Cylindrical Domain Research Articles

Influence of swirl ratio and radial Reynolds number on wind characteristics of multi-vortex tornadoes

In this study, systematic numerical simulations are conducted to investigate how swirl ratio and radial Reynolds number affect the wind characteristics of multi-vortex tornadoes. By properly controlling boundary conditions, multi-vortex tornadoes are produced in a cylindrical computational domain. Six cases with different swirl ratios are studied to examine the influence of swirl ratio, while five cases with different radial Reynolds number are studied to investigate the influence of radial Reynolds number. To facilitate the characterization, the core size and rotational speed of subvortices, as well as the relative distance between the subvortex and the core radius of the main vortex, are defined. The results demonstrate that the increase in swirl ratio leads to the increase in the number of subvortices. For the overall vortex, the increase in swirl ratio decreases the maximum tangential velocity but increases the core radius of the overall flow. For subvortices, for the case where four subvortices are produced, the increase in swirl ratio increases the core size of subvortices but decreases the rotational speed of subvortices. While the increase in radial Reynolds number does not change the number of subvortices produced, it decreases the core size of subvortices, but increases the rotational speed of each subvortex.

On the solvability of a fourth-order differential evolution equation on singular cylindrical domain in R4

Abstract In this paper, we investigate the solvability of a fourth-order differential evolution equation on singular cylindrical domain containing a cuspidal point. Some regularity results are obtained for the classical solutions by using the Dunford operational calculus.

STRICHARTZ ESTIMATES FOR THE WAVE EQUATION INSIDE CYLINDRICAL CONVEX DOMAINS

AbstractWe establish local-in-time Strichartz estimates for solutions of the model case Dirichlet wave equation inside cylindrical convex domains $\Omega \subset \mathbb {R}^ 3$ with smooth boundary $\partial \Omega \neq \emptyset $ . The key ingredients to prove Strichartz estimates are dispersive estimates, energy estimates, interpolation and $TT^*$ arguments. Strichartz estimates for waves inside an arbitrary domain $\Omega $ have been proved by Blair, Smith and Sogge [‘Strichartz estimates for the wave equation on manifolds with boundary’, Ann. Inst. H. Poincaré Anal. Non Linéaire26 (2009), 1817–1829]. We provide a detailed proof of the usual Strichartz estimates from dispersive estimates inside cylindrical convex domains for a certain range of the wave admissibility.

Weighted mixed norm estimates for fractional wave equations with VMO coefficients

This paper is a comprehensive study of Lp estimates for time fractional wave equations of order α∈(1,2) in the whole space, a half space, or a cylindrical domain. We obtain weighted mixed-norm estimates and solvability of the equations in both non-divergence form and divergence form when the leading coefficients have small mean oscillation in small cylinders.

On the logarithmic type boundary modulus of continuity for the Stefan problem: To the memory of Emmanuele DiBenedetto

A logarithmic type modulus of continuity is established for weak solutions to a two-phase Stefan problem, up to the parabolic boundary of a cylindrical space-time domain. For the Dirichlet problem, we merely assume that the spatial domain satisfies a measure density property, and the boundary datum has a logarithmic type modulus of continuity. For the Neumann problem, we assume that the lateral boundary is smooth, and the boundary datum is bounded. The proofs are measure theoretical in nature, exploiting De Giorgi's iteration and refining DiBenedetto's approach. Based on the sharp quantitative estimates, construction of continuous weak (physical) solutions is also indicated. The logarithmic type modulus of continuity has been conjectured to be optimal as a structural property for weak solutions to such partial differential equations.

Strongly Elliptic Differential-Difference Equations with Mixed Boundary Conditions in a Cylindric Domain

We consider strongly elliptic differential-difference equations with mixed boundary conditions in a cylindrical domain. We establish the connection between such problems and nonlocal mixed problems for strongly elliptic differential equations, and prove the uniqueness of solutions.

Highly Ordered Porous Inorganic Structures via Block Copolymer Lithography: An Application of the Versatile and Selective Infiltration of the "Inverse" P2VP-b-PS System.

