Asymptotic Solution Research Articles

A higher-order asymptotic solution for 3D sharp V-notch front tip fields in creeping solids

As one of the most important topics studied in creep fracture mechanics, mechanics fields at 3D sharp V-notch front tip in creeping solids have drawn remarkable attention at present. In this paper, the higher-order asymptotic analysis is carried out for 3D sharp V-notch front tip fields in creeping solids subjected to mode Ⅰ loading, and a 3D higher-order term solution is theoretically proposed by introducing the out-of-plane stress factor Tz to establish the 3D nonlinear governing equations of front tip fields. The present higher-order asymptotic solution for 3D sharp V-notch can naturally be degenerated to that for 3D crack. It is found that the stress exponents and angular distribution functions of higher-order terms for 3D notch and crack are highly related to Tz. The proposed 3D higher-order term solution overall shows better agreement with the FEA results than the existing 3D leading-term solution and 2D higher-order term solutions available in the literature, especially for smaller notch angle and shorter ligament lengths. Based on the 3D higher-order asymptotic solution, a new fracture parameter A2T is given and combined with Tz to characterize 3D constraint effect. It is noted that the effects of A2T and Tz on 3D constraint are highly interlinked rather than simply separated into in-plane and out-of-plane parts. The present 3D higher-order asymptotic solution can promote the understanding and characterization of 3D constraint effect and is of great significance in the evaluation of 3D fracture behavior and structural failure assessment under creep conditions.

Closed-form asymptotic solution for the transport of chlorine concentration in composite pipes

Recently, some researchers have revisited the analysis of chlorine transportation in cylindrical pipes by deploying a coupling between the Laplace transform method and the complex analysis’ residue approach for inverting complex integrals. This method yielded interesting results after the incorporation of root-finding numerical schemes. Thus, away from incorporating numerical tools, the present study makes consideration of the same formulation of chlorine transport in a single-layered pipe and further extends it to the case of a bi-layered pipe using the hybrid of the Laplace transform method and the asymptotic approximations method. The need for asymptotic approximations for the modified Bessel functions, which arise in the reduced ordinary differential equations, necessitates the quest for closed-form analytical solutions, which are largely considered benchmark solutions for numerical investigations. Moreover, the obtained closed-form asymptotic solutions have been examined graphically; where it was observed that both the radial diffusion coefficient η and the spatial radial variable are contributory in the transport of chorine concentration in the media.

On inherent hyperelastic crease

We present analyses of the inherent hyperelastic crease that entails an unstable degenerative singular perturbation field over a uniformly compressed hyperelastic half-space. The analyses reveal asymptotic far-field characteristics of the singular field that bring out admissible incremental elastic deformation mechanisms of the inherent creasing instability typically observed at a compressive strain of ∼0.354 in neo-Hookean solids. The admissible singular perturbation field has an asymptotic leading-order incremental-stress singularity decaying 1/r2 as the distance from the crease tip r→∞. In general, asymptotic incremental-deformation fields of 1/r2 stress singularity can be introduced by surface flaws. We found that the singular field creates a far-field eigenmode of three energy-release angular sectors separated by two energy-elevating sectors of incremental deformation. The far-field eigenmode braces the near-tip energy-release field of the surface flaw against the transition to an unstable self-similar expansion field of creasing. Two asymptotic far-field parameters can characterize the braced-incremental-deformation (bid) and crease fields. One is the Burgers vector of a projected subsurface dislocation of the asymptotically singular far field. The other is a dimensionless shape factor representing the ratio of the subsurface depth of the dislocation to the Burgers vector. Our analyses show that the shape factor gauges the transition from the stable bid field to the unstable inherent crease field at the crease limit point as the compression increases. Two tangential-manifold conservation integrals are revealed to identify the two parameters. At the crease-limit point, matched asymptotes lead to the ratio between the two separate scaling parameters of the asymptotic solutions of the crease-tip folding field and the leading-order far field. To this end, we introduced a novel finite element method (FEM) for simulating the crease field nearly in a hyperelastic half-space with a finite-size simulation domain. Furthermore, we uncovered with the Gent model (1996) that strain-stiffening in hyperelasticity alters the dependence of the shape factor on the compressive strain, raising crease resistance. The new findings in hyperelastic crease mechanisms will help study ruga mechanics of self-organization and design soft-material structures and skin conditions for high crease resistance.

