Asymptotic Analytical Solutions Research Articles

Eastward-moving convection-enhanced modons in shallow water in the equatorial tangent plane

We report a discovery of steady long-living slowly eastward moving large-scale coherent twin cyclones, the equatorial modons, in the shallow water model in the equatorial beta-plane, the archetype model of the ocean and atmosphere dynamics in tropics. We start by constructing analytical asymptotic modon solutions in the non-divergent velocity approximation and then show by simulations with a high-resolution numerical scheme that such configurations evolve into steady dipolar solutions of the full model. In the atmospheric context, the modons persist in the presence of moist convection, being accompanied and enhanced by specific patterns of water-vapour condensation.

Analytical and Numerical Investigation of Fe3O4–Water Nanofluid Flow over a Moveable Plane in a Parallel Stream with High Suction

In the current framework, a model is constituted to explore the impacts of high suction and partial slip on Fe3O4–water nanoliquid flow over a porous moveable surface in a parallel free stream. The mechanisms of heat transfer are also modeled in the existence of Newtonian heating effect. The obtaining PDEs are transformed into a non-linear ODE system employing appropriate boundary conditions to diverse physical parameters. The governing ODE system is solved using a singular perturbation technique that results in an analytical asymptotic solution as a function of the physical parameters. The obtained solution allows us to carry out an analytical parametric study to investigate the impact of the physical parameters on the nonlinear attitude of the system. The precision of the proposed method is verified by comparisons between the numerical and analytical results. The results confirm that the proposed technique yields a good approximation to the solution as well as the solution calculation has no CPU time-consuming or round off error. Numerical solutions are computed and clarified in graphs for the model embedded parameters. Moreover, profiles of the skin friction coefficient and the heat transfer rate are also portrayed and deliberated. The data manifests that both solid volume fraction and slip impact significantly alter the flow profiles. Moreover, an upward trend in temperature is anticipated for enhancing Newtonian heating strength. Additionally, it was found that both the nanofluid velocity and temperature distributions are decelerated when the solid volume fraction and suction parameters increase. Furthermore, a rise in slip parameter causes an increment in velocity profiles, and a rise in Biot number causes an increment in the temperature profiles.

Foam-liquid front motion in Eulerian coordinates.

A mathematical model formulated as a system of Hamilton-Jacobi equations describes implicitly the propagation of a foam-liquid front in an oil reservoir, as the zero-level set of the solution variable. The conceptual model is based on the 'pressure-driven growth' model in Lagrangian coordinates. The Eulerian mathematical model is solved numerically, where the marching is done via a finite volume scheme with an upwind flux. Periodic reinitialization ensures a more accurate implicit representation of the front. The numerical level set contour values are initially formed to coincide with an early time asymptotic analytical solution of the pressure-driven growth model. Via the simulation of the Eulerian numerical model, numerical data are obtained from which graphical representations are generated for the location of the propagating front, the angle that the front normal makes with respect to the horizontal and the front curvature, all of which are compared with the Lagrangian model predictions. By making this comparison, it is possible to confirm the existence of a concavity in the front shape at small times, which physically corresponds to an abrupt reorientation of the front over a limited length scale.

Partial Contact of a Rigid Multisinusoidal Wavy Surface with an Elastic Half-Plane

The interaction effects, arising at partial contact of rigid multisinusoidal wavy surface with an elastic half-plane, are considered in the assumption of continuous contact configuration. The analytical exact and asymptotic solutions for periodic and nonperiodic contact problems for wavy indenters are derived. Continuous contact configuration, appearing at small ratios of amplitude to wavelength for cosine harmonics, leads to continuous oscillatory contact pressure distribution and oscillatory relations between mean pressure and a contact length. Comparison of periodic and nonperiodic solutions shows that long-range elastic interaction between asperities does not depend on a number of cosine wavelengths.

