Tzyy-Leng Horng

Undular bore theory for the Gardner equation

We develop modulation theory for undular bores (dispersive shock waves) in the framework of the Gardner, or extended Korteweg-de Vries (KdV), equation, which is a generic mathematical model for weakly nonlinear and weakly dispersive wave propagation, when effects of higher order nonlinearity become important. Using a reduced version of the finite-gap integration method we derive the Gardner-Whitham modulation system in a Riemann invariant form and show that it can be mapped onto the well-known modulation system for the Korteweg-de Vries equation. The transformation between the two counterpart modulation systems is, however, not invertible. As a result, the study of the resolution of an initial discontinuity for the Gardner equation reveals a rich phenomenology of solutions which, along with the KdV-type simple undular bores, include nonlinear trigonometric bores, solibores, rarefaction waves, and composite solutions representing various combinations of the above structures. We construct full parametric maps of such solutions for both signs of the cubic nonlinear term in the Gardner equation. Our classification is supported by numerical simulations.

Facile simulation of carbon with wide pore size distribution for electric double-layer capacitance based on Helmholtz models

This study reports on a facile method based on Helmholtz models for simulating the electric double-layer capacitance of various forms of carbon in aqueous H2SO4 and KOH and organic tetraethylammonium tetrafluoroborate/acetonitrile electrolytes. The proposed method combines cylindrical pore models for macropores and mesopores with the slit-pore model for micropores exhibiting constant surface-based capacitance (C/S). The pore structures and pore size distribution of carbon are analyzed by using a method based on non-local density functional theory (NLDFT). We then used data related to the capacitance of microporous carbon to evaluate the constant C/S values produced by distinct electrolytes in carbon micropores and to determine the molecule-sieving effect of the micropores. The constant C/S values obtained from the micropores suggest that the effective static dielectric constant at the electrode–electrolyte interface is proportional to the thickness of the ion-solvating layer. The C/S values in mesopores decreased with a decrease in the pore size due to the effects of wall-curvature confinement. For an aqueous electrolyte, the C/S values in micropores are larger than those in mesopores and macropores due to the compactness of the ion-solvating layers, which account for the higher dielectric constant in the micropores. Our simulation results regarding the capacitance values of each carbon are in excellent agreement with experimental data, thereby verifying the reliability of the proposed model. This model is capable of providing reliable, precise predictions of capacitance values and also reveals the mechanism underlying the double-layer formation of distinct pores and the interfacial properties associated with capacitive performance.

Simulation analysis of airflow alteration in the trachea following the vascular ring surgery based on CT images using the computational fluid dynamics method

This study presents a computational fluid dynamics (CFD) model to simulate the three-dimensional airflow in the trachea before and after the vascular ring surgery (VRS). The simulation was based on CT-scan images of the patients with the vascular ring diseases. The surface geometry of the tracheal airway was reconstructed using triangular mesh by the Amira software package. The unstructured tetrahedral volume meshes were generated by the ANSYS ICEM CFD software package. The airflow in the tracheal airway was solved by the ESI CFD-ACE+ software package. Numerical simulation shows that the pressure drops across the tracheal stenosis before and after the surgery were 0.1789 and 0.0967 Pa, respectively, with the inspiratory inlet velocity 0.1 m/s. Meanwhile, the improvement percentage by the surgery was 45.95%. In the expiratory phase, by contrast, the improvement percentage was 40.65%. When the inspiratory velocity reached 1 m/s, the pressure drop became 4.988~Pa and the improvement percentage was 43.32%. Simulation results further show that after treatment the pressure drop in the tracheal airway was significantly decreased, especially for low inspiratory and expiratory velocities. The CFD method can be applied to quantify the airway pressure alteration and to evaluate the treatment outcome of the vascular ring surgery under different respiratory velocities.

Investigation of the Spatiotemporal Responses of Nanoparticles in Tumor Tissues with a Small-Scale Mathematical Model

The transport and accumulation of anticancer nanodrugs in tumor tissues are affected by many factors including particle properties, vascular density and leakiness, and interstitial diffusivity. It is important to understand the effects of these factors on the detailed drug distribution in the entire tumor for an effective treatment. In this study, we developed a small-scale mathematical model to systematically study the spatiotemporal responses and accumulative exposures of macromolecular carriers in localized tumor tissues. We chose various dextrans as model carriers and studied the effects of vascular density, permeability, diffusivity, and half-life of dextrans on their spatiotemporal concentration responses and accumulative exposure distribution to tumor cells. The relevant biological parameters were obtained from experimental results previously reported by the Dreher group. The area under concentration-time response curve (AUC) quantified the extent of tissue exposure to a drug and therefore was considered more reliable in assessing the extent of the overall drug exposure than individual concentrations. The results showed that 1) a small macromolecule can penetrate deep into the tumor interstitium and produce a uniform but low spatial distribution of AUC; 2) large macromolecules produce high AUC in the perivascular region, but low AUC in the distal region away from vessels; 3) medium-sized macromolecules produce a relatively uniform and high AUC in the tumor interstitium between two vessels; 4) enhancement of permeability can elevate the level of AUC, but have little effect on its uniformity while enhancement of diffusivity is able to raise the level of AUC and improve its uniformity; 5) a longer half-life can produce a deeper penetration and a higher level of AUC distribution. The numerical results indicate that a long half-life carrier in plasma and a high interstitial diffusivity are the key factors to produce a high and relatively uniform spatial AUC distribution in the interstitium.