Viscoelastic Hertzian Impact

The problem of normal impact of a rigid sphere on a Maxwell viscoelastic solid half-space is considered. The first-order asymptotic solution is constructed in the framework of Hunter’s model of viscoelastic impact. In particular, simple analytical approximations have been derived for the maximum contact force and the time to achieve it. A linear regression method is suggested for evaluating the instantaneous elastic modulus and the mean relaxation time from a set of experimental data collected for different spherical impactors and impact velocities.

Applications of Asymptotic Solutions of the Diffusivity Equation to Infinite Acting Pressure Transient Analysis

Summary Understanding how pressure propagates in a reservoir is fundamental to the interpretation of pressure and rate transient measurements at a well. Unconventional reservoirs provide unique technical challenges as the simple geometries and flow regimes [wellbore storage (WBS) and radial, linear, spherical, and boundary-dominated flow] applied in well test analysis are now replaced by nonideal flow patterns due to complex multistage fracture completions, nonplanar fractures, and the interaction of flow with the reservoir heterogeneity. In this paper, we introduce an asymptotic solution technique for the diffusivity equation applied to pressure transient analysis (PTA), in which the 3D depletion geometry is mapped to an equivalent 1D streamtube. This allows the potentially complex pressure depletion geometry within the reservoir to be treated as the primary unknown in an interpretation, compared with the usual method of interpretation in which the depletion geometry is assumed and parameters of the formation and well are the unknown properties. The construction is based upon the solution to the Eikonal equation, derived from the diffusivity equation in heterogeneous reservoirs. We develop a Green’s function that provides analytic solutions to the pressure transient equations for which the geometry of the flow pattern is abstracted from the transient solution. The analytic formulation provides an explicit solution for many well test pressure transient characteristics such as the well test semi-log pressure derivative (WTD), the depth of investigation (DOI), and the stabilized zone (SZ) (or dynamic drainage area), with new definitions for the limit of detectability (LOD), the transient drainage volume, and the pseudosteady-state (PSS) limit. Generalizations of the Green’s function approach to bounded reservoirs are possible (Wang et al. 2017) but are beyond the scope of the current study. We validate our approach against well-known PTA solutions solved using the Laplace transform, including pressure transients with WBS and skin. Our study concludes with a discussion of applications to unconventional reservoir performance analysis for which reference solutions do not otherwise exist.

The asymptotic solution of near-limit chain-branching premixed flames with the Zel’dovich–Liñán two-step mechanism in the linear and fast recombination regime

Near-limit characteristics of chain-branching premixed flames is investigated numerically and asymptotically by employing the Zel’dovich–Liñán two-step mechanism with linear and (sufficiently) fast recombination, for which both chain-branching and -recombination steps take place in a thin reactive layer of O(β−1), with β being the chain-branching Zel’dovich number. The numerical and asymptotic results reveal that the rate-ratio eigenvalue r between the chain branching and recombination steps approaches a minimum critical value of rmin→e as the chain-recombination step becomes infinitely fast. The existence of the finite asymptotic value of r=e indicates that no steadily propagating flame solution can be found for fuel concentrations lower than its critical value, corresponding to the criticality of r=e, thereby explaining the (lean) flammability limit. This intrinsic flammability characteristics with linear recombination is the most outstanding contrast to the quadratic recombination counterpart, for which flammability is achieved only with external heat loss added to the model.Novelty and significance statementNear-limit combustion characteristics of chain-branching flames is investigated numerically and asymptotically by employing the Zel’dovich–Linan two-step chemical kinetics with linear and fast recombination. This work demonstrates that the rate-ratio eigenvalue between the chain-branching and -recombination steps asymptotically approaches a critical minimum value, corresponding to the flammability limit. The existence of flammability limit with the linear recombination is found to be the most significant contrast to the quadratic recombination model, which is incapable of predicting flammability limit. By this study, a formal asymptotic procedure, by which the crossover temperature and lean flammability limit condition can be determined without assuming a steady-state approximation of intermediate species, is established and can be applied to analyze flame structures with multi-step reduced mechanisms.

