by red blood cells (RBCs), however, are largely unknown, and important questions remain. Is blood viscosity influenced by vessel tortuosity? How do RBC dynamics affect wall shear stress (WSS) patterns and the near-wall cell-free layer (CFL) over a range of conditions? The objective of this work was to parameterize hemodynamic characteristics unique to a tortuous microvessel. RBC-resolved simulations were performed using an immersed boundary method-based 3D fluid dynamics solver. A representative tortuous microvessel was selected from a stimulated angiogenic network obtained from imaging of the rat mesentery and digitally reconstructed for the simulations. The representative microvessel was a venule with a diameter of approximately 20 μm. The model assumes a constant diameter along the vessel length and does not consider variations due to endothelial cell shapes or the endothelial surface layer. Microvessel tortuosity was observed to increase blood apparent viscosity compared to a straight tube by up to 26%. WSS spatial variations in high curvature regions reached 23.6 dyne/cm2 over the vessel cross-section. The magnitudes of WSS and CFL thickness variations due to tortuosity were strongly influenced by shear rate and negligibly influenced by tube hematocrit levels. New findings from this work reveal unique tortuosity-dependent hemodynamic characteristics over a range of conditions. The results provide new thought-provoking information to better understand the contribution of tortuous vessels in physiological and pathological processes and help improve reduced-order models.