Terms Of Series Expansion Research Articles

Analysis of a Reiner–Rivlin liquid sphere enveloped by a permeable layer

The present article investigates the axisymmetric flow of a steady incompressible Reiner–Rivlin liquid sphere enveloped by a spherical porous layer using the cell model technique. The Brinkman-extended Darcy model is deployed for the porous medium hydrodynamics, and isotropic permeability is considered. The stream function solutions of the governing equations are obtained, which involves the Gegenbauer functions and the modified Bessel functions. An asymptotic series expansion in terms of the Reiner–Rivlin liquid parameter S has been employed to determine the expression of the flow field for the Reiner–Rivlin liquid. Boundary conditions on the cell surface corresponding to the Happel, Kuwabara, Kvashnin, and Cunningham models are considered. Analytical expressions are derived for dimensionless pressure, tangential stress, and the couple stress components using the method of separation of variables and Gegenbauer functions/polynomial. The integration constants are evaluated with appropriate boundary conditions on the inner and outer boundary of the porous zone with the aid of Mathematica symbolic software. Solutions for the drag force exerted by the Reiner–Rivlin fluid on the porous sphere are derived with corresponding expressions for the drag coefficient. Mathematical expression of the drag coefficient, pressure distribution, velocity profile, and separation parameter is established. On the basis of viscosity ratio, permeability parameter, dimensionless parameter, and the volume fraction, variations of the drag coefficient, velocity profiles, fluid pressure, and separation parameter (SEP) are investigated with their plots. The effects of the streamline pattern make the flow more significant for the Mehta–Morse when compared to the other models. Additionally, the mathematical expression of the separation parameter (SEP) is also calculated, which shows that no flow separation occurs for the considered flow configuration and is also validated by its pictorial depiction. This problem is motivated by emulsion hydrodynamics in chemical engineering where rheological behavior often arises in addition to porous media effects.

A computational method for a thermoacoustic boundary layer in a gas-filled tube.

This paper demonstrates a computational method to solve equations for a thermoacoustic boundary layer in a gas-filled tube subjected to a temperature gradient axially. With applications to numerical simulations of thermoacoustic systems, the purpose is to establish a method to evaluate numerically a velocity vb at the edge of a boundary layer directed into a core region in the outside of the layer. The computational method exploits a series expansion in terms of Chebyshev polynomials. Solved is a benchmark example for which analytical solutions are available. Assuming that a quiescent gas is started by an impulse and an acoustic field in the core region is given, an axial velocity and a temperature disturbance in the boundary layer are sought, from which vb is derived. Comparing the numerical solutions obtained against the analytical ones, it is found that the acoustic field and ultimately vb are well obtained with good accuracy. It is thus concluded that the present computational method will apply to numerical simulations based on the boundary-layer theory.

Fourier method for the Neumann problem on a torus

The Fourier method approach to the Neumann problem for the Laplacian operator in the case of a solid torus contrasts in many respects with the much more straight forward situation of a ball in 3-space. Although the Dirichlet-to-Neumann map can be readily expressed in terms of series expansions with toroidal harmonics, we show that the resulting equations contain undetermined parameters which cannot be calculated algebraically. A method for rapidly computing numerical solutions of the Neumann problem is presented with numerical illustrations. The results for interior and exterior domains combine to provide a solution for the Neumann problem for the case of a shell between two tori.

Nonlinear vibrations of doubly curved micropanels reinforced by graphene nanoplates in length direction under non-uniform thermal loading

Structures made of composites are quite common in different industries entailing automotive, marine, and aircraft industries, thanks to their favorable lightweight and strength. This study investigated the nonlinear vibrations of the various shapes of size-dependent graphene nanoplatelets reinforced micropanels under non-uniform thermal loading. In diverse ways in the axial direction, the material characteristics of graphene nanoplates (GPLs) alter. Taylor’s series expansion terms in curvature coordinates are used to illustrate the displacement field. Nonlinear von Karman theory and Hamiltonian principle work together to derive the nonlinear form of the pertinent equations and boundary conditions. Strain gradient theory with consideration for inherent length scale characteristics is investigated to account for size effects. By discretizing the spatial domain equations, deriving Duffing-type equations, and using Kronecker and Hadamard products, the governing equations and numerous nonlinear boundary edges are resolved. Using the techniques and answers outlined above, the results section is produced. There are four sections for this research. The findings are cross-checked with those from other studies that have been published in the first section. The effects of geometric and physical factors are shown in Parts 2, 3, and 4 on the nonlinear to linear frequency curve, the nonlinear frequency curve, and the sensitivity of the nonlinear frequency-to-size effect (SNFSE).

