Search Results (13)

In this paper, we extend the theory of planar pseudo knots to the theories of annular and toroidal pseudo knots. Pseudo knots are defined as equivalence classes under Reidemeister-like moves of knot diagrams characterized by crossings with undefined over/under information. In the theories of annular and toroidal pseudo knots, we introduce their respective lifts to the solid and the thickened torus. Then, we interlink these theories by representing annular and toroidal pseudo knots as planar $\mathrm{O}$-mixed and $\mathrm{H}$-mixed pseudo links. We also explore the inclusion relations between planar, annular and toroidal pseudo knots, as well as of $\mathrm{O}$-mixed and $\mathrm{H}$-mixed pseudo links. Finally, we extend the planar weighted resolution set to annular and toroidal pseudo knots, defining new invariants for classifying pseudo knots and links in the solid and in the thickened torus. Full article

In this paper, we present two different ways for computing the Kauffman bracket skein module of $S^1\times S^2$, $\mathrm{KBSM}\left(S^1\times S^2\right)$, via braids. We first extend the universal Kauffman bracket type invariant V for knots and links in the Solid Torus ST, which is obtained via a unique Markov trace constructed on the generalized Temperley–Lieb algebra of type B, to an invariant for knots and links in $S^1\times S^2$. We do that by imposing on V relations coming from the braid band moves. These moves reflect isotopy in $S^1\times S^2$ and they are similar to the second Kirby move. We obtain an infinite system of equations, a solution of which is equivalent to computing $\mathrm{KBSM}\left(S^1\times S^2\right)$. We show that $\mathrm{KBSM}\left(S^1\times S^2\right)$ is not torsion free and that its free part is generated by the unknot (or the empty knot). We then present a diagrammatic method for computing $\mathrm{KBSM}\left(S^1\times S^2\right)$ via braids. Using this diagrammatic method, we also obtain a closed formula for the torsion part of $\mathrm{KBSM}\left(S^1\times S^2\right)$. Full article

The nonlinear pressure–flux relationship in the xylem of a conifer is attributed to the fluid–solid interaction within the bordered pits. However, the fluid–solid interactions between the torus–margo structure and the water flow within the pit lack comprehensive understanding. Herein, a fluid–solid interaction model was developed including the deformation of the torus–margo and the flow of water transportation. Nine pit samples were reconstructed using scanning electron microscopy (SEM) and transmission electron microscopy (TEM) pictures. Fluid–solid coupling models for pits in the stems of oriental arborvitae (Platycladus orientalis) were developed. The deflection of the torus was roughly proportional to the pressure difference between adjacent tracheids, while the pit resistance exhibited a considerable nonlinear increase. From 250 to 1750 Pa, the pit resistance increased from 4.1466 × 10¹⁶ Pa·s/m³ to 8.8344 × 10¹⁶ Pa·s/m³. The pit resistance decreased, and the pit’s ability to regulate water flow enhanced when the pit diameter increased. Both the pit resistance and the pit’s ability to regulate water decreased when the pit depth increased. The decrease in Young’s modulus for the margo promoted the nonlinear pressure–flux relationship in bordered pits. The findings provide theoretical evidence for the nonlinear relationship between pressure and flux in bordered pits, as well as for the prevention of gas bubble transit through a bordered pit during tracheid cavitation. The passive hydraulic regulation of bordered pits could increase flow resistance and reduce the water flow rate in the xylem, inhibiting tree transpiration. Full article

Bordered pits in conifers have been recognized as a significant evolutionary characteristic that served to impede the spread of embolisms between tracheids. Nevertheless, there was a lack of comprehensive understanding regarding the mechanical properties of the torus and the pit border in relation to the formation of contact seals in aspirated pits. A solid mechanics model was developed to study aspirated bordered pits, incorporating the elastic deformation of the torus–margo structure and the contact behavior between the torus and the pit border. Ten pit samples were reconstructed using transmission electron microscopy (TEM) and scanning electron microscopy (SEM) pictures in order to develop mechanical models for pits in the stems of Platycladus orientalis. Due to the limited contact area on the torus surface, the contact pressure between the torus and the pit border exceeded the air pressure in aspirated bordered pits. The external force and the duration required to seal pits decreased when the pit diameter increased and the pit depth decreased. The augmentation of the torus and margo mass necessitated a lengthier duration for the closure of cavities. The findings presented in this study offered theoretical support for the capillary-seeding hypothesis. The aspiration in bordered pits mechanically depended on the structural and material characteristics of the torus and margo. Full article

