Terms Of Asymptotic Series Research Articles

Transversal field and heat transfer in the flow of non-linear viscoelastic fluids in tear-drop shaped tubes including viscous dissipation

Steady, laminar flow of a class of non-linear viscoelastic fluids in tear-drop shaped ducts is investigated for constant heat flux at the wall including viscous dissipation. Field equations are solved analytically through expanding the field variables in double asymptotic series in terms of the Weissenberg number and a mapping parameter coupled with the shape factor method, a continuous and one-to-one mapping, to map the circular contour onto the tear-drop shaped tube boundary. Analytical solution for the transversal and axial velocity fields is derived for the first time, and the influence of the viscoelastic constitutive parameters in shaping the transversal vortex structure is discussed in detail. The temperature field and the dependence of the Nusselt Nu = f (Br, Pe, Re, We) number on the Brinkman Br, Péclet Pe, Reynolds Re and Weissenberg We numbers as well as combinations of the material parameters are investigated. At the lowest order in the Weissenberg number the velocity and temperature fields of Newtonian fluids in tear-drop shaped straight tubes as well as in straight round tubes with viscous dissipation are recovered. The approach the analysis is based on can be easily adapted to yield the velocity and temperature fields of Newtonian and constitutively non-linear viscoelastic fluids in non-circular tubes other than tear-drop shaped including viscous dissipation.

Критическое поведение систем с конкуренцией между близкодействием и дальнодействием

Critical behaviour of a range of ferromagnetic materials deviates from the predictions of the Ising, XY and Heisenberg models. Additional long-range forces competing with regular exchange interaction may explain this deviation. These competing interactions lead to new universality classes of critical behaviour. The paper uses the field theory approach to investigate critical behaviour in those systems in which long-range and short-range forces compete. We consider the case when a power function of distance r-D-σ, when 1.5 < σ < 2.0, can describe the long-range forces. There exists a distinctive critical behaviour mode for these values. We derived vertex functions using a two-loop approximation directly in three-dimensional space (D = 3) and, for all values, obtained a linear approximation of asymptotic series in terms of long-range interaction parameters. We applied the Pade --- Borel summation technique to these asymptotic series. We computed stable fixed points and critical exponents as functions of long-range interaction parameters for low relativeefficiency of the long-range interaction. We investigated how critical exponents depend on the factor in the power law and relative long-range interaction intensity. We compared our results to the critical exponent values found experimentally for manganites. We used the experimental critical exponent γ values to compute long-range interaction parameters and then used the long-range interaction parameters to derive the ß exponent values, which we then compared to the experimental values. We show good agreement between our theoretical results and experimental data.

Elastoviscoplastic fluid flow in non-circular tubes: Transversal field and interplay of elasticity and plasticity

• An elastoviscoplastic fluid model which allows non-rectilinear flow is developed. • Velocity fields in triangular and square cross-sectional geometries are computed. • The strength of the secondary flow vortices decreases with increasing yield stress. • Plasticity pushes out the in-plane vortices from the center of the cross section towards the edges and the corners. • Viscoelasticity increases axial flow and flow rate for given plastic conditions. An analytical study of elastoviscoplastic fluid flow in tubes of non-circular cross section is presented. The constitutive structure of the fluid is described by a linear frame invariant combination of the Phan-Thien−Tanner model of viscoelastic fluids and the Bingham model of plastic fluids. Non-circular tube cross sections are modeled by the shape factor method a one-to-one mapping of the circular base contour into a wide spectrum family of arbitrary tube contours. Field variables are expanded into asymptotic series in terms of the elasticity measure, the Weissenberg number We , coupled with an asymptotic expansion in terms of the geometrical mapping parameter ε leading to a set of hierarchical momentum balance equations which are solved successively up to and including the third order in We when the secondary field appears for the first time. The computational algorithm developed is applied to the study of the non-rectilinear flow in tubes with triangular and square cross sections. We find that the presence of the yield stress dampens the intensity of the purely viscoelastic vortices, the higher the yield stress the lower the intensity of the vortices in the cross-section, and the further away the vortices are from the center of the cross section as compared to the purely viscoelastic vortices. The results also evidence that viscoelasticity increases the axial flow for given viscoplastic conditions and pressure drop, and consequently increases the rate of flow, a phenomenon that may find applications in optimizing material transportation.

