Cylindrical Coordinates Research Articles

Studying a cylindrical moving boundary problem in ductal carcinoma in situ

In this paper, a mathematical model of ductal carcinoma in situ (DCIS) is considered as a moving boundary problem in cylindrical coordinates. Using the front-fixing technique, first, we transform the problem into a nonlinear parabolic problem on a fixed domain, and then some essential properties of the solution, such as the uniqueness of the solution, are shown. A numerical approach based on the finite difference method is established for solving the fixed domain problem. The solvability of the proposed finite difference method is proved, and the stability and convergence of the numerical solution are shown. For a test case problem, analytical and numerical results are compared to demonstrate the ability and effectiveness of the numerical method.

Electroosmotic mechanism of Ellis fluid with Joule heating, viscous dissipation and magnetic field effects in a pumping microtube.

The dynamics of electro-osmotically generated flow of biological viscoelastic fluid in a cylindrical geometry are investigated in this paper. This flux is the result of walls contracting and relaxing sinusoidally in a magnetic environment. The rheology of the fluid is accurately captured with the Ellis fluid (blood) model. Both Joule heating and viscous dissipation are accounted for during thermal analysis. The electric potential induced in the EDL is obtained by applying the Debye-Huckel linearization to the non-linear Poisson-Boltzmann equation. Mathematical modelling is incorporated in cylindrical coordinates in wave frame of reference. Assuming a long wavelength characterized by a low Reynolds number, the Ellis fluid model's governing equations are simplified. Subsequently, the differential equations that result are computed numerically utilizing the NDSolve utility that is integrated into Mathematica. Graphical representations are utilized to visually and comprehensively assess the thermal characteristics, flow features, heat transfer coefficient, and skin friction coefficient. Various factors are taken into consideration, including the impact of Ellis fluid parameters, electric double layer, magnetic field, Brinkman number, and Ohmic dissipation. Ellis fluid's axial velocity boosts with a rise of the electroosmotic parameter and power-law index while decreasing with an increase in the Hartmann number and material fluid parameter. The fluid temperature is directly proportional to EDL parameter and parameters of Ohmic and viscous dissipation. The current model may be used in clinical scenarios involving the gastrointestinal system and capillaries, electro-hydrodynamic therapy, delivery of drugs in pharmacological, and biomedical devices.

Disc distortion revisited

Abstract We revisit the dynamics of razor-thin, stone-cold, self-gravitating discs. By recasting the equations into standard cylindrical coordinates, the linearised vertical dynamics of an exponential disc can be followed for several gigayears on a laptop in a few minutes. An initially warped disc rapidly evolves into a flat inner region and an outward-propagating spiral corrugation wave that rapidly winds up and would quickly thicken a disc with non-zero radial velocity dispersion. The Sgr dwarf galaxy generates a similar warp in the Galactic disc as it passes through pericentre, and the warp generated by the dwarf’s last pericentre ∼35 Myr ago is remarkably similar to the warp traced by the Galaxy’s HI disc. The resemblance to the observed warp is fleeting but its timing is perfect. For the adopted parameters the amplitude of the model warp is a factor 3 too small, but there are several reasons for this being so. The marked flaring of our Galaxy’s low-α disc just outside the solar circle can be explained as a legacy of earlier pericentres.

Evaporating capillary bridges of pure and binary liquids

We present a numerical and experimental study on the evaporation of microliter capillary bridges of both pure and binary liquids. Specifically, we focused on capillary bridges of a binary liquid composed of water and isopropanol confined between poly-dimethylsiloxane coated surfaces. We developed a finite-element method-based numerical model to solve Laplace equations for vapor diffusion of the two species present in the capillary bridge, considering quasi-steady and diffusion-limited evaporation. We applied a modified version of Raoult's law, incorporating activity coefficients for binary liquids. The Galerkin finite element method was employed in axisymmetric cylindrical coordinates. The numerical model was validated against in-house experiments of side visualization on an evaporating capillary bridge. We quantified the effect of confinement from the plates on slowing down the diffusion of liquid vapor. The volume evolution of the binary liquid capillary bridge was found to be nonlinear, strongly influenced by the initial concentration of isopropanol in the capillary bridge. This nonlinearity is attributed to the faster diffusion of isopropanol vapor compared to water vapor. We examined the effects of height, substrate radius, contact angle, and composition on the evaporation characteristics. We proposed a computationally efficient reduced-order model for determining evaporation kinetics, which yields predictions very close to those of the numerical model.