A bubble expanding and collapsing near a solid boundary develops a liquid jet toward the boundary. The jet leaves a torus bubble and induces vortices in the liquid that persist long after the bubble oscillations have ceased. The vortices are studied numerically in axial symmetry and compared to experiments in the literature. The flow field is visualized with different methods: vorticity with superimposed flow-direction arrows for maps at a time instant and colored-liquid-layer flow-field maps (dye advection) for following the complete long-term fluid flow up to a chosen time since bubble generation. Bubbles with equal energy—maximum radius in a free liquid $R_{\max }^{\infty }$= 500 µm—are studied for different distances $D_{\mathrm{init}}$ from the solid boundary. The interval of normalized distances $D^{*}$ = $D_{\mathrm{init}}$/$R_{\max }^{\infty }$ from 0.4 to 1.8 is covered. Two types of vortices were reported in experiments, one moving toward the solid boundary and one moving away from it. This finding is reproduced numerically with higher resolution of the flow field and in more detail. The higher detail reveals that the two types of vortices have different rotation directions and coexist with individually varying vorticity amplitude throughout the interval studied. In a quite narrow part of the interval, the two types change their strength and extent with the result of a reversal of the dominating rotational direction of the fluid flow. Thereby, the experimentally found transition interval could be reproduced and refined. It is interesting to note that in the vortex transition interval, the erosion of a solid surface is strongly augmented. Full article

In the chemical and water treatment industries, it is necessary to achieve maximum contact between the solid and liquid phase, thus promoting the mass and heat transfer, to obtain a homogeneous solution. Increasing stirring speed is the most recommended solution in different types of reactors: stirred tank, column, and tubular. However, this inadvertently increases the energy consumption of the industry. Determination of the minimum speed, labeled the just suspended speed (N_js) and crucial to attaining homogeneity, has been widely investigated. Numerous studies have been carried out to assess formulas for determining the solid particle speed in various reactor types. Given the limitations of the existing formulations based on a generalization of a unique equation for computing N_js for all soil classifications, it appears that most formulas can only approximate complex phenomena that depend on several parameters. A novel formula was developed, and the results given in this paper demonstrate the effectiveness of generating significant uncertainties for the estimation of N_js. The purpose of this study was the elaboration of experiment-based data-driven formulas to calculate N_js for different particle size classes. Nonlinear multiple regression (MNLR) models were used to generate the new formulas. The gradient descent optimization algorithm was employed to solve the hyperparameters of each novel equation, utilizing supervised learning. A comparison of the data indicated that the unique formulas presented in this study outperformed empirical formulas and provide a useful means for lowering energy consumption, while increasing the heat and mass transfer in torus type reactors. Full article

In the process of aerospace service, circular mesh antennas generate large nonlinear vibrations under an alternating thermal load. In this paper, the Smale horseshoe and Shilnikov-type multi-pulse chaotic motions of the six-dimensional non-autonomous system for circular mesh antennas are first investigated. The Poincare map is generalized and applied to the six-dimensional non-autonomous system to analyze the existence of Smale horseshoe chaos. Based on the topological horseshoe theory, the three-dimensional solid torus structure is mapped into a logarithmic spiral structure, and the original structure appears to expand in two directions and contract in one direction. There exists chaos in the sense of a Smale horseshoe. The nonlinear equations of the circular mesh antenna under the conditions of the unperturbed and perturbed situations are analyzed, respectively. For the perturbation analysis of the six-dimensional non-autonomous system, the energy difference function is calculated. The transverse zero point of the energy difference function satisfies the non-degenerate conditions, which indicates that the system exists Shilnikov-type multi-pulse chaotic motions. In summary, the researches have verified the existence of chaotic motion in the six-dimensional non-autonomous system for the circular mesh antenna. Full article

