Asymptotic Solution Method Research Articles

Asymptotic method of solution to the problem of electromagnetic field scattering on a set of small impedance particles

Asymptotic method of solution to the problem of electromagnetic field scattering on a set of small impedance particles

NUMERICAL STUDY OF THE INFLUENCE OF TEMPERATURE-DEPENDENT VISCOSITY ON THE UNSTEADY LAMINAR FLOW AND HEAT TRANSFER OF A VISCOUS INCOMPRESSIBLE FLUID DUE TO A ROTATING DISC

In this article, the effect of temperature-dependent viscosity (TVD) on the unsteady laminar flow and heat transfer (HT) of a viscous incompressible fluid due to a rotating disc (RD) has been investigated numerically by exploiting an in-house numerical code. A set of time-dependent, axisymmetric, and non-linear partial differential equations which govern the fluid flows and heat transfer are reduced to non-linear local non-similarity ordinary differential equations by introducing a newly developed group of transformations for different time regimes. Three different solution methods, such as, (i) perturbation solution method for small t, (ii) asymptotic solution method for large t, and (iii) implicit finite difference method for the entire t regime, have been applied to solve the resulting equations treating t as the time-dependent rotating parameter. The local radial skin friction, tangential skin friction and the heat transfer are computed at the surface of the disc for different numerical parameters, such as, Prandtl number, Pr and the viscosity-variation parameter, e. Besides, the key dimensionless quantities such as velocity and temperature profiles, which are inherently linked with the boundary layer thickness, are presented graphically for different values of e while Pr = 0.72. It is found that the dimensionless radial, tangential and axial velocity profiles decrease as e increases, and consequently, the momentum boundary layer thickness is decreased. On the other hand, the non-dimensional temperature profiles are increased owing to the increasing values of e, and this effect eventually leads to a small increment in the thermal boundary layer thickness.

Asymptotic method for solution of oscillatory fractional derivative

In the paper an oscillatory system with liquid dampers is considered, when the mass of the head is large enough. By means of expedient transformations, the equation of motion with fractional derivatives is reduced to an equation of fractional order containing a small parameter. The corresponding nonlocal boundary value problem is solved and the zero and first approximations of solutions of the relative small parameter are constructed. The results are illustrated on the concrete example, where the solution differs from the analytical solution by 10^-2 order.

Mathematical modeling of multi-region premixed combustion of moist bamboo particles

In this study, a mathematical modeling using asymptotic solution method was performed to a multi-region premixed combustion of moist moso bamboo particles under adiabatic condition. The analytical model assumes and divides the modeling system into multi-regions as preheating, drying, pyrolysis, and homogeneous and heterogeneous reactions. The formulated mass and energy conservation equations were written for each region and solved analytically using specified jump and boundary conditions. The experimental validation using temperatures of homogeneous flame and heterogeneous reaction fronts confirmed that the prediction accuracy is promising. Then, combustion characteristics such as distributions of temperature and species mass fractions were clarified. Influences of particle diameter, gaseous fuel Lewis number, and equivalence ratio were finally explored on crucial quantities such as homogeneous flame temperature and heterogeneous reaction temperatures, burning velocity, and pyrolysis front location. The results showed that increasing bamboo particle diameter leads to lower burning velocity, lower flame temperatures, and prolonged reaction fronts. Fuel Lewis number showed similar trends for burning velocity and flame and reaction temperatures as those of particle diameter, while opposite conclusions were found for reaction front locations. The impacts of equivalence ratio are opposite for burning velocity, flame temperature, and reaction front locations as those of particle diameter. • Multi-region premixed combustion of moist moso bamboo is mathematically modeled. • Preheating, drying, pyrolysis, combustion reactions are considered in the model. • Mass and energy equations are analytically solved using jump conditions. • Temperatures and places of homogeneous and heterogeneous flames are obtained. • Effects of key parameters on combustion characteristics of the system are evaluated.