Analysis of fluid flow and heat transfer in CNT-infused spiraling disk

This study investigates the fluid flow and heat transfer characteristics influenced by the thermophysical properties of carbon nanotube suspension in engine oil over a stretching-spiraling disk. The surface comprises Homan stagnation point flow impinging normal to a rotating, radially stretching disk, which gives velocity a form of logarithmic spiral. Similarity ansatz transformed coupled nonlinear differential equation system is derived and solved numerically using MATLAB’s bvp4c collocation method. Numerical and asymptotic solutions are also obtained against the controlled parameters via graphical representations and tabular data. Findings indicate that the magnetic field enhances temperature distribution while reducing the velocity field. At the same time, stretching increases radial velocity but decreases azimuthal velocity and temperature profiles, regardless of carbon nanotube type. The asymptotic values of skin friction decline with an increase in the spiralisng angle and in the absence of a magnetic field for both SWCNTs and MWCNTs cases.

Attractors in [formula omitted]-gravity

We investigate the asymptotic behavior of the cosmological field equations in Symmetric Teleparallel General Relativity, where a nonlinear function of the boundary term is introduced instead of the cosmological constant to describe the acceleration phase of the universe. Our analysis reveals constraints on the free parameters necessary for the existence of an attractor that accurately represents acceleration. However, we also identify asymptotic solutions depicting Big Rip and Big Crunch singularities. To avoid these solutions, we must impose constraints on the phase-space, requiring specific initial conditions.

Exact Solution to a Generalised Lillo–Mike–Farmer Model with Heterogeneous Order-Splitting Strategies

The Lillo–Mike–Farmer (LMF) model is an established econophysics model describing the order-splitting behaviour of institutional investors in financial markets. In the original article (Lillo et al. in Phys Rev E 71:066122, 2005), LMF assumed the homogeneity of the traders’ order-splitting strategy and derived a power-law asymptotic solution to the order-sign autocorrelation function (ACF) based on several heuristic reasonings. This report proposes a generalised LMF model by incorporating the heterogeneity of traders’ order-splitting behaviour that is exactly solved without heuristics. We find that the power-law exponent in the order-sign ACF is robust for arbitrary heterogeneous order-submission probability distributions. On the other hand, the prefactor in the ACF is very sensitive to heterogeneity in trading strategies and is shown to be systematically underestimated in the original homogeneous LMF model. Our work highlights that predicting the ACF prefactor is more challenging than the ACF exponent because many microscopic details (complex ingredients in actual data analyses) start to matter.

Asymptotic solution for turbulent variances: application to convergence of averages and particle dispersion

Asymptotic solution for turbulent variances: application to convergence of averages and particle dispersion

Rate-limiting factors in thin-film evaporative heat transfer processes

In this paper, we present a theoretical study aimed at investigating the rate-limiting factors in thin-film evaporative heat transfer processes, considering the finite-rate evaporation kinetics. The problems of evaporation of a flat thin-film in either pure vapours or vapour-inert-gas mixtures are analysed based on the non-dimensionalised macroscopic transport equations for continuum fluids, coupled with out-of-equilibrium kinetic boundary conditions. Both the full numerical solutions and asymptotic analytical solutions at slow evaporation limit are provided and applied to analyse thin water film evaporation. Existing solutions, assuming negligible heat transfer in the gas domain, or negligible temperature jump across the non-equilibrium kinetic layer, or more boldly a thermodynamically equilibrial interface (i.e. its temperature is at the saturation temperature), can be fully recovered from the more general solutions presented here. Our results show that while these assumptions hold in special cases, they can lead to significant errors in many conditions, especially when the film thickness δ is reduced to a few micrometers or thinner. We show that the conventional views that the rate-limiting factors in thin-film evaporative heat transfer is either the heat diffusion through the liquid film or the mass transfer in the gas domain only apply to thick film (i.e. δ≫λ where λ is the mean free path in the vapour phase). As δ decreases to a few micrometer or smaller (more precisely when the Knudsen number Kn increases beyond O(1) in an pure vapour environment or when the kinetic Peclet number Pe is reduced below O(1) in inert gases), the interfacial thermal resistance due to the evaporation kinetics can be on the same orders of magnitude as the thermal resistance in the liquid film. The analysis also allow us to compare the heat transfer processes during the evaporation of a thin-film in pure vapours to those in inert gases, providing deeper insight into the effectiveness of various strategies for exploring the evaporation process in practical thermal management.