Aspects of thermal one-point functions and response functions in AdS black holes

We revisit the problem of analytically computing the one-point functions for scalar fields in planar AdS black holes of arbitrary dimension, which are sourced by the Weyl squared tensor. We analyze the problem in terms of power series expansions around the boundary using the method of Frobenius. We clarify the pole structure of the final answer in terms of operator mixing, as argued previously by Grinberg and Maldacena. We generalize the techniques to also obtain analytic results for slowly modulated spatially varying sources to first nontrivial order in the wave vector for arbitrary dimension. We also study the first-order corrections to the one-point function of the global AdS black hole at large mass, where we perturb in terms that correspond to the curvature of the horizon.

Simulation of a plasmonic sensor using kinetic theory of plasma with the Vlasov equation in MATLAB

This research proposes a mathematical model for a plasmonic sensor using kinetic theory of plasma with the Vlasov equation. A nanoantenna cavity of a plasmonic material is driven by an input electromagnetic wave, which changes the charge density and current flow in the cavity, resulting in a change in the Fermi distribution function of the charged particles. The results are achieved in terms of current density and conductivity by solving the Boltzmann transport equation, Maxwell’s equations, and Taylor series expansion in terms of perturbed electric fields with linear integro differential equations. The results are simulated using MATLAB. The changes in current density and conductivity are validated by experimental analysis of graphene plasmonic material using patch antenna with the dielectric substrates SiO2 and Al2O3. By varying the applied electric fields, current changes at the output of the plasmonic antenna are analyzed using signal-processing techniques. Wavelet transforms are used to find the space-scale behavior of the output signals, such as current density variation, voltage variation, and susceptibility change with sub-band coding techniques in terms of wavelet coefficients.

Creeping flow of non-Newtonian fluid through membrane of porous cylindrical particles: A particle-in-cell approach

The present study is an attempt to deal with hydrodynamic and thermal aspects of the incompressible Carreau fluid flow past a membrane consisting of uniformly distributed aggregates of porous cylindrical particles enclosing a solid core which aims to provide a comprehensive study of the impact of non-Newtonian nature of Carreau fluid in the filtration process through membranes. The non-Newtonian characteristic of Carreau fluid is adopted to describe the mechanism of the pseudoplastic flow through membranes. The layout of the fluid flow pattern is separated into two distinct areas in which the area adjacent to the solid core of the cylindrical particle is considered as porous. However, the region surrounding the porous cylindrical particle is taken as non-porous (clear fluid region). The Brinkman equation governs the porous region, whereas the non-porous region is regulated by the Stokes equation. The nonlinear governing equations of the Carreau fluid flow in the different regions are solved using an asymptotic series expansion in terms of the small parameters, such as Weissenberg number ( We ≪ 1 ) and a non-dimensional parameter ( S ≪ 1 ), for the higher permeability of the porous material. For large permeability, the expression of velocity is derived, and the same has been used to compute the hydrodynamic permeability, Kozeny constant, and temperature profile. The numerical scheme (NDSolve in Mathematica) is used to solve the singularly perturbed boundary value problems in the case of small permeability of the porous medium [i.e., ( S ≫ 1 )]. The graphical analysis illustrating the outcomes of the effects of varying control parameters such as the power-law index, viscosity ratio parameter, permeability of the porous medium, Weissenberg number, and Nusselt number on the membrane permeability, Kozeny constant and temperature profile are discussed comprehensively and validated with previously published works on the Newtonian fluid in the limiting cases. The notable determination of the present study is that the Carreau fluid parameters, such as the Weissenberg number, power-law index, and viscosity ratio parameter, have a significant impact on the velocity, and hence, the membrane permeability, Kozeny constant, and temperature profile. The results showed a significant increase in the flow velocity and hydrodynamic permeability as the dominance of elastic forces over viscous forces increased in the case of high permeability ( S ≪ 1 ). The velocity gets a slight reduction for lower permeability of the porous material ( S ≫ 1 ); however, the hydrodynamic permeability behaves similar to the higher permeability of the porous material. The findings of the proposed work may be instrumented in analyzing various processes, including wastewater treatment filtration processes, and blood flow through smooth muscle cells. The proposed work, however, requires experimental verification.