A facile and versatile strategy was developed to produce highly ordered porous metal oxide structures via block copolymer (BCP) lithography. Phase separation of poly(2-vinylpyridine)-b-polystyrene (P2VP-b-PS) was induced by solvent vapor annealing in a nonselective solvent environment to fabricate cylindrical arrays. In this work, we thoroughly analyzed the effects of the film thickness, solvent annealing time, and temperature on the ordering of a P2VP-majority system for the first time, resulting in "inverse" structures. Reflectometry, atomic force microscopy, scanning electron microscopy, energy-dispersive X-ray spectroscopy, X-ray photoelectron spectroscopy (XPS), and transmission electron microscopy were used to characterize the formation of the highly ordered BCP morphology and the subsequently produced metal oxide film. At 40 min solvent annealing time, hexagonally close packed structures were produced with cylinder diameters ∼40 nm. Subsequently, the BCP films were infiltrated with different metal cations. Metal ions (Cr, Fe, Ni, and Ga) selectively infiltrated the P2VP domain, while the PS did not retain any detectable amount of metal precursor. This gave rise to a metal oxide porous structure after a UV/ozone (UVO) treatment. The results showed that the metal oxide structures demonstrated high fidelity compared to the BCP template and cylindrical domains presented a similar size to the previous PS structure. Moreover, XPS analyses revealed the complete elimination of the BCP template and confirmed the presence of the metal oxides. These metal oxides were used as hard masks for pattern transfer via dry etching as a further application. Silicon nanopores were fabricated mimicking the BCP template and demonstrated a pore depth of ∼50 nm. Ultimately, this strategy can be applied to create different inorganic nanostructures for a diverse range of applications, for example, solar cells, diodes, and integrated circuits. Furthermore, by optimizing the etching parameters, deeper structures can be obtained via ICP/RIE processes, leading to many potential applications.

Dynamical and chaotic behaviors of natural convection flow in semi-annular cylindrical domains using energy-conserving low-order spectral models

This paper presents a comprehensive theoretical study on the dynamical behavior of natural convection flow in the confined region between horizontal half-cylinders. For this purpose, a low-order spectral model with three modes will produce for the fluid flow system using the Galerkin technique. It proved that the generated model is physically meaningful, as it conserves energy in the dissipationless limit and has bounded solutions in the phase space. Analytical procedures indicate that the system has three stationary points and the onset of instability in the flow is when the Rayleigh number reaches a critical value. With an appropriate Lyapunov function is proved that for the Rayleigh numbers below the critical value, the flow is globally stable. As the Rayleigh number gets higher and reaches a fixed value, a Hopf bifurcation occurs, and chaotic motion appears in the system. The critical and Hopf Rayleigh numbers relation are derived parametrically based on dynamical system theories. Also, numerical simulations will carry on the presented low-order model. Different dynamical behaviors of this flow and its transition from regular to chaotic motion are explained, with phase portraits and velocity-temperature diagrams obtained by numerical solutions. This parametric study can pave the way for future researchers to determine at around values of critical parameters should an experiment or direct numerical simulation be performed to have more accurate data without resorting to tests at all operating conditions.

A 1D Reduced-Order Model (ROM) for a Novel Latent Thermal Energy Storage System

Phase change material (PCM)-based thermal energy storage (TES) systems are widely used for repeated intermittent heating and cooling applications. However, such systems typically face some challenges due to the low thermal conductivity and expensive encapsulation process of PCMs. The present study overcomes these challenges by proposing a lightweight, low-cost, and low thermal resistance TES system that realizes a fluid-to-PCM additively manufactured metal-polymer composite heat exchanger (HX), based on our previously developed cross-media approach. A robust and simplified, analytical-based, 1D reduced-order model (ROM) was developed to compute the TES system performance, saving computational time compared to modeling the entire TES system using PCM-related transient CFD modeling. The TES model was reduced to a segment-level model comprising a single PCM-wire cylindrical domain based on the tube-bank geometry formed by the metal fin-wires. A detailed study on the geometric behavior of the cylindrical domain and the effect of overlapped areas, where the overlapped areas represent a deviation from 1D assumption on the TES performance, was conducted. An optimum geometric range of wire-spacings and size was identified. The 1D ROM assumes 1D radial conduction inside the PCM and analytically computes latent energy stored in the single PCM-wire cylindrical domain using thermal resistance and energy conservation principles. The latent energy is then time-integrated for the entire TES, making the 1D ROM computationally efficient. The 1D ROM neglects sensible thermal capacity and is thus applicable for the low Stefan number applications in the present study. The performance parameters of the 1D ROM were then validated with a 2D axisymmetric model, typically used in the literature, using commercially available CFD tools. For validation, a parametric study of a wide range of non-dimensionalized parameters, depending on applications ranging from pulsed-power cooling to peak-load shifting for building cooling application, is included in this paper. The 1D ROM appears to correlate well with the 2D axisymmetric model to within 10%, except at some extreme ranges of a few of the non-dimensional parameters, which lead to the condition of axial conduction inside the PCM, deviating from the 1D ROM.