Migration of a slip-spin solid spherical particle in a micropolar fluid-filled circular cylindrical tube

A combined study analytically and numerically for the axially symmetric creeping flow due to a slip-spin solid sphere surface moving in a microstructure fluid of micropolar type along the centreline of a circular cylindrical tube is introduced. This investigation was presented under low Reynolds number conditions. A general solution is obtained by superposing the essential solutions in both spherical and cylindrical coordinates to solve the Eringen micropolar field equations. The condition of the microrotation, along with couple stress, is used at the surface of the solid particle; while the microrotation is used at the inner cylindrical surface. Boundary conditions are imposed initially on the inner cylindrical surface using Fourier transforms and subsequently on the outer surface of the solid particle using a collocation technique. This paper aims to study the wall interaction problem of a translating slip-spin solid spherical particle in a micropolar fluid along the centreline of a circular cylindrical tube. The study also investigates the effect of the addition of slip conditions for velocity and microrotation on the surface of a solid particle. There is good convergence in the numerical findings obtained for the normalised hydrodynamic drag force (the tube-corrected factor) applying on the surface of the solid particle for several values of the micropolarity coefficient, slip-spin coefficients, and the ratio between the radius of the solid particle and tube. Regarding the flow of a solid spherical particle along the centreline of a cylindrical tube, our drag findings compare favourably to the solutions found in the literature. We found that the normalised drag force acting on the solid particle monotonically increases with the increase of the particle-to-tube radius ratio and reaches infinity in the limitless, with the increase of the micropolarity coefficient, and with the increase of the slip-spin coefficients for a steady ratio of particle-to-tube radius.

Second order conservative Lagrangian DG schemes for compressible flow and their application in preserving spherical symmetry in two-dimensional cylindrical geometry

In this paper, we construct a class of second-order cell-centered Lagrangian discontinuous Galerkin (DG) schemes for the two-dimensional compressible Euler equations on quadrilateral meshes. This Lagrangian DG scheme is based on the physical coordinates rather than the fixed reference coordinates, hence it does not require studying the evolution of the Jacobian matrix for the flow mapping between the different coordinates. The conserved variables are solved directly, and the scheme can preserve the conservation property for mass, momentum and total energy. The strong stability preserving (SSP) Runge-Kutta (RK) method is used for the time discretization. Furthermore, there are two main contributions. Firstly, differently from the previous work, we design a new Lagrangian DG scheme which is truly second-order accurate for all the variables such as density, momentum, total energy, pressure and velocity, while the similar DG schemes in the literature may lose second-order accuracy for certain variables, as shown in numerical experiments. Secondly, as an extension and application, we develop a particular Lagrangian DG scheme in the cylindrical geometry, which is designed to be able to preserve one-dimensional spherical symmetry for all the linear polynomials in two-dimensional cylindrical coordinates when computed on an equal-angle-zoned initial grid. The distinguished feature is that it can maintain both the spherical symmetry and conservation properties, which is very important for many applications such as implosion problems. A series of numerical experiments in the two-dimensional Cartesian and cylindrical coordinates are given to demonstrate the good performance of the Lagrangian DG schemes in terms of accuracy, symmetry and non-oscillation.