Supercritical regime for the kissing polynomials

We study a family of polynomials which are orthogonal with respect to the varying, highly oscillatory complex weight function eniλz on [−1,1], where λ is a positive parameter. This family of polynomials has appeared in the literature recently in connection with complex quadrature rules, and their asymptotics have been previously studied when λ is smaller than a certain critical value, λc. Our main goal is to compute their asymptotics when λ>λc.We first provide a geometric description, based on the theory of quadratic differentials, of the curves in the complex plane which will eventually support the asymptotic zero distribution of these polynomials. Next, using the powerful Riemann–Hilbert formulation of the orthogonal polynomials due to Fokas, Its, and Kitaev, along with its method of asymptotic solution via Deift–Zhou nonlinear steepest descent, we provide uniform asymptotics of the polynomials throughout the complex plane.Although much of this asymptotic analysis follows along the lines of previous works in the literature, the main obstacle appears in the construction of the so-called global parametrix. This construction is carried out in an explicit way with the help of certain integrals of elliptic type. In stark contrast to the situation one typically encounters in the presence of real orthogonality, an interesting byproduct of this construction is that there is a discrete set of values of λ for which one cannot solve the model Riemann–Hilbert problem, and as such the corresponding polynomials fail to exist.

A two-scale asymptotic expansion method for periodic composite Euler beams

This paper develops a two-scale asymptotic expansion solution method for periodic composite Euler beams for the first time. In this method, a two-scale solution in the asymptotic expansion form is achieved for the fourth-order uniformly elliptic differential equation with periodic oscillating coefficients, and the first-order perturbed displacement is zero. The analytical solutions of unit cell problems are found with the help of four normalization conditions proposed in this work, and it follows that the homogenized elastic modulus of periodic composite beam is exactly the harmonic mean of material moduli. Another contribution of this work is to mathematically verify the convergence of the two-scale asymptotic expansion solution through the two-scale convergence method. In addition, the paper reveals the effects of higher-order expansion terms on the two-scale displacements and concludes that the second order or even higher-order perturbed displacements are necessary for obtaining accurate curvatures relevant to micro stresses. Finally, numerical experiments validate that the present method is rigorous in mathematics and physically acceptable.

Calculation of Natural Frequencies on Transverse Vibration for drill strings in Non-Uniform Temperature Field

It is necessary to consider the effect of temperature which changes the mechanical properties of materials when drill strings work in a non-uniform temperature field. In this paper, the natural frequencies of transverse vibration are studied under axial compression in the nonuniform temperature field. Using an asymptotic solution method, first-order approximation formula of the nonlinear vibration equation is obtained. The effects of temperature changes and axial pressure on the natural frequencies are discussed by an example. The results show that the influence of temperature on the critical load is small; change rate of natural frequency increases with the increase of axial pressure; while the axial pressures become larger, the influence of temperature on the first-order frequency is greater than on the second-order’s.

Non-local effects and size-dependent properties in Stefan problems with Newton cooling

We model the growth of a one-dimensional solid by considering a modified Fourier law with a size-dependent effective thermal conductivity and a Newton cooling condition at the interface between the solid and the cold environment. In the limit of a large Biot number, this condition becomes the commonly used fixed-temperature condition. It is shown that in practice the size of this non-dimensional number is very small. We study the effect of a small Biot number on the solidification process with numerical and asymptotic solution methods. The study indicates that non-local effects become less important as the Biot number decreases.

Transient Mixed Convection Boundary Layer Flow of an Incompressible Fluid Past a Wedge in Presence of Magnetic Field

In this paper, an analysis is performed to explorethe transient, laminar two-dimensional, mixed convection boundary layer flow of a viscous and incompressible fluid past a vertical wedge taking into account the effect of magnetic field. With appropriate transformations the boundary layer equations are reduced to a local nonsimilarity equations and the solutions are obtained employing three distinct methods, namely, (i) perturbation method for small time; (ii) asymptotic solution method for large time; (iii) straight forward finite difference method for any time. The agreement between the solutions obtained from prescribed methods is found to be excellent. In this study the evaluation of skin-friction coefficient and the local Nusselt number with the effects of different governing parameters such as different time, τ, the exponent, m (= 0.4, 0.5, 1.0), mixed convection parameter, λ (= 0.0, 0.2, 0.4) and magnetic field parameter, M (=0.0, 1.0) for fluids having Prandtl number, Pr= 0.72, 1.0 and 7.0have been discussed. It is observed that both the local skin friction and local Nusseltnumber decreases due to an increase in the value of M. It is also found that an increase in the value of Prandtl number, Pr, leads to a decrease in the value of local skin friction coefficient and the value of local Nusselt number coefficient increases with the increasing values of Prandtl number.