Wiener–Hopf Difference Equations and Semi-Cardinal Interpolation with Integrable Convolution Kernels

Let Hsubset {mathbb {Z}}^d be a half-space lattice, defined either relative to a fixed coordinate (e.g. H = {mathbb {Z}}^{d-1}!times !{mathbb {Z}}_+), or relative to a linear order preceq on {mathbb {Z}}^d, i.e. H = {jin {mathbb {Z}}^d: 0preceq j}. We consider the problem of interpolation at the points of H from the space of series expansions in terms of the H-shifts of a decaying kernel phi . Using the Wiener–Hopf factorization of the symbol for cardinal interpolation with phi on {mathbb {Z}}^d, we derive some essential properties of semi-cardinal interpolation on H, such as existence and uniqueness, Lagrange series representation, variational characterization, and convergence to cardinal interpolation. Our main results prove that specific algebraic or exponential decay of the kernel phi is transferred to the Lagrange functions for interpolation on H, as in the case of cardinal interpolation. These results are shown to apply to a variety of examples, including the Gaussian, Matérn, generalized inverse multiquadric, box-spline, and polyharmonic B-spline kernels.

Global approximate solution of SIR epidemic model with constant vaccination strategy

Providing a global semi-analytical method has the advantage of offering a global and accurate solution to the epidemic model, which can help in studying and controlling the spread of the disease. Unlike numerical methods such as the Runge–Kutta–Fehlberg method, global semi-analytical methods can provide an explicit expression of the solution over the entire time period, including obtaining the peak time, which is crucial information for understanding disease spread. In this paper, a global semi-analytical method based on the two-point Padé approximants for solving the SIR epidemic model of childhood diseases is presented. The objective of this study is to examine the temporal dynamics of a childhood disease when a preventive vaccine is present. For this purpose, we have first derived the solution of the SIR model of childhood diseases in terms of series expansions for small and large values. Then, the theory of two-point Padé approximations is used to provide the global approximate solution. The peak time related to this model is also obtained via these global approximants. Furthermore, in order to show the efficiency of our study, some graphs have been given to compare our results with those obtained using the classical Padé approximations and the numerical Runge–Kutta–Fehlberg method.

On the properties of some inversion methods of the Laplace transform

The problem of inversion of the integral Laplace transform, which belongs to the class of ill-posed problems, is considered. Integral equations are reduced to ill-conditioned systems of linear algebraic equations, in which the unknowns are either the coefficients of the series expansion in terms of special functions, or the approximate values of the desired original at a number of points. Described method of inversion using special quadrature formulas of the highest degree of accuracy and the characteristics of the accuracy and stability of this method are indicated. Quadrature inversion formulas are constructed, which are adapted for the inversion of long-term and slow processes of linear viscoelasticity. A method of deformation of the integration contour in the Riemann-Mellin inversion formula is proposed, which leads the problem to the calculation of certain integrals and allows obtaining error estimates. A method is described for determining the possible breakpoints of the original and calculating the magnitude of the jump at these points.

Improved convergence based on two-point Padé approximants: Simple shear, uniaxial elongation, and flow past a sphere

We investigate the performance of two-point Padé approximants on the truncated series solutions for the shear and elongational viscosities (obtained directly from the exact solutions or by asymptotic methods) for viscoelastic flows modeled with the Giesekus constitutive equation. Τhe major dimensionless groups are the Weissenberg number ( Wi), the Giesekus mobility parameter ( α), and the ratio of the polymer viscosity to the total viscosity of the fluid ( ηp) in terms of which the exact analytical solutions for the shear and elongational viscosities are found. The exact solutions are used to derive truncated series expansions in terms of the Weissenberg number at the Newtonian limit (zero Wi) and in terms of the inverse of the Weissenberg number at the purely elastic limit (infinite Wi). These series are then used to construct low- and high-order two-point diagonal Padé approximants. It is found that both the low- and high-order formulas predict the right trends over the entire range of Wi, from the Newtonian limit to the pure elastic limit. As expected, the high-order formula is much more accurate than the low-order formula. The results show that viscoelastic phenomena, such as shear thinning and extensional thickening, at any value of the Weissenberg number can be predicted analytically with acceptable accuracy using the proposed two-point Padé non-linear analysis. Finally, a fourth-order formula in terms of the Weissenberg number (or in terms of the Deborah number) for the drag force on a rigid sphere that translates with constant speed in a Giesekus fluid is also provided along with a discussion regarding the missing information from the literature, that is, required to perform the proposed non-linear analysis.