Estimating Interplanetary Magnetic Field Conditions at Mercury's Orbit From MESSENGER Magnetosheath Observations Using a Feedforward Neural Network

AbstractMercury's small magnetosphere is embedded in the dynamic and intense solar wind environment characteristic of the inner heliosphere. Both the magnitude and orientation of the interplanetary magnetic field (IMF) significantly influence the solar wind‐magnetospheric interaction at Mercury, driving phenomena such as magnetic reconnection. The MErcury Surface, Space Environment, Geochemistry and Ranging (MESSENGER) spacecraft provided in‐situ magnetic field measurements of the solar wind, the magnetosheath, and the magnetosphere along each orbit. However, it is a challenge to directly assess the IMF's impact on Mercury's plasma environment due to the temporal separation between observations within the solar wind and the magnetosphere, especially in the absence of an upstream monitor. Here, we present a feedforward neural network (FNN) trained on a subset of magnetosheath observations to estimate the strength and orientation of the IMF upstream of the bow shock. Utilizing magnetosheath magnetic field, cylindrical spatial coordinates, and heliocentric distance measurements, the FNN predicts upstream IMF conditions with an score of 0.70 and mean averaged error of 5.3 nT, thereby greatly decreasing the temporal separation between IMF estimates and magnetospheric measurements throughout the MESSENGER mission. This approach yields IMF estimates for all magnetosheath data measured by MESSENGER, providing a useful tool for future investigations of the IMF impact on Mercury's magnetosphere. This method will be integrable with the dual‐spacecraft BepiColombo magnetosheath measurements, providing useful estimates of upstream IMF conditions particularly during the extended periods in which neither spacecraft sample the solar wind. Our results demonstrate the utility of machine learning techniques on advancing space science research.

On the non-flatness nature of noncommutative Minkowski spacetime and the singular behavior of probes

It is more than a century-old concept that the Minkowski spacetime is flat. From the pure geometric point of view, we explicitly address the issue of whether a noncommutative Minkowski spacetime is flat or not. In the framework of the twisted-diffeomorphism approach to noncommutative gravity with canonical type noncommutative (NC) coordinate structure, one important result that we get is that the NC Minkowski spacetime parametrized either with spherical polar coordinates or with parabolic coordinates has nontrivial NC corrections to Riemann curvature tensor, Ricci tensor and curvature scalars. Another crucial result is that the curvature scalars have singular behavior at certain points, and these singularities are not coordinate singularities. The nature of these singularities clearly points towards the idea of high-energetic probes turning into black holes. The absence of any such noncommutative corrections, and thus any such singularities, in the cases of Cartesian coordinates and cylindrical coordinates leads to the conclusion that high-energetic probes do not turn into black holes when the canonical NC structure is considered for these coordinates.

Spectral Galerkin Approximation and Error Analysis Based on a Mixed Scheme for Fourth‐Order Problems in Complex Regions

ABSTRACTIn this article, an efficient spectral Galerkin method, which is based on a mixed scheme, is proposed and studied for solving fourth‐order problems in complex regions. The fundamental idea behind this approach is to transform the initial problem into an equivalent form in cylindrical coordinates and to reshape the computational domain into a product‐type rectangular one, which facilitates the utilization of spectral methods. However, when considering the equivalent fourth‐order form directly in cylindrical coordinates, it introduces intricate pole conditions and variable coefficients, posing challenges to both theoretical analysis and algorithm implementation. To address this, we employ the orthogonality of Fourier series to further decompose it into a sequence of decoupled two‐dimensional fourth‐order eigenvalue problems. For each such problem, we introduce an auxiliary function to transform it into an equivalent second‐order coupled system. Building on this, we formulate a mixed variational formulation and discrete scheme, and prove the error estimates for eigenvalue and eigenfunction approximations. Furthermore, we extend this algorithm to the two‐dimensional complex domains. Finally, a series of numerical examples are presented, and the numerical results validate the effectiveness of the algorithm and the correctness of the theoretical results.