Well-posedness of the main mixed problem for the multidimensional Lavrentiev — Bitsadze equation

It is known that the oscillations of elastic membranes in space are modelled with partial differential equations. If the deflection of the membrane is considered as a function of u(x; t); x = (x1; :::; xm);m 2; then, according to the Hamilton principle, we arrive to a multidimensional wave equation.Assuming that the membrane is in equilibrium in the bending position, we also obtain the multidimensional Laplace equation from the Hamiltons principle.Consequently, the oscillations of elastic membranes in space can be modelled with a multidimensional Lavrentiev Bitsadze equation.The main mixed problem in the cylindrical domain for multidimensional hyperbolic equations in the space of generalized functions is well studied. In the works of the author, the well-posedness of this problem for multidimensional hyperbolic and elliptic equations is proved, and the explicit forms of classical solutions are obtained.As far as we know, these questions for multidimensional hyperbolic-elliptic equations have not been studied.The mixed problem with boundary-value conditions for the multidimensional Lavrentiev Bitsazde equation is ill-posed.In this paper, we prove the unique solvability and obtain an explicit form of classical solution of themain mixed problem with boundary and initial conditions for the multidimensional Lavrentiev Bitsadze equation.

A Rapid, Low-Cost, and High-Precision Multifrequency Electrical Impedance Tomography Data Acquisition System for Plant Phenotyping

Plant phenotyping plays an important role for the thorough assessment of plant traits such as growth, development, and physiological processes with the target of achieving higher crop yields by the proper crop management. The assessment can be done by utilizing two- and three-dimensional image reconstructions of the inhomogeneities. The quality of the reconstructed image is required to maintain a high accuracy and a good resolution, and it is desirable to reconstruct the images with the lowest possible noise. In this work, an electrical impedance tomography (EIT) data acquisition system is developed for the reconstruction and evaluation of the inhomogeneities by utilizing a non-destructive method. A high-precision EIT system is developed by designing an electrode array sensor using a cylindrical domain for the measurements in different planes. Different edible plant slices along with multiple plant roots are taken in the EIT domain to assess and calibrate the system, and their reconstructed results are evaluated by utilizing an impedance imaging technique. A non-invasive imaging is carried out in multiple frequencies by utilizing a difference method of reconstruction. The performance and accuracy of the EIT system are evaluated by measuring impedances between 1 and 100 kHz using a low-cost and rapid electrical impedance spectroscopy (EIS) tool connected to the sensor. A finite element method (FEM) modeling is utilized for image reconstruction, which is carried out using electrical impedance and diffuse optical tomography reconstruction software (EIDORS). The reconstruction is made successfully with the optimized results obtained using Gauss–Newton (GN) algorithms.

Hadamard’s example and solvability of the mixed Cauchy problem for the multidimensional Gellerstedt equation

Abstract In the theory of partial differential equations, an example constructed by J. Hadamard, which shows the instability of the solution of the Cauchy problem for the Laplace equation with respect to small changes in the initial data, is of great importance. Hadamard’s example served as the beginning of a systematic study of ill-posed problems in mathematical physics. On the other hand, the study of the Cauchy problem for the Laplace equation arises from problems of geophysics. At the same time, the question arises whether the Cauchy problem is correct for other elliptic equations including degenerate elliptic equations. We have constructed analogs of Hadamard’s example and established the incorrectness of the solution of the Cauchy problem for the Gellerstedt equation in two-dimensional and multidimensional cases. The condition of strong solvability of the mixed Cauchy problem for the multidimensional Gellerstedt equation in a cylindrical domain is found. The proof is based on the spectral properties of the Laplace operator and the properties of special functions.