Expansion of some well-known functions and classical polynomials in series of the generalized Laguerre polynomials.1

The generalized Laguerre polynomials form a complete set (orthogonal and normalized) in the space ** with respect to a certain weighting function because they are the Eigen functions of a second-order differential operator. Here we shall show how to expand some well-known classical polynomials such as the Legendre and the Hermite polynomials in series of generalized Laguerre polynomials. Since the generalized Laguerre orthogonal polynomials are set of polynomials that are mutually orthogonal to each other with respect to a measure of weighting function that is just the integrand of the gamma function, thus it grants us the guarantee of the ability to expand the first kind of Bessel functions and the gamma function in terms of the generalized Laguerre polynomials. The series expansion in terms of the generalized Laguerre polynomials can be achieved by following various approaches. For instance, the series expansion of the first kind Bessel functions is gained by the generalized hypergeometric function approach, whereas the series expansion of gamma function was obtained directly by the usual way that is by calling the orthogonality and orthonormality properties of the generalized Laguerre polynomials. Another powerful technique to gain the series expansion of the classical polynomials is the generating function approach as it has been followed here to obtain a series expansion of the Legendre and the Hermite polynomials in terms of the generalized Laguerre polynomials. The series expansion in series of the generalized Laguerre polynomials has a variety of applications in mathematics, physics, and engineering

Numerical computation of the effective thermal conductivity of two-phase composite materials by digital image analysis

We consider the numerical determination of the effective thermal conductivity of 2D, two-phase composite materials, whose microstructural geometries are described by digital images. More specifically, key microstructural data pertaining to the spatial distribution and volume fractions of the constituent phases in the material are extracted, along with the shapes of the interphase boundaries, using computer vision techniques. For inclusions of simple shapes (polygons, circles, ellipses), the extracted interphase boundary data is easily obtained and can be exploited immediately (e.g. polygonal vertices) or can be fitted with smooth curves, using least square error minimization techniques in order to reconstruct parametric representations in the case of circular or elliptical boundaries. For more intricate microstructure geometries characterized by clustered or agglomerated intersecting inclusions, piecewise cubic spline functions are adopted to obtain a least-square fitting of the interphase boundaries. The effective thermal conductivities are then calculated based on a boundary integral formulation defined over the reconstructed inclusion boundaries. They are expressed in terms of power series expansions of a conductivity contrast parameter of the constituent phases in the material. To accelerate the convergence of these expansions, nonlinear transformations (known as Padé Approximants) are used to yield rational approximations of the effective thermal conductivity tensor. Tests conducted on a variety of 2D, two-phase microstructures yielded accurate results over a wide range of microstructure image resolution, regardless of the contrast parameters of the constituent phases, thus, validating the efficacy and robustness of the proposed scheme.

An Artificial Neural Network Approach for Solving Space Fractional Differential Equations

The linear algebraic system generated by the discretization of fractional differential equations has asymmetry, and the numerical solution of this kind of problems is more complex than that of symmetric problems due to the nonlocality of fractional order operators. In this paper, we propose the artificial neural network (ANN) algorithm to approximate the solutions of the fractional differential equations (FDEs). First, we apply truncated series expansion terms to replace unknown function in equations, then we use the neural network to get series coefficients, and the obtained series solution can make the norm value of loss function reach a satisfactory error. In the part of numerical experiments, the results verify that the proposed ANN algorithm can make the numerical results achieve high accuracy and good stability.

Effect of ring-source release on dispersion process in Poiseuille flow with wall absorption

The analysis of Taylor dispersion in a fluid flowing through a narrow pipe is of considerable interest among the scientific community. Most of the studies on dispersion primarily considered a transversely uniform release of contaminant at the initial moment, while only a few have addressed the influence of the initial condition on the dispersion process in a tube flow. The present work investigates the transient dispersion process with a ring-source release of solute by means of an unsteady two-dimensional transport model. An irreversible absorptive type of a first-order reaction is also considered. The transport model is solved using the series expansion technique of W. N. Gill [“A note on the solution of transient dispersion problems,” Proc. R. Soc. A 298, 335 (1967)], known as the generalized dispersion technique. The whole transport model is explained using the transport coefficients obtained from the first four terms of Gill's series expansion. Meanwhile, the concept of statistical moments is also used for finding the transport coefficients. The study outcomes are well established, satisfying the result of existing literature under limiting conditions. The inclusion of the ring-source release of solute in the study of dispersion has a significant effect at the initial time. In addition, the view of the cross-sectional distribution of solute concentration gives more detailed information on solute scattering under the proposed model.