This work studied the feasibility of using a novel microreactor based on torus geometry to carry out a sample pretreatment before its analysis by graphite furnace atomic absorption. The miniaturized retention of total arsenic was performed on the surface of a magnetic sorbent material consisting of 6 mg of magnetite (Fe₃O₄) confined in a very small space inside (20.1 µL) a polyacrylate device filling an internal lumen (inside space). Using this geometric design, a simulation theoretical study demonstrated a notable improvement in the analyte adsorption process on the solid extractant surface. Compared to single-layer geometries, the torus microreactor geometry brought on flow turbulence within the liquid along the curvatures inside the device channels, improving the efficiency of analyte–extractant contact and therefore leading to a high preconcentration factor. According to this design, the magnetic solid phase was held internally as a surface bed with the use of an 8 mm-diameter cylindric neodymium magnet, allowing the pass of a fixed volume of an arsenic aqueous standard solution. A preconcentration factor of up to 60 was found to reduce the typical “characteristic mass” (as sensitivity parameter) determined by direct measurement from 53.66 pg to 0.88 pg, showing an essential improvement in the arsenic signal sensitivity by absorption atomic spectrometry. This methodology emulates a miniaturized micro-solid-phase extraction system for flow-through water pretreatment samples in chemical analysis before coupling to techniques that employ reduced sample volumes, such as graphite furnace atomic absorption spectroscopy. Full article

This paper proposes and researches a novel cylinder-type FOWT using a neutrally buoyant double-layer torus structure with annular flow; its oscillatory motion in severe sea conditions is controlled by a spinning top device designed as a neutrally buoyant double-layer torus structure with annular flow water in a torus structure with a small internal radius, and welded to the periphery of the cylinder-type FOWT underwater buoyancy-providing part. The rotational axis retention effect and the gyroscopic effect are considered appropriate approaches to suppress the oscillating motion of FOWT. To obtain a better hydrodynamic response, the scale of the torus structure, such as its radius, the radius of the internal annular flow water, and the angular velocity of the annular flow water are taken as the design parameters, and a large number of comparative calculations based on the fluid–solid coupling theory of potential flow are carried out to determine the appropriate design parameters. Eventually, on the basis of the obtained suitable design parameters, the proposed conceptual design approach is demonstrated to be feasible in view of the energy consumption. Full article

Pterostilbene (3,5-dimethoxy-4′-hydroxystilbene, PTB) is a natural dietary stilbene, occurring primarily in blueberries and Pterocarpus marsupium heartwood. The interest in this compound is related to its different biological and pharmacological properties, such as its antioxidant, anti-inflammatory, and anticarcinogenic activities and its capacity to reduce and regulate cholesterol and blood sugar levels. Nevertheless, its use in therapy is hindered by its low aqueous solubility; to overcome this limitation we studied the feasibility of the use of cyclodextrins (CDs) as solubility-enhancing agents. CDs are natural macrocyclic oligomers composed of α-d-glucose units linked by α-1,4 glycosidic bonds to form torus-shaped molecules, responsible for inclusion complex formation with organic molecules. In particular, the aim of this study was to evaluate the feasibility of complexation between PTB and native CDs using various preparative methods. The isolated solid products were characterized using differential scanning calorimetry (DSC), simultaneous thermogravimetric/DSC analysis (TGA/DSC), Fourier transform infrared (FT-IR) spectroscopy, and X-ray diffraction (XRD) on powder and single crystals. The results indicated little or no evidence of the affinity of PTB to complex with α-CD using the kneading method. However, with β-CD and γ-CD thermal analysis revealed an interaction which was also corroborated by FT-IR and ¹H-NMR spectroscopy. With β-CD, a hydrated complex of PTB was isolated and its characterization by single-crystal XRD revealed, for the first time, the mode of inclusion of the PTB molecule in the cavity of a CD. To complement the solid-state data, liquid-phase studies were carried out to establish the effect of CDs on the aqueous solubility of PTB and to determine the complex stoichiometries and the association constants for complex formation. Phase-solubility studies showed A_L-type profiles for α- and β-CD and a B_S profile for γ-CD, with K_1:1 values of 1144, 4950, and 133 M⁻¹ for α-CD·PTB, β-CD·PTB, and γ-CD·PTB, respectively. The stoichiometry of CD·PTB complexes, determined by Job’s method, revealed for each system a 1:1 molar ratio. The dissolution rate of PTB was approximately doubled just by employing simple physical mixtures, but the best performance was achieved by products obtained via kneading and co-precipitation, which effected the complete dissolution of PTB in 40 and 20 min for β-CD and γ-CD, respectively. Full article