Parallel channels' fracturing mechanism during ice management operations. Part I: Theory

It is frequently observed that long cracks in sea-ice tend to form between parallel channels during ice management operations. The long cracks that develop play an important role in reducing the size of the managed ice floes, which is one of the main goals in an ice management operation. However, the fracture mechanism behind these long cracks remains unclear. To address this issue, a comprehensive study is reported here in two associated papers. In the current paper (i.e., Paper I), an edge-crack theoretical model is proposed to elucidate the parallel channels' fracture mechanism. The proposed theoretical model is partially based on theories regarding ship – level ice interactions and partially based on previous studies on the general ice fracturing mechanism. The edge-crack theoretical model is extensively examined using a separately developed numerical scheme based on the eXtended Finite Element Method (XFEM), which allows for the existence of a singularity field and displacement jump within conventional Finite Elements (i.e., FEM). The numerical scheme is benchmarked against known asymptotic analytical solutions and field experiments. Afterwards, with the developed numerical scheme, through fitting numerical simulation results in terms of the edge crack's Stress Intensity Factors (SIFs) and a relevant asymptotical analysis, we managed to derive a group of closed-form formulae with wide application ranges. For the current engineering problem, this set of formulae quantifies the maximum parallel channel spacing hmax, beyond which the observed parallel channels' fracturing events cease to occur. Moreover, the same numerical scheme is utilised to study parallel channels' fracturing paths. Based on the XFEM-based crack path simulations, a second group of formulae and a numerical recipe were obtained to characterise a simplified crack path. This set of equations enables us to quantify the maximum floe size LMCD that can be generated between two parallel channels and its corresponding floe size ratio. In the sequel paper (i.e., Paper II), these equations are validated by a series of well-controlled field experiments undertaken during the Oden Arctic Technology Research Cruise of 2015 (OATRC2015).

Oxygen diffusion in liquid (over)stoichiometric corium

Abstract Within the framework of pressurized water reactor severe accidents, the ICE research program is devoted to studies on a variety of fundamental mechanisms involved in Fuel-Coolant Interaction (FCI). Among them, the oxidation of liquid corium is expected to impact the premixing/fragmentation stage, thus possibly influencing the subsequent steam explosion. As a first step towards the exhaustive description of oxidation phenomena occurring in typical FCI layouts, the 1-D oxygen diffusion in liquid (over)stoichiometric spherical corium droplets is investigated in this article. A mesoscopic description is adopted, based on the evolution of oxygen concentration c O , depending on space, time, corium composition and temperature. The underlying physical model is coupled with CALPHAD-based thermodynamic calculations. An asymptotic analytical solution is first determined, assuming that the apparent diffusion coefficient D O does not depend on c O . Then, the general variational form of governing equations is derived, leading to the implementation of a related numerical approach using the finite-element method, in COMSOL® software. This numerical approach is first benchmarked with the analytical calculation, and then extrapolated to the more general case where D O varies with c O . Finally, some case studies representative of FCI issues are selected, in order to determine typical oxygen diffusion transient times within a variety of liquid corium droplets.

A modified extended finite element method for fluid-driven fractures incorporating variable primary energy loss mechanisms

A coupled extended finite element method (XFEM) is presented here for modeling propagation of fluid-driven fractures in different regimes including toughness- and viscosity-dominated regimes. The extended finite element method allows to model growth, and coalescence of arbitrary discontinuities (fractures) without requiring the mesh to conform to discontinuities nor significant refinement near the fractures. Fluid-driven fractures propagation is a coupled, nonlinear and non-local problem with moving boundary conditions. The proposed method is based on the extended finite element method with modifications to incorporate variable stress singularity at the crack tips for the transition between toughness-dominated and viscosity-dominated regimes. These modifications consist of enriched functions that are initially inspired by the asymptotic analytical solutions. The standard extended finite element approximation is enriched by adding near tip asymptotic solutions just for displacements, however the proposed method introduces a consistent enriched function for fluid pressure calculations close to the fracture tips to catch the singularity. Additionally, a technique is presented to remove singularity issue for required numerical integrations. Green's functions concept is proposed here to expedite calculations. To circumvent violation of partition of unity and parasitic terms in the approximation space induced by the blending elements at the edge of the enriched domain, the ramp function is utilized to improve the convergence rate. Stress intensity factors are calculated using a new contour integral method that can handle cases with different tip singularities. The proposed technique is verified with the cases that have analytical solutions. Some examples are presented to show the advantages of this technique in comparison to the regular XFEM.