Electromagnetic Surface Waves in a Relativistic Magnetized Plasma Propagating in a Cylindrical Column

Dispersion relations of electromagnetic surface waves in a magnetized relativistic plasma propagating in a cylindrical column surrounded by vacuum are derived by means of the relativistic fluid equations. The zero order relativistic beam is assumed to stream along the direction of the axial magnetic field. The coupled equations of the E- and B-wave modes are solved analytically in terms of the cylindrical coordinates in weak or intermediate values of the magnetic field in which the cyclotron frequency is much smaller than the relativistic Doppler-shifted frequency. The entangled field components of the E- and B-waves are decoupled by assuming that the flow is incompressible. The surface wave dispersion relations of the E- and B-wave modes show a strong similarity and symmetry. The static magnetic field splits the E- and B-waves, thus doubling the modes in unmagnetized plasma. By taking the radius of the cylinder infinite, the dispersion relations in a semi-infinite plasma are derived.

Numerical calculation of heat transfer in cylindrical coordinates using finite-difference method in evaluating temperature of different points of radiopharmaceutical vials

Choosing the type of vial of radiopharmaceuticals, the materials of canned goods and food, as well as the thickness of their walls, plays an important role in the transportation of companies' products. Regularly, the optimal design of the type of food wall has a significant effect on the shelf life of the products inside it, which can be more resistant to heat. Especially, carrying vials of radiopharmaceuticals, which plays a vital role in medical centers and hospitals in hot cities around the world. Therefore, in this research, different points of a container containing biological material are analyzed numerically to estimate the heat transfer and the temperature distribution in it.

Numerical studies of hydrodynamic relativistic astrophysical jet: The modification of vorticity transport equation

In this Letter, we have carried out the two-dimensional numerical simulation of axisymmetric relativistic jet in cylindrical coordinates by employing higher order finite volume method in PLUTO [Mignone et al., “PLUTO: A numerical code for computational astrophysics,” Astrophys. J. Suppl. Ser. 170, 228 (2007)] solver. The modified vorticity transport equation has been proposed for relativistic flow by taking the curl of the momentum equation, which shows significant change in the baroclinic vorticity production term due to relativistic effect. Both mathematical analysis and numerical results show that the vorticity production term due to baroclinic torque is heavily influenced due to the presence of specific enthalpy gradient and square of Lorentz factor gradient in a relativistic fluid flow.

Deformation and enclosed volume change of axisymmetric non-linear dielectric elastomer membrane pump: a theoretical study

Abstract This work investigates the change in the enclosed volume of an axisymmetric non-linear hyper-elastic membrane pump subjected to dielectric actuation. The equilibrium equations in cylindrical coordinates, coupled with non-linear constitutive relations, are solved numerically to obtain the deformed membrane geometry under pressure loading (using Variational Calculus). The effect of dielectric actuation is then incorporated by considering the change in material properties and deformation induced by the applied electric field. The deformed geometry under dielectric actuation is determined, and the change in enclosed volume is calculated by comparing the deformed states with and without the applied electric field. The proposed approach enables quantifying the volumetric change in non-linear hyper-elastic membrane pumps due to dielectric actuation for potential applications in soft robotics, adaptive optics, and microfluidics. By non-dimensionalizing variables, key dimensionless parameters governing the problem are identified for broader applicability. Furthermore, this methodology can estimate volume changes for different electric actuations when a specific material is chosen with experimentally derived parameter trends, facilitating material selection or synthesis to meet targeted volumetric requirements.