Emergence of paraelectric, improper antiferroelectric, and proper ferroelectric nematic phases in a liquid crystal composed of polar molecules.

We have elaborated a theoretical approach for the description of polar nematic phases observed by Nishikawa etal. [Adv. Mater. 29, 1702354 (2017)0935-964810.1002/adma.201702354], their structures, and transitions between them. Specific symmetry contributions to the pair molecular potentials provide the molecular mechanisms responsible for the formation of proper and improper polarity on the macroscopic level. An improper antiferroelectric nematic M2 phase can arise between paraelectric nematic M1 and proper ferroelectric nematic MP in the temperature scale. The local polarization in M2 arises mostly due to the local splay deformation. The director distribution in M2 represents the conjugation of cylindrical waves with opposite splay and polarization signs. The director and polarization are parallel to the cylindrical domain axes in the middle of each cylinder but exhibit considerable (mostly radial) deformation on the periphery of each cylinder. Polarization vectors are mostly stacked antiparallel on the borders between the domains without the director disruption. The domain size decreases with the decreasing temperature, the percentage of the antiferroelectric decouplings increases, and M2 exhibits the first-order phase transition into proper ferroelectric MP. With the increasing temperature the domain size in the M2 phase increases, the domination of particular polar orientation of molecules reduces, and finally, the domain size diverges at particular temperature corresponding to the second-order phase transition from M2 to paraelectric M1. Variations of the polar and nonpolar orientational order parameters are estimated within each phase and between the phases. Our experimental and computer simulation results (also presented in the paper) fully support our theoretical findings.

On the Approximation of Solutions to the Heat Equation in the Lebesgue Class $$L^2$$ by More Regular Solutions

A criterion for the approximability of all solutions of the heat equation in a bounded cylindrical domain that belong to the Lebesgue class by more regular (e.g., Sobolev) solutions of the same equation in a bounded cylindrical domain with larger base is obtained. Namely, the complement of the smaller base to the larger one must have no (nonempty connected) compact components. As an important corollary, we prove a theorem on the existence of a doubly orthogonal basis for the corresponding pair of Hilbert spaces.

On the Stokes System in Cylindrical Domains

The existence of solutions to some initial-boundary value problem for the Stokes system is proved. The result is shown in Sobolev–Slobodetskii spaces such that the velocity belongs to \(W_r^{2+\sigma ,1+\sigma /2}(\Omega ^T)\) and the gradient of pressure to \(W_r^{\sigma ,\sigma /2}(\Omega ^T)\), where \(r\in (1,\infty )\), \(\sigma \in (0,1)\), \(\Omega ^T=\Omega \times (0,T)\). These are special Besov spaces: \(B_{r,r}^{2+\sigma ,1+\sigma /2}(\Omega ^T)\) and \(B_{r,r}^{\sigma ,\sigma /2}(\Omega ^T)\), respectively. The existence is proved by the technique of regularizer.

Precise dispersive estimates for the wave equation inside cylindrical convex domains

In this work, we establish precise local in time dispersive estimates for solutions of the model case Dirichlet wave equation inside cylindrical convex domains Ω ⊂ R 3 \Omega \subset \mathbb {R}^3 with smooth boundary ∂ Ω ≠ ∅ \partial \Omega \neq \emptyset . This result is the improved estimates established by Len Meas [C. R. Math. Acad. Sci. Paris 355 (2017), pp. 161–165]. Let us recall that dispersive estimates are key ingredients to prove Strichartz estimates. Strichartz estimates for waves inside an arbitrary domain Ω \Omega have been proved by Blair, Smith, Sogge [Proc. Amer. Math. Soc. 136 (2008), pp. 247–256; Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), pp.1817–1829]. Optimal estimates in strictly convex domains have been obtained by Ivanovici, Lebeau, and Planchon [Ann. of Math. 180 (2014), pp. 323–380]. Our case of cylindrical domains is an extension of the result of Ivanovici, Lebeau, and Planchon [Ann. of Math. 180 (2014), pp. 323–380] in the case when the nonnegative curvature radius depends on the incident angle and vanishes in some directions.