Analytical solution of the Graetz problem for non-linear viscoelastic fluids in tubes of arbitrary cross-section

Graetz problem is solved analytically for steady laminar flow of non-linear viscoelastic fluids in straight tubes of arbitrary cross-section. The one-to-one mapping method used to model the cross-sectional geometry satisfies the no-slip and thermal boundary conditions for a wide range of arbitrary tube contours. Field variables are expanded in asymptotic series in terms of the Weissenberg number Wi, leading to a set of linear hierarchical equations which are solved successively up to and including the third order in Wi. Exact analytical solution for the Graetz problem for Newtonian fluids in round as well as arbitrary cross-sectional tubes is recovered at the lowest order. Secondary flows arise at the third order of the analysis as shown previously by Siginer and Letelier [1]. Their effect on the convection field is fully taken into account. The temperature distributions for triangular and square cross-sectional tubes are computed as particular cases of the general analysis together with the variation of the asymptotic Nusselt number for the triangular cross-section as a function of Wi to demonstrate the substantial enhancement of the heat transfer rates due to the non-linear viscoelastic constitutive structure of the fluid. The analytical algorithm presented is very versatile and can be easily applied to a wide spectrum of non-circular tube contours.

Resurgent analysis of localizable observables in supersymmetric gauge theories

Localization methods have recently led to a plethora of new exact results in supersymmetric gauge theories, as certain observables may be computed in terms of matrix integrals. These can then be evaluated by making use of standard large N techniques, or else via perturbative expansions in the gauge coupling. Either approximation often leads to observables given in terms of asymptotic series, which need to be properly defined in order to obtain nonperturbative results. At the same time, resurgent analysis has recently been successfully applied to several problems, e.g., in quantum, field and string theories, precisely to overcome this issue and construct nonperturbative answers out of asymptotic perturbative expansions. The present work uses exact results from supersymmetric localization to address the resurgent structure of the free energy and partition function of Chern-Simons and ABJM gauge theories in three dimensions, and of $$ \mathcal{N}=2 $$ supersymmetric Yang-Mills theories in four dimensions. For each case, the complete structure of Borel singularities is exactly determined, and the relation of these singularities with the large-order behavior of (multi-instanton) perturbative expansions is made fully precise.

Universal unfolding of symmetric resonances

We present the analysis of the bifurcation sequences of a family of resonant 2-DOF Hamiltonian systems invariant under spatial mirror symmetry and time reversion. The phase-space structure is investigated by a singularity theory approach based on the construction of a universal deformation of the detuned Birkhoff–Gustavson normal form. Thresholds for the bifurcations of periodic orbits in generic position are computed as asymptotic series in terms of physical parameters of the original system.

Heat Flow Out of a Compact Manifold

We discuss the heat content asymptotics associated with the heat flow out of a smooth compact manifold in a larger compact Riemannian manifold. Although there are no boundary conditions, the corresponding heat content asymptotics involve terms localized on the boundary. The classical pseudo-differential calculus is used to establish the existence of the complete asymptotic series, and methods of invariance theory are used to determine the first few terms in the asymptotic series in terms of geometric data. The operator driving the heat process is assumed to be an operator of Laplace type.

Renormalized Surface Charge Density for a Strongly Charged Plate in Asymmetric Electrolytes: Exact Asymptotic Expansion in Poisson Boltzmann Theory

The Poisson-Boltzmann equation for a strongly charged plate inside a generic charge-asymmetric electrolyte is solved using the method of asymptotic matching. Both near field and far field asymptotic behaviors of the potential are systematically analyzed. Using these expansions, the renormalized surface charge density is obtained as an asymptotic series in terms of the bare surface charge density.

Laminar flow of non-linear viscoelastic fluids in straight tubes of arbitrary contour

The fully developed steady velocity field in pressure gradient driven laminar flow of non-linear viscoelastic fluids with instantaneous elasticity constitutively represented by a class of single mode, non-affine quasilinear constitutive equations is investigated in straight pipes of arbitrary contour ∂D. A continuous one-to-one mapping is used to obtain arbitrary tube contours from a base tube contour ∂D 0. The analytical method presented is capable of predicting the velocity field in tubes with arbitrary cross-section. The base flow is the Newtonian field and is obtained at O (1). Field variables are expanded in asymptotic series in terms of the Weissenberg number Wi. The analysis does not place any restrictions on the smallness of the driving pressure gradients which can be large and applies to dilute and weakly elastic non-linear viscoelastic fluids. The velocity field is investigated up to and including the third order in Wi. The Newtonian field in general arbitrary contours is obtained and longitudinal velocity field components due to shear-thinning and to non-linear viscoelastic effects are identified. Third order analysis shows a further contribution to the longitudinal field driven by first normal stress differences. Secondary flows driven by unbalanced second normal stresses in the cross-section manifest themselves as well at this order. Longitudinal equal velocity contours, the secondary flow field structure, the first and the second normal stress differences as well as wall shear stress variations are discussed for several non-circular contours some for the first time.