An analytic element model for highly fractured elastic media

AbstractWe present an analytic formulation for an elastic medium under uniform stress and in plane strain, with a very large number of cracks that are allowed to intersect. The singularities at the tips of the cracks are represented exactly; the solution is written in terms of series expansions of negative powers of a complex variable defined outside the unit circle in a complex reference plane. We modeled an assembly of 10,000 cracks and show stress trajectories both for the whole model and for individual cracks, chosen at random from the assembly.

Behavior of Coefficients of Series of Exponentials of Finite Order Near the Boundary

Let G be a bounded convex domain with a smooth boundary in which a given system of exponents is not complete. For a class of analytic functions in G that can be represented in G by a series of exponentials, we examine the behavior of coefficients of the series expansion in terms of the growth order near the boundary ∂G. We establish two-sided estimates for the order through characteristics depending only on the indices of the series of exponentials and the support function of the domain (these estimates are strong). As a consequence, we obtain a formula for calculating the growth order through the coefficients.

DiffExp, a Mathematica package for computing Feynman integrals in terms of one-dimensional series expansions

DiffExp is a Mathematica package for integrating families of Feynman integrals order-by-order in the dimensional regulator from their systems of differential equations, in terms of one-dimensional series expansions along lines in phase-space, which are truncated at a given order in the line parameter. DiffExp is based on the series expansion strategies that were explored in recent literature for the computation of families of Feynman integrals relevant for Higgs plus jet production with full heavy quark mass dependence at next-to-leading order. The main contribution of this paper, and its associated package, is to provide a public implementation of these series expansion methods, which works for any family of integrals for which the user provides a set of differential equations and boundary conditions (and for which the program is not computationally constrained). The main functions of the DiffExp package are discussed, and its use is illustrated by applying it to the three loop equal-mass and unequal-mass banana graph families. Program summaryProgram title: DiffExpCPC Library link to program files:https://doi.org/10.17632/j43fkg8vjt.1Developer's repository link:https://gitlab.com/hiddingm/diffexpLicensing provisions: GPLv3Programming language: Mathematica (Wolfram Language)Nature of problem: Obtaining high-precision numerical results for families of Feynman integrals in dimensional regularization, at arbitrary points in phase-space and in an efficient manner, from the system of differential equations satisfied by the integrals and a set of suitable boundary conditions.Solution method: The program provides a public implementation of the integration strategy outlined in Ref. [1]. A line is taken from a given boundary point to a point in phase-space at which numerical results are desired. The line is subdivided into multiple segments, and along each line segment the differential equations are solved in terms of one-dimensional series expansions with a radius of convergence that covers the respective line segment, and up to high order in the line parameter. Boundary conditions are transported by matching results along neighboring line segments. Branch points are crossed by analytically continuing logarithms and square roots that appear in the series expansions. By evaluating the series solutions of the last line segment at the endpoint, the desired numerical results are obtained.Additional comments including restrictions and unusual features: Depending on the complexity of the differential equations of the integrals, the package is limited by CPU speed and/or memory. The prefactors of the Feynman integrals in the basis of master integrals for which the differential equations are given, should only contain rational functions and/or square roots. Fine-tuning of the configuration options may be required for families involving sectors with a large numbers of coupled integrals. Reference[1]F. Moriello, J. High Energy Phys. 2020 (2020) 150. arXiv:1907.13234, https://doi.org/10.1007/JHEP01(2020)150.

AN ITERATIVE FORMULA FOR SIMULTANEOUS LOCATION OF THE ZEROS OF A POLYNOMIAL

Needless to say that the search for efficient algorithms for determining zeros of polynomials has been continually raised in many applications. In this paper we give a cubic iteration method for determining simultaneously all the zeros of a polynomial – assumed distinct – starting with ‘reasonably close’ initial approximations – also assumed distinct. The polynomial – in question – is expressed in its Taylor series expansion in terms of the initial approximations and their correction terms. A formula with cubic rate of convergence – based on retaining terms up to 2ndorder of the expansion in the correction terms – is derived.

Apparent contact angle of drops on liquid infused surfaces: geometric interpretation.

We theoretically investigate the apparent contact angle of drops on liquid infused surfaces as a function of the relative size of the wetting ridge and the deposited drop. We provide an intuitive geometrical interpretation whereby the variation in the apparent contact angle is due to the rotation of the Neumann triangle at the lubricant-drop-gas contact line. We also derive linear and quadratic corrections to the apparent contact angle as power series expansion in terms of pressure differences between the lubricant, drop and gas phases. These expressions are much simpler and more compact compared to those previously derived by Semprebon et al. [Soft Matter, 2017, 13, 101-110].