Preparation of bismuth oxide photocatalyst and its application in white-light LEDs

Bismuth oxide photocatalysts were synthesized and coated on the front surface of phosphor-converted white light-emitting diodes to produce a safe and environmentally benign lighting source. Bismuth oxide photocatalyst powders were synthesized with a spray pyrolysis method at 500°C, 600°C, 700°C, and 800°C. Using the absorption spectrum in the blue and UV regions of the bismuth oxide photocatalysts, the blue light and UV leakage problems of phosphor-converted white LEDs can be significantly reduced. The experimental results showed that bismuth oxide photocatalyst synthesized at 700°C exhibited the most superior spectrum inhibiting ability. The suppressed ratio reached 52.33% in the blue and UV regions from 360 to 420 nm. Related colorimetric parameters and the photocatalyst decomposition ability of fabricated white-light LEDs were tested. The CIE chromaticity coordinates (x, y) were (0.349, 0.393), and the correlated color temperature was 4991K. In addition, the coating layer of photocatalyst can act as an air purifier and diffuser to reduce glare. A value of 66.2 ± 0.60 ppmv of molecular formaldehyde gas can be decomposed in 120mins.

Transcritical flow of a stratified fluid over topography: analysis of the forced Gardner equation

Transcritical flow of a stratified fluid past a broad localised topographic obstacle is studied analytically in the framework of the forced extended Korteweg–de Vries (eKdV), or Gardner, equation. We consider both possible signs for the cubic nonlinear term in the Gardner equation corresponding to different fluid density stratification profiles. We identify the range of the input parameters: the oncoming flow speed (the Froude number) and the topographic amplitude, for which the obstacle supports a stationary localised hydraulic transition from the subcritical flow upstream to the supercritical flow downstream. Such a localised transcritical flow is resolved back into the equilibrium flow state away from the obstacle with the aid of unsteady coherent nonlinear wave structures propagating upstream and downstream. Along with the regular, cnoidal undular bores occurring in the analogous problem for the single-layer flow modeled by the forced KdV equation, the transcritical internal wave flows support a diverse family of upstream and downstream wave structures, including kinks, rarefaction waves, classical undular bores, reversed and trigonometric undular bores, which we describe using the recent development of the nonlinear modulation theory for the (unforced) Gardner equation. The predictions of the developed analytic construction are confirmed by direct numerical simulations of the forced Gardner equation for a broad range of input parameters.

Quantum crystals in a trapped Rydberg-dressed Bose-Einstein condensate

Spontaneously crystalline ground states, called quantum crystals, of a trapped Rydberg-dressed Bose-Einstein condensate are numerically investigated. As a result described by a mean-field order parameter, such states simultaneously possess crystalline and superfluid properties. A hexagonal droplet lattice is observed in a quasi-two-dimensional system when dressing interaction is sufficiently strong. Onset of these states is characterized by a drastic drop of the non-classical rotational inertia proposed by Leggett [Phys. Rev. Lett. 25, 1543 (1970)]. In addition, an AB stacking bilayer lattice can also be attained. Due to an anisotropic interaction possibly induced by an external electric field, transition from a hexagonal to a nearly square droplet lattice is also observed.

Equilibrium vortex formation in ultrarapidly rotating two-component Bose-Einstein condensates

Equilibrium vortex formation in rotating binary Bose gases with a rotating frequency higher than the harmonic trapping frequency is investigated theoretically. We consider the system being evaporatively cooled to form condensates and a combined numerical scheme is applied to ensure the binary system being in an authentic equilibrium state. To keep the system stable against the large centrifugal force of ultrafast rotation, a quartic trapping potential is added to the existing harmonic part. Using the Thomas-Fermi approximation, a critical rotating frequency {Omega}{sub c} is derived, which characterizes the structure with or without a central density hole. Vortex structures are studied in detail with rotation frequency both above and below {Omega}{sub c} and with respect to the miscible, symmetrically separated, and asymmetrically separated phases in their nonrotating ground-state counterparts.

Spontaneous crystallization of skyrmions and fractional vortices in fast-rotating and rapidly quenched spin-1 Bose-Einstein condensates

We investigate the spontaneous generation of crystallized topological defects via the combining effects of fast rotation and rapid thermal quench on spin-1 Bose-Einstein condensates (BECs). By solving the stochastic projected Gross-Pitaevskii equation, we show that, when the system reaches equilibrium, a hexagonal lattice of skyrmions and a square lattice of half-quantized vortices can be formed in a ferromagnetic and antiferromagnetic spinor BEC, respectively, which can be imaged by using the polarization-dependent phase-contrast method.

Numerical Prediction of Hydrodynamic Loading on Circular Cylinder Array in Oscillatory Flow Using Direct-Forcing Immersed Boundary Method

Cylindrical structures are commonly used in offshore engineering, for example, a tension-leg platform (TLP). Prediction of hydrodynamic loadings on those cylindrical structures is one of important issues in design of those marine structures. This study aims to provide a numerical model to simulate fluid-structure interaction around the cylindrical structures and to estimate those loadings using the direct-forcing immersed boundary method. Oscillatory flows are considered to simulate the flow caused by progressive waves in shallow water. Virtual forces due to the existence of those cylindrical structures are distributed in the fluid domain in the established immersed boundary model. As a results, influence of the marine structure on the fluid flow is included in the model. Furthermore, hydrodynamic loadings exerted on the marine structure are determined by the integral of virtual forces according to Newton’s third law. A square array of four cylinders is considered as the marine structure in this study. Time histories of inline and lift coefficients are provided in the numerical study. The proposed approach can be useful for scientists and engineers who would like to understand the interaction of the oscillatory flow with the cylinder array or to estimate hydrodynamic loading on the array of cylinders.