Asymptotic method for solution of identification problem of the nonlinear dynamic systems

A dynamic system, when the motion of the object is described by the system of nonlinear ordinary differential equations, is considered. The right part of the system involves the phase coordinates as a unknown constant vector-parameter and a small number. The statistical data are taken from practice: the initial and final values of the object coordinates. Using the method of quasilinearization the given equation is reduced to the system of linear differential equations, where the coefficients of the coordinate and unknown parameter, also of the perturbations depend on a small parameter linearly. Then, by using the least-squares method the unknown constant vector-parameter is searched in the form of power series on a small parameter and for the coefficients of zero and the first orders the analytical formulas are given. The fundamental matrices both in a zero and in the first approach are constructed approximately, by means of the ordinary Euler method. On an example the determination of the coefficient of hydraulic resistance (CHR) in the lift in the oil extraction by gas lift method is illustrated, as the obtained results in the first approaching coincides with well-known results on order of 10-2.

Approximated Analytical Solution of a Problem on Indentation of an Electro-Elastic Half-Space with Inhomogeneous Coating by a Conductive Punch

The paper addresses to the development of the bilateral asymptotic method of solution of contact problems taking into account electro-mechanical fields coupling. Approximated analytical solutions are provided for contact problems on indentation of an electroelastic piezoelectric half-space with inhomogeneous in depth piezoelectric coating by a rigid conductive circular flat, spherical or conical punch.

An Efficient Approach of Homotopic Asymptotic for System Differential Equations of Non Integer Order

The momentous objective of this paper is to using a method with a free parameter which named optimum asymptotic homotopic method for finding solution of system differential equations featuring fractional derivative. To better understand the methodology and view its advantages compared to similar examples are presented.

Lubricated point heavily loaded contacts of functionally graded materials. Part 2. Lubricated contacts

The paper presents a study of the effectiveness of functionally graded materials in heavily loaded point elastohydrodynamically lubricated contacts with straight lubricant entrainment. As part of the study, some criteria of coating effectiveness are introduced and discussed. The approach used for achieving this goal is based on the recently developed asymptotic solution method for steady isothermal elastohydrodynamically lubricated problems of heavily loaded point contacts, as well as on a semi-analytical method for the solutions of contact problems for elastic functionally graded materials without the presence of lubrication in the contacts (dry contacts). For line elastohydrodynamically lubricated contacts such a problem has already been studied. The solution process can be subdivided into two stages: the solution of the contact problems for dry contacts and a solution of the problem for lubricated contacts. The solution of dry contact problems for functionally graded materials is described in detail in Part 1 of the paper. As in the case of homogeneous contact materials in the analysis of the elastohydrodynamically lubricated problem for functionally graded elastic materials, it is shown that the whole contact region can be subdivided into three subregions: the central one which is adjacent to the other two regions occupied by the ends of the zones. The central region can be subdivided into the Hertzian region and adjacent to it the inlet and exit zones which, in turn, are adjacent to the inlet and exit boundaries of the contact, respectively. In the Hertzian region the elastohydrodynamically lubricated problem solution is very close to the solution of the corresponding dry (i.e. non-lubricated) contact problem for functionally graded elastic materials which have been analyzed in Part 1 of the paper. In the central region in the inlet and exit zones of the heavily loaded point elastohydrodynamically lubricated contact, the elastohydrodynamically lubricated problem for functionally graded elastic materials using certain similarity transforms can be reduced to asymptotically valid equations identical to the ones obtained in the inlet and exit zones of the heavily loaded line elastohydrodynamically lubricated contacts for homogeneous elastic materials. Therefore, many of the well known properties of heavily loaded line elastohydrodynamically lubricated contacts for homogeneous elastic materials are also valid for heavily loaded point elastohydrodynamically lubricated contacts for functionally graded elastic materials. These asymptotically valid equations can be analyzed and numerically solved based on stable methods using a specific regularization approach which was developed for lubricated line contacts. Also, this asymptotic analysis leads to the easy analytical derivation of equations for the lubrication film thickness which take into account the inhomogeneity of the elastic materials. As a result of this analysis, criteria for an increase in lubrication film and friction force reduction are proposed. These criteria depend on lubricant properties, as well as the properties of functionally graded elastic materials involved in lubricated contacts.

Asymptotic iteration method for solution of the Kratzer potential in D-dimensional Klein-Gordon equation

The solution of D-dimensional Klein-Gordon equation for Kratzer potential was presented using asymptotic iteration method. The D-dimensional Klein-Gordon equation was reduced to one-dimensional radial differential equation. The Kratzer central potential was used in radial part solution to obtain the energy eigenvalues and radial wave function. The bound-state energy eigenvalues were calculated for various values of quantum numbers and dimensions using Matlab software.