Heat transfer asymptote in laminar flow of non-linear viscoelastic fluids in straight non-circular tubes

The fully developed thermal field in constant pressure gradient driven laminar flow of a class of non-linear viscoelastic fluids with instantaneous elasticity in straight pipes of arbitrary contour ∂D with constant wall flux is investigated. The non-linear fluids considered are constitutively represented by a class of single mode, non-affine constitutive equations. The driving forces can be large. Asymptotic series in terms of the Weissenberg number Wi are employed to expand the field variables. A continuous one-to-one mapping is used to obtain arbitrary tube contours from a base tube contour ∂D 0. The analytical method presented is capable of predicting the velocity and temperature fields in tubes with arbitrary cross-section. Heat transfer enhancement due to shear-thinning is identified together with the enhancement due to the inherent elasticity of the fluid. The latter is to a very large extent the result of secondary flows in the cross-section but there is a component due to first normal stress differences as well. Increasingly large enhancements are computed with increasing elasticity of the fluid as compared to its Newtonian counterpart. Order of magnitude larger enhancements are possible even with slightly viscoelastic fluids. The coupling between inertial and viscoelastic non-linearities is crucial to enhancement. Isotherms for the temperature field are discussed for non-circular contours such as the ellipse and the equilateral triangle together with the behavior of the average Nusselt number Nu, a function of the Reynolds Re, the Prandtl Pr and the Weissenberg Wi numbers. Analytical evidence for the existence of a heat transfer asymptote in laminar flow of viscoelastic fluids in non-circular contours is given for the first time. Nu becomes asymptotically independent from elasticity with increasing Wi, Nu = f( Pe, Wi) → Nu = f( Pe). This asymptote is the counterpart in laminar flows in non-circular tubes of the heat transfer asymptote in turbulent flows of viscoelastic fluids in round pipes. A different asymptote corresponds to different cross-sectional shapes in straight tubes. The change of type of the vorticity equation governs the trends in the behavior of Nu with increasing Wi and Pe. The implications on the heat transfer enhancement is discussed in particular for slight deviations from Newtonian behavior where a rapid rise in enhancement seems to occur as opposed to the behavior for larger values of the Weissenberg number where the rate of increase is much slower. The asymptotic independence of Nu from elasticity with increasing Wi is related to the extent of the supercritical region controlled by the interaction of the viscoelastic Mach number M and the Elasticity number E, which mitigates and ultimately cancels the effect of the increasingly strong secondary flows with increasing Wi to level off the enhancement. The physics of the interaction of the effects of the Elasticity E, viscoelastic Mach M, Reynolds Re and Weissenberg Wi numbers on generating the heat transfer enhancement is discussed.

Vertex singularities associated with conical points for the 3D laplace equation

The solution u to the Laplace equation in the neighborhood of a vertex in a three-dimensional domain may be described by an asymptotic series in terms of spherical coordinates . For conical vertices, we derive explicit analytical expressions for the eigenpairs νi and fi(θ, φ), which are required as benchmark solutions for the verification of numerical methods. Thereafter, we extend the modified Steklov eigen-formulation for the computation of vertex eigenpairs using p/spectral finite element methods and demonstrate its accuracy and high efficiency by comparing the numerically computed eigenpairs to the analytical ones. Vertices at the intersection of a crack front and a free surface are also considered and numerical eigenpairs are provided. The numerical examples demonstrate the efficiency, robustness, and high accuracy of the proposed method, hence its potential extension to elasticity problems. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011

Risk theory insight into a zone-adaptive control strategy

The main purpose of this paper is a risk theory insight into the problem of asset–liability and solvency adaptive management. In the multiperiodic insurance risk model composed of chained classical risk models, a zone-adaptive control strategy, essentially similar to that applied in Directives [Directive 2002/13/EC of the European Parliament and of the Council of 5 March 2002, Brussels, 5 March 2002], is introduced and its performance is examined analytically. That examination was initiated in [Malinovskii, V.K., 2006b. Adaptive control strategies and dependence of finite time ruin on the premium loading. Insurance: Math. Econ. (in press)] and is based on the application of the explicit expression for the finite-time ruin probability in the classical risk model. The result of independent interest in the paper is the representation of that finite-time ruin probability in terms of asymptotic series, as time increases.