Freezing and thawing magnetic droplet solitons.

Magnetic droplets are non-topological magnetodynamical solitons displaying a wide range of complex dynamic phenomena with potential for microwave signal generation. Bubbles, on the other hand, are internally static cylindrical magnetic domains, stabilized by external fields and magnetostatic interactions. In its original theory, the droplet was described as an imminently collapsing bubble stabilized by spin transfer torque and, in its zero-frequency limit, as equivalent to a bubble. Without nanoscale lateral confinement, pinning, or an external applied field, such a nanobubble is unstable, and should collapse. Here, we show that we can freeze dynamic droplets into static nanobubbles by decreasing the magnetic field. While the bubble has virtually the same resistance as the droplet, all signs of low-frequency microwave noise disappear. The transition is fully reversible and the bubble can be thawed back into a droplet if the magnetic field is increased under current. Whereas the droplet collapses without a sustaining current, the bubble is highly stable and remains intact for days without external drive. Electrical measurements are complemented by direct observation using scanning transmission x-ray microscopy, which corroborates the analysis and confirms that the bubble is stabilized by pinning.

A probabilistic study of the kinetic Fokker–Planck equation in cylindrical domains

We consider classical solutions to the kinetic Fokker–Planck equation on a bounded domain \({\mathcal {O}} \subset ~{\mathbb {R}}^d\) in position, and we obtain a probabilistic representation of the solutions using the Langevin diffusion process with absorbing boundary conditions on the boundary of the phase-space cylindrical domain \(D = {\mathcal {O}} \times {\mathbb {R}}^d\). Furthermore, a Harnack inequality, as well as a maximum principle, are provided on D for solutions to this kinetic Fokker–Planck equation, together with the existence of a smooth transition density for the associated absorbed Langevin process. This transition density is shown to satisfy an explicit Gaussian upper-bound. Finally, the continuity and positivity of this transition density at the boundary of D are also studied. All these results are in particular crucial to study the behavior of the Langevin diffusion process when it is trapped in a metastable state defined in terms of positions.

Closed vortex filament in a cylindrical domain: Circulation quantization

This article investigates quantum oscillations of a vortex ring with zero thickness that evolves in a cylindrical domain V=D×[0,L]. The symbol D denotes the planar domain, which is bounded by some closed connected curve S. The quantization scheme of this dynamical system is based on the approach proposed by the author earlier [“Small oscillations of a vortex ring: Hamiltonian formalism and quantization,” Eur. J. Mech. B/Fluids 92, 100–106 (2022); arXiv:Math-ph/2112.04859v1]. As result, we find the discrete values Γn for circulation Γ. In contrast to the traditional approach where such quantities are usually postulated, the values Γn are deduced rigorously as the consequence of the conventional scheme of quantum theory. The model demonstrates the splitting of levels also. In particular, the levels correction values depend on the domain V: both the cylinder height L and the form of the curve S affect the final formula for the quantities Γn. Moreover, we prove that the basic circulation levels demonstrate a “fine structure.” These anomalous terms, which are proportional to the value ℏ2, are calculated in the article as well. The conclusions are compared with some results of numerical simulations by other authors “Intermittency of velocity circulation in quantum turbulence,” Phys. Rev. X 11, 011053 (2021).

Droplet-like Defect Annihilation Mechanisms in Hexagonal Cylinder-Forming Block Copolymers.

The annihilation of typical individual defects in hexagonal cylinder-forming block copolymers is investigated using the self-consistent field theory (SCFT) in conjunction with the string method. Usually, defect removal in two-dimensional hexagonal patterns involves reorganizing the cylindrical domains. Unlike atoms in solid crystals, the self-assembled cylindrical domains of block copolymers are "soft". Thus, the kinetic motions of the cylindrical domains resemble liquid droplets. Dislocations in hexagonal patterns are eliminated via creating and removing cylindrical domains. Our results show that new cylindrical domains are created via either a nucleation-like process or a fission-like process, whereas excessive domains are eliminated via a fusion-like or evaporation-like process. For weakly segregated block copolymers, the nucleation-like and evaporation-like processes are preferred.