Mixed convection flow over a horizontal plate and the horizontal wake far downstream

The laminar mixed convection flow over a heated or cooled horizontal plate of finite length is a surprisingly intriguing problem. The hydrostatic pressure jump at the trailing edge and across the wake is to be compensated by induced circulation in the outer potential flow, similar to a plate at a non-zero angle of attack. The mixed convection flows over semi-infinite and finite horizontal plates, respectively, are thus substantially different. As the hydrostatic pressure jump remains finite along the infinitely extended wake, the induced outer potential flow around the plate would grow beyond bounds in a domain extending to infinity. Thus, boundary conditions must be imposed at the boundaries of a finite domain. However, the numerical solution of the Navier-Stokes equations remains a challenging problem due to the lack of appropriate outflow conditions. The traditional outflow condition cannot comply with the finite hydrostatic pressure jump across the wake. Thus, an asymptotic expansion for the wake far downstream from the plate is performed. Remarkably, the perturbation of the flow does not decay with increasing distance from the plate as in a classical wake. The asymptotic solution is implemented as an outflow boundary condition for the numerical solution of the Navier-Stokes equations. The results are compared with a boundary-layer solution obtained in previous work for the limiting case of vanishing Prandtl number.

Analysis of dynamic wave model for unsteady flow and sediment transport in alluvial rivers

The coupling interactions between flood propagation, sediment transport, and river morphology in alluvial rivers are mathematically described by the high-order dynamic wave model. The coupling capability of currently used dynamic wave models is systematically conducted. The results indicate that the propagation of a dynamic flood wave only depends on the Froude number, but is independent of the coupling of sediment transport and river mobility. Furthermore, based on the continuum hypothesis, the dynamic equations describing the motion of the active bed layer are obtained. A renewed dynamic wave model is established. Four families of asymptotic solutions to the eigenvalues of the renewed four-order hyperbolic system are obtained by means of the singular-perturbation technology. The results demonstrate that the interactions between flood propagation, sediment transport, and riverbed mobility are coupled. Propagation of the main dynamic flood wave and the dynamic sediment wave will be slower with the increasing deposition rate, but will be faster when the erosion intensity is enhanced. These mainly occur in the lower flow regime. In the process of deposition, the second dynamic flood wave and the dynamic bed wave will propagate both upward and downstream. Besides, the dynamic bed wave will propagate downstream and the second dynamic flood wave will only propagate upstream, regardless of the flow regime.

Asymptotic analysis of acoustic black hole effect in cylindrical shells.

The acoustic black hole (ABH) effect is investigated within the framework of thin shell theory. Asymptotic solutions to the dispersion equation for the thin cylindrical shell are obtained, and the ABH effect is examined using analytical formulas for group velocities and anti-derivatives of the asymptotic expansions of wave numbers. It is shown that the ABH effect is achievable in thin cylindrical shells with variable thickness, in a similar manner as for beams and plates. However, it should not be expected to exist in the low-frequency range where the flexural wave motion in the wall of a shell is strongly coupled with uniform longitudinal wave motion.

Mass transport in Couette flow

In this study we examine the large Péclet number, Pe, limit of a concentration boundary layer in Couette flow. The boundary layer has a thickness of order Pe−1/2. The asymptotic concentration is asymptotically obtained as an integral solution up to order Pe−1/2 using the Fourier sine transform. The asymptotic solution is found to be in good agreement with the full numerical solution for large Péclet numbers. Further, the effective diffusivity obtained from the asymptotic solution is found to be in good agreement with the effective diffusivity obtained from the full numerical solution for large Péclet numbers.