Investigation of thermal diffusivity measurement technique based on photoacoustic signal response under the influence of three-dimesional heat transfer

In this study, we investigated the feasibility of measuring the thermal diffusivity of thermally thick solid material using the photoacoustic (PA) signal response in a three-dimensional heat diffusion state. For this purpose, we developed a heat conduction simulation model based on cylindrical coordinates to replicate the PA effect on three-dimensional heat diffusion. By numerical analysis, it was found that the phase of PA signal deviates from one-dimensional theory prediction when the lateral heat diffusion of the solid sample reaches the chamber wall of PA cell. Specifically, it was found that the phase lag inflects at the frequency where the sample’s thermal diffusion length μs is 0.91 mm at a chamber radius of 3 mm. By utilizing this correlation, the infection frequency can potentially provide the thermal diffusivity of the thermally thick sample. The experimental validation qualitatively confirmed that the phenomena observed in the numerical analysis occur in practice. However, quantitatively, the experimental results showed discrepancies from the numerical analysis, which is likely due to the influence of the radial distribution of laser intensity. Nonetheless, this issue can be resolved in practical applications through calibration using a reference sample. Numerical analysis suggested that for materials with thermal diffusivity lower than that of air, measuring thermal diffusivity could be seemingly impossible. However, the experimental results demonstrated that calibration with a reference sample having sufficiently lower thermal diffusivity than that of air effectively compensates for the influence of air’s thermal diffusion. The significant factor affecting the accuracy of thermal diffusivity determination with the proposed method is identifying the inflection frequency. It will be necessary to explore an appropriate method for determining the frequency by integrating multiple sources of information, such as simultaneously observing the amplitude of the PA signal.

Extended King integral for modeling of parametric array loudspeakers with axisymmetric profiles.

Parametric array loudspeakers have been widely used in audio applications for generating directional audio beams. However, accurately calculating audio sound with a low computational load remains challenging, even for basic axisymmetric source profiles. This work addresses this challenge by extending the King integral in linear acoustics to incorporate both cumulative and local nonlinear effects, under the framework of the quasilinear solution without the paraxial approximation. The proposed method exploits the azimuthal symmetry in cylindrical coordinates to simplify modeling. To further improve computational efficacy, fast Hankel and Fourier transforms are employed for the radial and beam radiation directions, respectively. Numerical results with both uniform and focusing profiles demonstrate the advantages of the proposed approach over the traditional spherical wave expansion and direct integration methods, especially for larger aperture sizes. Specifically, for typical configurations with source aperture size of 0.2 m, we observe at least a 24-fold improvement in computational speed and a 227-fold reduction in memory requirements. These advancements allow us, for the first time, to present the sound field radiated by parametric array loudspeakers with a large aperture size of up to 0.5 m, without paraxial approximations. The implementation codes are available on https://github.com/ShaoZhe-LI/PAL_King.

Stratified equatorial flows in cylindrical coordinates with surface tension

This paper considers a mathematical model of steady flows of an inviscid and incompressible fluid moving in the azimuthal direction. The water density varies with depth and the waves are propagating under the force of gravity, over a flat bed and with a free surface, on which acts a force of surface tension. Our solution pertains to large scale equatorial dynamics of a fluid with free surface expressed in cylindrical coordinates. We also prove a regularity result for the free surface.

Solution scheme development of the nonhomogeneous heat conduction equation in cylindrical coordinates with neumann boundary condition by finite difference method

Partial differential heat conduction equations are typically used to determine temperature distribution within any solid domain. The difficulty and complexity of the solution of the equation depend on differential equation characteristics, boundary conditions, coordinate systems, and the number of dependent variables. In the current study, the numerical solution schemes were developed by the Explicit Finite Difference and the Implicit Method- the Crank-Nicolson techniques for the partial differential heat conduction equation including heat generation term described as one-dimensional, time-dependent with the Neumann boundary conditions. The solution schemes were, then, applied to the battery problem including highly varying heat generation. Besides, the solution of the problem was performed by using Matlab pdepe solver to verify the developed schemes. Results suggest that the Crank-Nicolson scheme is unconditionally stable, whereas the explicit scheme is only stable when the Courant-Friedrichs-Lewy condition requirement is less than 0.3404. Comparing the developed schemes to the results obtained from the pdepe solver, the schemes are as reliable as the pdepe solver with certain grid structures. Besides, the developed numerical schemes allow for shorter computational times than the pdepe solver at the same grid structures when considering CPU times.