Pulsating linear in situ combustion: why do we often observe oscillatory behavior?

We have studied simplified, pulsating, one-dimensional, in situ combustion processes. For two cases, with different reaction stoichiometry, oscillations in temperature, flue gas rate, and flue gas composition are demonstrated and the parameter space resulting in oscillatory behavior is identified. To understand the role of different parameters, linear stability of the problem is studied. Because linear stability analysis requires the solution of uniform front propagation, we investigated an asymptotic analytical solution of the problem. We found an original formula for the front propagation velocity. The analytical solution enabled us to define four dimensionless parameters including Zeldovich (Ze) number, Damkohler (Da) number, a specialized air-fuel ratio (B), and a ratio incorporating air and rock heat capacities (Δ1). Using linear stability analysis, we show that the stability of the problem is also governed by these four parameters. Because Δ1 ≈ 1 for typical laboratory conditions, the set of (Ze, Da, B) is used to construct the stability plane; consequently, several important design considerations are suggested. Both larger air injection rate and air enriched in oxygen increase the front propagation speed but push the system toward oscillatory behavior. Conversely, the introduction of catalysts and metal additives, that decrease the activation energy of reactions, increases the front speed and stability. Similarly, increasing the amount of fuel available for the combustion makes the design more stable and drives the combustion front to propagate more quickly.

An accurate computational method for the diffusion regime verification

The diffusion regime (sub-diffusive, standard, or super-diffusive) is defined by the order of the derivative in the corresponding transport equation. We develop an accurate computational method for the direct estimation of the diffusion regime. The method is based on the derivative order estimation using the asymptotic analytic solutions of the diffusion equation with the integer order and the time-fractional derivatives. The robustness and the computational cheapness of the proposed method are verified using the experimental methane and methyl alcohol transport kinetics through the catalyst pellet.

Stress analysis of functional gradient strip due to transient thermal heat

AbstractThe work is mainly focuses on the analysis of thermal gradients and its effect on thermal stresses in FGM (functional gradient material) grading‐strip made up of TiC‐Ni due to thermal heat with different rate: thermal shock, transient exponential and linear heat. In this paper, asymptotic analytical solution for exponential and linear heat will be investigated. Few models of material grading are considered: (a) models which depend on different volume content of TiC and (b) models with homogeneous layers. Thermal and mechanical properties of the material are assumed different along axis. It is assumed that Poisson's ratio and Young's modulus varies exponentially with the coordinate of the strip. The dependence on different material property idealizations is also investigated. As well for research maximum and minimal thermal stresses in paper compared laminated non‐FGM and FGM strip. The maximum tensile stress obtained for FGM case. At the same time compressive stresses near heated surface for two‐layered strip lager then for three‐layered and less or absent for FGM strip. Heating/cooling opposite edge to thermal shocked ones leads to increase/decrease compressive stress in metal phase part in FGM and layered strip. At the beginning of heat the differences in compressive stress for thermal shock and exponential heat could differ in ten times. Compressive stress in linear heat case increase from beginning and stay stable as maximum from three heat cases as long as edge heat up to maximum temperature. The results obtained in this research paper allow us to analyze the validity of this method and to understand principles of thermal behavior on FGM‐strip in a case of different conditions and different heat transfer rate.

Microscale Gaseous Slip Flow in the Insect Trachea and Tracheoles

An analytical investigation into compressible gas flow with slight rarefactions through the insect trachea and tracheoles during the closed spiracle phase is undertaken, and a complete set of asymptotic analytical solutions is presented. We first obtain estimates of the Reynolds and Mach numbers at the channel terminal ends where the tracheoles directly deliver respiratory gases to the cells, by comparing the magnitude of the different forces in the compressible gas flow. The 2D Navier-Stokes equations with a slip boundary condition are used to investigate compressibility and rarefied effects in the trachea and tracheoles. Expressions for the velocity components, pressure gradients and net flow inside the trachea are then presented. Numerical simulations of the tracheal compressible flow are performed to validate the analytical results from this study. This work extends previous work of Arkilic et al. (J Microelectromech Syst 6(2):167-178, 1997) on compressible flows through a microchannel. Novel devices for microfluidic compressible flow transport may be invented from results obtained in this study.