A novel technique for low-velocity impact of shallow arches

In the current study, an analytical model to analyze the low-velocity impact (LVI) response of metamaterial shallow arches subjected to rigid body impact is presented. The presented nonlinear model considers transverse deformation of the cross-section and geometric nonlinearity based on the higher-order shear theory and von Kármán nonlinearity. The research delves into three key aspects. The first is concerned with the establishment of a contact model to capture the contact characteristics between the impactor and the arch. The second aspect deals with the design of the member with Negative Poisson’s Ratio (NPR). The third aspect is concerned with the asymptotic solution of dynamic equations by using a two-perturbation technique. The proposed model is used to analyze the effects of auxetic, initial deformation, and foundation on the impact response and post-impact vibration of the arch. The results show that the contact area between the sphere impactor and the arch is elliptical. It is also noted that the contact force and indentation are highly dependent on the laminated configuration and auxetic properties. The study reveals that the presented model is a valid technique to evaluate the impact characteristics of metamaterial arches along with optimal design impact resistance of arches.

Heavy tails and probability density functions to any nonlinear order for the surface elevation in irregular seas

The probability density function (PDF) for the free surface elevation in an irregular sea has an integral formulation when based on the cumulant generating function. To leading order, the result is Gaussian, whereas nonlinear extensions have long been limited to Gram–Charlier series approximations. As shown recently by Fuhrman et al. (J. Fluid Mech., vol. 970, 2023, A38), however, the second-order integral can be represented exactly in closed form. The present work extends this further, enabling determination of this PDF to even higher orders. Towards this end, a new ordinary differential equation (ODE) governing the PDF is first derived. Asymptotic solutions in the limit of large surface elevation are then found, utilizing the method of dominant balance. These provide new analytical forms for the positive tail of the PDF beyond second order. These likewise clarify how high-order cumulants (involving statistical moments such as the kurtosis) govern the tail, which is shown to get heavier with each successive order. The asymptotic solutions are finally utilized to generate boundary conditions, such that the governing ODE may be solved numerically, enabling novel determination of the PDF at third and higher order. Successful comparisons with challenging data sets confirm accuracy. The methodology thus enables the PDF of the surface elevation to be determined numerically, and the asymptotic tail analytically, to any desired order. Results are worked out explicitly up to fifth order. The theoretical probability of extreme surface elevations (typical of rogue waves) may thus be assessed quantitatively for highly nonlinear irregular seas, requiring only relevant statistical quantities as input.

Asymptotic solutions of differential equations with singular impulses

The paper considers impulsive systems with singularities. The main novelty is that beside the singularity of the differential equation, the impulsive equation is a singular one. The method of boundary functions is applied to obtain the main result. Examples with simulations confirming the theoretical results are given.

On Generalised Hankel Functions and a Bifurcation of Their Asymptotic Expansion

The generalised Bessel differential equation has an extra parameter relative to the original Bessel equation and its asymptotic solutions are the generalised Hankel functions of two kinds distinct from the original Hankel functions. The generalised Bessel differential equation of order ν and degree μ reduces to the original Bessel differential equation of order ν for zero degree, μ = 0. In both cases the differential equations have a regular singularity near the origin and the the point at infinity is the other singularity. The point at infinity is an irregular singularity of different degree, namely one for the original and two for the generalised Bessel differential equation. It follows that in the limit of degree being equal to zero the generalised Hankel functions do not converge to the original ones. The implication is that the generalised Bessel differential equation has a Hopf-type bifurcation for the asymptotic solution. In the case of a real variable and parameters the asymptotic solution is: (i) oscillatory when the degree of generalised Hankel function is zero (corresponding in this case to original Hankel functions); (ii) diverging hence unstable for the generalised Hankel functions with positive degree; (iii) decaying hence stable for the generalised Hankel functions with negative degree.

Inertial coalescence of drops with some viscosity

When two fluid drops touch, they coalesce due to surface tension. At early times, there is only a relatively small fluid bridge joining the drops. An asymptotic solution is presented for an inertial regime of early-time coalescence, in which inertial forces balance surface tension at leading order. It is demonstrated that viscosity nevertheless has a leading-order effect. Radial momentum is created at the tightly curved edge of the fluid bridge by the net force $2\gamma$ (per unit length) due to surface tension. This momentum is left behind the radially expanding bridge edge in a thin viscous wake. The divergent volume flux in the wake entrains fluid from above and below the bridge, and drives an inviscid irrotational flow in the drops on the scale of the bridge radius. This flow widens the gap between the drops ahead of the bridge, and the larger gap width results in a lower rate of coalescence. Including viscosity in this way improves the agreement between theory and the available experimental and numerical data.