Stability in a mathematical model of neurite elongation

We have developed a continuum partial differential equation model of tubulin-driven neurite elongation and solved the steady problem. For non-zero values of the decay coefficient, the authors identified three different regimes of steady neurite growth, small, moderate and large, dependent on the strength of the tubulin flux into the neurite at the soma. Solution of the fully time-dependent moving boundary problem is, however, hampered by its analytical intractibility. A linear instability analysis, novel to moving boundary problems in this context, is possible and reduces to finding the zeros of an eigen-condition function. One of the system parameters is small and this permits solutions to the eigen-condition equation in terms of asymptotic series in each growth regime. Linear instability is demonstrated to be absent from the neurite growth model and a Newton-Raphson root-finding algorithm is then shown to corroborate the asymptotic results for some selected examples. By numerically integrating the fully non-linear time-dependent system, we show how the steady solutions are non-linearly stable in each of the three growth regimes with decay and oscillatory behaviour being as predicted by the linear eigenvalue analysis.

Boundary-Integral method for calculating poloidal axisymmetric AC magnetic fields

This paper presents a boundary-integral equation (BIE) method for the calculation of poloidal axisymmetric magnetic fields applicable in a wide range of ac frequencies. The method is based on the vector potential formulation, and it uses the Green's functions of Laplace and Helmholtz equations for the exterior and interior of conductors, respectively. The paper focuses on a calculation of axisymmetric Green's function for the Helmholtz equation which is both simpler and more accurate than previous approaches. We use three different techniques for calculation of Green's function, depending on the parameter range. For low and high ac frequencies, we use a power series expansion in terms of elliptical integrals and an asymptotic series in terms of modified Bessel functions of second kind, respectively. For the intermediate frequency range, we use Gauss-Chebyshev-Lobatto quadratures. We verify the accuracy of the method by comparing its results with the analytical solution for a sphere in a uniform external ac field. We illustrate the application of the method by analyzing a composite model inductor containing an external secondary circuit.

Matching of Asymptotic Expansions for Wave Propagation in Media with Thin Slots I: The Asymptotic Expansion

In this series of two articles, we consider the propagation of a time harmonic wave in a medium made of the junction of a half-space (containing possibly scatterers) with a thin slot. The Neumann boundary condition is considered along the boundary on the propagation domain, which authorizes the propagation of the wave inside the slot, even if the width of the slot is very small. We perform a complete asymptotic expansion of the solution of this problem with respect to the small parameter $\varepsilon/\lambda$, the ratio between the width of the slot, and the wavelength. We use the method of matched asymptopic expansions which allows us to describe the solution in terms of asymptotic series whose terms are characterized as the solutions of (coupled) boundary value problems posed in simple geometrical domains, independent of $\varepsilon/\lambda$: the (perturbed) half-space, the half-line, a junction zone. In this first article, we derive and analyze, from the mathematical point of view, these boundary value problems. The second one will be devoted to establishing error estimates for truncated series.

Variant Planck's over 2pi expansion for the periodic orbit quantization of chaotic systems.

We report the results of a periodic orbit quantization of classically chaotic billiards beyond Gutzwiller approximation in terms of asymptotic series in powers of the Planck constant (or in powers of the inverse of the wave number kappa in billiards). We derive explicit formulas for the kappa(-1) approximation of our semiclassical expansion. We illustrate our theory with the classically chaotic scattering of a wave on three disks. The accuracy on the real parts of the scattering resonances is improved by one order of magnitude.

Asymptotic theory of guided modes in two parallel, identical dielectric waveguides

The guided modes supported by two parallel, identical planar dielectric waveguides are investigated exactly by a quasi-optical technique and approximately by using the improved coupled-mode theory. The wave numbers obtained by the two methods are expanded into asymptotic series in terms of a suitable small parameter. By comparing the asymptotic series obtained by the two methods term by term, it is shown that the improved coupled-mode theory is asymptotically exact in the limit of weak coupling and not so in the strong-coupling regime. The merits and the defects of the improved coupled-mode theory are discussed.