Connecting high-resolution 3D chromatin maps with cell division and cell differentiation at the root apical meristem

Key messageWe used marker-free technologies to study chromatin at cellular resolution. Our results show asymmetric chromatin distribution, explore chromatin dynamics during mitosis, and reveal structural differences between trichoblast and atrichoblast cell.The shapes, sizes, and structural organizations of plant nuclei vary considerably among cell types, tissues, and species. This diversity is dependent on various factors, including cellular function, developmental stage, and environmental or physiological conditions. The differences in nuclear structure reflect the state of chromatin, which, in turn, controls gene expression and regulates cell fate. To examine the interrelationship between nuclear structure, cell morphology, and tissue-specific cell proliferation and differentiation processes, we conducted multiple visualizations of H3K4me1, H3K9me2, 4′,6-diamidino-2-phenylindole, 5-ethynyl 2′-deoxyuridine, and SCRI Renaissance 2200, followed by subsequent quantitative analysis of individual cells and nuclei. By assigning cylindrical coordinates to the nuclei in the iRoCS toolbox, we were able to construct in situ digital three-dimensional chromatin maps for all the tissue layers of individual roots. A detailed analysis of the nuclei features of H3K4me1 and H3K9me2 in the mitotic and the elongation zones in trichoblast and atrichoblast cells at the root apical meristem revealed cell type-specific chromatin dynamics with asymmetric distribution of euchromatin and heterochromatin marks that may be associated with cell cycle and cell differentiation characteristics of specific cells. Furthermore, the spatial distribution of nuclei stained with 5-ethynyl 2′-deoxyuridine in the epidermis and cortex tissues suggests short-range coordination of cell division and nuclear migration in a linear sequence through an unknown regulatory mechanism.

Design of adaptive barrier function-based backstepping finite time guidance control for interceptor-target systems

This study introduces an optimal finite-time backstepping sliding mode guidance law, leveraging an adaptive continuous barrier function to examine the interceptor-target system’s motion in cylindrical coordinates. The integration of sliding mode and backstepping controllers enhances the robustness and performance of the proposed guidance method amidst external disturbances and model uncertainties. This combination also improves transient response, reduces chattering, and lowers control efforts. The adaptive continuous barrier function employed in this method eliminates the need for prior knowledge of the upper bounds of uncertainties and disturbances. Additionally, it ensures that adaptation gains are not overestimated, leading to more accurate performance and complete elimination of chattering. The Lyapunov stability theory is used to prove the finite-time convergence of the state trajectories of the interceptor-target system to a predefined neighborhood around the origin, despite external disturbances and uncertainties. A genetic optimization algorithm is utilized to optimally select the parameters of the guidance method. The efficacy of the proposed technique is validated through simulation results and real-time experiments on the Baseline Speedgoat Real-Time Target Machine platform. The method’s effectiveness and performance are further demonstrated through analysis and simulation in the MATLAB/Simulink environment.

Circular phase shift migration based photoacoustic computational tomography

Abstract In the recent several decades, photoacoustic technique has proved to be capable of noninvasive biomedical imaging with scalable spatial resolution and high penetration depth. To achieve faithful image reconstruction, various method has been proposed, such as back projection, phase shift migration, model-based method, etc. These methods have simple implementations in homogeneous media, whereas in the cases of heterogeneous media (e.g., layered media), complicated modifications with respect to wave propagation at layer interface are always necessary. Here, we propose a frequency domain algorithm extension of phase shift migration in polar coordinates, which is obtained by making second order approximation to the optoacoustic point spread function in cylindrical coordinates and thereafter step by step wavenumber-frequency domain wave field extrapolation along the axial direction. Theoretical simulation and phantom experimental results demonstrate that the proposed circular phase shift migration method can well solve the layered heterogeneous reconstructive problem without modifying the wave propagation model. The acoustic diffraction limited acoustic resolution (up to 225 μm) and high imaging SNR (at least 16 dB) are obtained.