Asymptotic iteration method for analytical solution of Klein-Gordon equation for trigonometric Pӧschl-Teller potential in D-dimensions

Klein-Gordon equation for Trigonometric Pöschl-Teller Potential in D-dimensions was obtained within framework of a centrifugal term approximation. Asymptotic iteration method was used to obtain the relativistic energy spectrum and wave functions. The value of relativistic energy was calculated numerically and the results have shown that in higher dimension the energy level is increased with positive energy states. The wave functions were expressed in hypergeometric term.

Directional characteristics of cylindrical radiators with an arc-shaped acoustic screen

Directional sound radiation by a cylinder surrounded with an arc-shaped (open circular) acoustic screen is examined. The corresponding boundary problem is solved with the partial domain technique. The method of asymptotic solutions was used to verify the conditions of solving this problem using the method of simple reduction. As a result of the analysis, the quantitative characteristics of the acoustic field of the cylindrical radiator inside the arc-shaped screen were evaluated.

Description of grain-boundary diffusion in nanostructured materials for thin-film diffusion source

Using asymptotic solution methods of differential equations, the grain-boundary diffusion at transition (between regimes C and B) stages of annealing has been described for nano- and submicrocrystalline materials characterized by nonequilibrium grain boundaries and, thus, by a strong coordinate dependence of the diffusion coefficient near them. As a continuation of [1], the solution of the diffusion problem has been obtained for the condition of thin-film (instantaneous) exhaustible diffusion source frequently used in experiments. Based on the performed numerical calculations and qualitative estimates, expressions have been suggested for the average layer concentration and the region of their applicability has been established. The features of grain-boundary diffusion connected with allowance for near-boundary regions and type of the diffusion source have been treated analytically. Using the literature data the depth of penetration of diffusant into the near-boundary zone has been estimated for some materials.

Periodic contact problem for an elastic wedge

A three-dimensional periodic contact problem in the theory of elasticity for a wedge is investigated when an infinite system of identical punches are arranged on one face of the wedge along the edge (at an equal distance from the edge and with equal intervals between neighbouring punches). Identical normal embedding forces are applied to the punches and friction forces are neglected. The other edge of the wedge is rigidly clamped. Without loss of generality of the approaches, the wedge material is assumed to be incompressible. The problem is reduced to an integral equation, from the kernel of which the principal part corresponding to an elastic half-space is separated out. The effect of the edge of a three-dimensional wedge on the contact pressure distribution and the mechanical characteristics of the contact is studied. A regular asymptotic method of solution is used for specified elliptic contact areas. When the contact areas are not known in advance, the method of non-linear boundary value integral equations is used. Calculations were carried out for elliptic and conical punches.

Modeling and simulation of ion transport in dielectric liquids - Fundamentals and review

This review paper introduces fundamental knowledge about and presents recent developments in modeling and simulation of ion transport and electrohydrodynamic flow in dielectric liquids. Starting with the formulation of the governing equations, we introduce background ideas as well as the related modeling of ion dissociation, recombination, injection, and discharge, which play key roles in the charge creation and transport problem. Numerical methods and solutions are then presented for the onedimensional problem over the domain between a pair of parallel electrodes. In particular, we show how the electrical properties of a nonpolar liquid, m-xylene, can be derived from the matching of the numerical solutions with Novotny's experimental data. Two asymptotic solution methods, one for the overlap-layer and the other for the thin-layer regimes, are then presented; in particular, we show a remarkably high accuracy for the solutions. The current-voltage relation with ion dissociation and recombination is shown to be completely different from that without them. A historical review of the modeling and simulation of ion transport and electrohydrodynamic flow is finally touched upon.

Fluctuating hydromagnetic natural convection flow past a magnetized vertical surface in the presence of thermal radiation

The effect of radiation on fluctuating hydro-magnetic natural convection flow of viscous, incompressible, electrically conducting fluid past a magnetized vertical plate is studied; when the magnetic field and surface temperature oscillates in magnitude about a constant non zero mean. The numerical solutions have been obtained for different values of radiation parameter (Rd) magnetic Prandtl number (Pm) magnetic force parameter (S) Prandtl number (Pr) and surface temperature ( w q ) in terms of amplitude and phase angle of coefficients of skin friction, rate of heat transfer and current density at the surface of the plate. Moreover, the effects of these parameters on transient coefficients of skin friction, rate of heat transfer and current density have been discussed. The finite difference method for primitive variable transformation and asymptotic series solution for stream function formulation has been used to obtain the numerical solution of the boundary layer flow field.