Clausius-Mosotti limit of the quantum theory of the electronic dielectric constant

It is shown that an unambiguous discussion of the limit of independent atoms in the theory of the dielectric constant has to be formulated in terms of asymptotic series in the ratio of the Bohr radius to the lattice spacing. In this sense, a derivation of the classical Clausius-Mosotti (or Lorentz-Lorenz) formula is given, starting from the quantum-mechanical theory including "local-field" corrections, at the self-consistent Hartree-Fock level. Our approach clarifies the origin of the many diverging results on the subject and eliminates most of the unnecessary approximations and/or assumptions.

Asymptotic techniques in elastic-plastic analysis of structures

Elastic-plastic structures can nowadays be analyzed with the powerful numerical procedures of the finite element method. Nevertheless, in many engineering applications, analytical expressions capable of predicting with sufficient accuracy the stress distributions, the extent of the plastic zones and the load displacement behavior could be of great practical value. For simple structures and loading stages not too far from the elastic limit, such analytical expressions may be obtained by using perturbation methods and asymptotic expansions. A small dimensionless parameter ϵ is defined as the ratio of a length characterizing the extent of the narrow plastic zone, to a conveniently chosen typical dimension of the structure. Stresses and displacements are formally expanded as asymptotic series in terms of powers of ϵ. For each order of magnitude, the exact basic relations lead to a separate set of simplified differential equations which can be integrated analytically or numerically by using standard procedures. The method is very general and can be applied to several classes of plastic behaviour and of structural problems. Three examples of very simple structures are chosen in particular to illustrate the applicability of the perturbation method to engineering problems: (1) For a simply supported beam under a concentrated load, ϵ is chosen as the ratio of the largest axial extension of the plastic zone to twice the beam length. (2) The case of a long thin cylindrical shell subject to a gradually increasing radial ring of load is algebraically more involved but can be treated similarly. (3) For a simply supported circular plate under a ring of load, ϵ is defined as the thickness ratio of the central plastic zone which is assumed to be sufficiently thin in the first stage of loading beyond the elastic limit. Both Tresca's and von Mises' yield criteria with the associated flow rule (rate theory) can be used. The method can also be applied to more involved geometries (in conjunction with linear FE-procedures) and more complex material behaviour.

Collapse of a one-dimensional cavity

Isentropic potential flows arising when a one-dimensional cavity collapses into an ideally polytropic continuous medium are examined. The analysis is carried out up to the instant of focussing or up to the instant infinite gradient arise in the flow. As a result of the investigation of the analytic solutions in special variables for the polytropy exponent 1 < γ < 3, it is proved that a free boundary separating the medium and the vacuum moves for some time with a constant velocity. Next, the solution is sought in physical space as a series in a neighborhood of the free boundary. When γ > 1 it is proved that the series converges and the free boundary's acceleration begins only from the instant of origin of an infinite gradient. An ordinary differential equation is obtained, governing the behavior of the gradient on the free boundary. The solutions of this equation are studied by numerical calculations and particular solutions are found. It turned out that the instant of origin of the singularity depends in an essential manner on the initial data. Exponents γ ∗ are introduced, such that when γ ⩽ γ ∗ there are no singularities on the free boundary up to the focussing instant if at the initial instant the medium was homogeneous and at rest. When γ > γ ∗ the function t ∗ = t ∗ (γ) , viz., the instant of origin of an infinite gradient on the free boundary, is obtained. Calculations carried out by a difference scheme showed that when γ ⩽ γ ∗ up to the focussing instant and when γ > γ ∗ up to instant t ∗ there are no large gradient inside the whole flow region. The exponents γ ∗ coincide with those found earlier papers ( ∗∗) in which the problem being investigated was investigated by means of constructing several terms of asymptotic series. It is concluded that when 1< γ < 3 the free boundary moves for some time t ∗ > 0 at a constant velocity, and when γ ⩽ γ ∗ = 1 + 2 i the time t ∗ coincides with the focussing instant t 1 = (γ −1) 2 and t ∗ < t 1 when γ > γ ∗ The complete construction of the asymptotic expansions and the exact estimates of these expansions was not carried out. Power series solving the original problem in the exact statement are constructed recurrently in this paper. All the facts obtained are proved on the basis of the study of the convergence domains of these series. When 1< γ < 3 they coincide with earlier derivations.