Slurry transport systems are often limited in run length by the life of the pump internals, especially the impeller. The present work investigated abrasive wear of the impeller of a Hayward Gordon XR2(7) Torus Recessed Impeller slurry pump in a flow loop. The stock stainless steel impeller was replaced by a set of plastic test impellers with the same nominal geometry, fabricated by additive manufacturing (3D printing). A parametric set of abrasive wear experiments was conducted at five pump rotational speeds and three solid concentrations of garnet slurry in a pipe flow loop. Pump performance tests were conducted using impellers with progressive wear conditions, to investigate how a worn impeller affects hydraulic power delivery. A parabolic fit was imposed to describe the relationship between head and flow rate, and an empirical model was proposed to predict the pump head with damaged impellers. When the rotational speed is high, the damaged impeller has a larger effect on the pump’s performance than when the rotational speed is low. The head difference between the undamaged impeller and a 7.62%-mass-loss damaged impeller was 1.5 m at 1750 rpm rotational speed, however, for 850 rpm, the head difference was 1 m. Implications for pump diagnostics in other types of systems are discussed. This experiment gives a method for rapidly assessing wear locations, and provides a tool to predict wear rates on harder materials if scaling parameters are available. This first attempt at a scaling law is not reliable enough to accurately predict the wear rate for specific conditions, but shows the relative wear as a function of pertinent parameters. Full article

3D printing technology has been applied to various fields and its medical applications are expanding. Here, we fabricated implantable 3D bio-printed hydrogel patches containing a nanomedicine as a future tailored cancer treatment. The patches were prepared using a semi-solid extrusion-type 3D bioprinter, a hydrogel-based printer ink, and UV-LED exposure. We focused on the composition of the printer ink and semi-synthesized fish gelatin methacryloyl (F-GelMA), derived from cold fish gelatin, as the main component. The low viscosity of F-GelMA due to its low melting point was remarkably improved by the addition of carboxymethyl cellulose sodium (CMC), a pharmaceutical excipient. PEGylated liposomal doxorubicin (DOX), as a model nanomedicine, was incorporated into the hydrogel and liposome stability after photo-polymerization was evaluated. The addition of CMC inhibited particle size increase. Three types of 3D-designed patches (cylinder, torus, gridlines) were produced using a 3D bioprinter. Drug release was dependent on the shape of the 3D-printed patches and UV-LED exposure time. The current study provides useful information for the preparation of 3D printed nanomedicine-based objects. Full article

The three-dimensional (3D) flow below the interface of an evaporating liquid at a low pressure is visualized and quantified using scanning particle image velocimetry. The technique presented highlights the use of a single camera and a relatively fast moving laser sheet to image the flow for an application where using more than one camera is difficult. The technique allows collection of the full three-dimensional velocity vector map over the whole liquid volume. The out-of-plane component of the velocity has been determined using two different processing approaches: (i) deriving the full vector from a 3D cross-correlation of the particle volumes and (ii) applying the continuity equation to determine out-of-plane velocities from the calculated in-plane velocity vector fields. The results obtained from both methods showed good agreement with each other. The 3D velocity field reveals the existence of a torus shaped vortex below the evaporating meniscus that was induced by the exposure of the cold liquid to the warmer solid walls. The velocity data also shows that the maximum velocity occurs below the interface, not at the interface which highlights that the observed vortex is not driven by thermocapillary forces that usually govern the flow during evaporation at smaller scales. Full article