Resonant three-dimensional nonlinear sloshing in a square-base basin. Part 4. Oblique forcing and linear viscous damping

Faltinsenet al.(J. Fluid Mech., vol. 487, 2003, pp. 1–42) (henceforth, Part 1) examined an undamped nonlinear resonant steady-state sloshing in a square-base tank by developing an approximate (asymptotic) Narimanov–Moiseev-type multimodal theory. The focus was on longitudinal and diagonal harmonic tank excitations. Neglecting the linear viscous boundary-layer damping was justified for model tanks with breadths of the order of metres. However, nonlinear sloshing in clean tanks of smaller size (count in centimetres) may be affected by damping in finite depth conditions. Qualitative and quantitative properties of the damped resonant steady-state sloshing in a square-base tank are now studied by using the modal theory from Part 1 equipped with the linear damping terms. The tank harmonically oscillates along an arbitrary horizontal (oblique) direction. An analytical asymptotic steady-state undamped solution is derived and the corresponding response curves are analysed versus the forcing direction. When the tank width$=$breadth$=$ $L\sim 10$ cm, the surface tension effect on the free-surface dynamics can be neglected but the linear viscous damping should be included into the Narimanov–Moiseev nonlinear asymptotic modal theory. We analytically show that the steady-state damped sloshing possesses a series of distinguishing features so that, e.g. the square-like standing wave regime fully disappears and becomes replaced by swirling. Typical response curves of the damped steady-state resonant sloshing are studied for the liquid depth-to-width ratio exceeding 0.5. The computational results of the steady-state resonant response amplitudes are in a satisfactory agreement with observations and measurements by Ikedaet al.(J. Fluid Mech., vol. 700, 2012, pp. 304–328), which were conducted with a relatively small laboratory container.

Analytical study on stress-induced phase transitions in geometrically graded shape memory alloy layers. Part I: Asymptotic equation and analytical solutions

In this paper, an analytical approach is proposed to study the stress-induced phase transitions in geometrically graded shape memory alloy (SMA) layers. First, within a two-dimensional setting, the governing PDE system is formulated, which contains the mechanical equilibrium equations and the phase transition criteria. By using the coupled series-asymptotic expansion method, the equilibrium equations can be reduced into one single ODE, which contains the leading-order term of the axial strain and the phase state functions. Besides that, the function representing the varying width of the SMA layer is also involved in this ODE. By further using the phase transition criteria, the phase state functions can be expressed in terms of the axial displacement for the pure loading and pure unloading processes. Then one asymptotic equation with variable coefficients is obtained. For linearly tapered SMA layers, the asymptotic equation is simplified and the WKB method is adopted to solve this equation. The obtained analytical solutions can capture the key features of the mechanical responses of the SMA layers and reveal the underlying mechanisms. The inhomogeneous configurations of the SMA layers during the phase transition process can also be simulated.

On Testing for Impropriety of Multivariate Complex-Valued Random Sequences

We consider the problem of testing whether a complex-valued vector random sequence is proper, i.e., if the sequence is uncorrelated with its complex conjugate. Previous nonparametric approaches to impropriety testing are limited to a sequence of independent random vectors, typically assumed to be Gaussian. In this paper, we extend the results to stationary vector sequences that can be non Gaussian. A binary hypothesis testing approach is formulated and a generalized likelihood ratio test (GLRT) is derived using the power spectral density estimator of an augmented sequence. An asymptotic analytical solution for calculating the test threshold is provided. The performance of the GLRT is analyzed by deriving an approximate asymptotic distribution of the GLRT statistic under the alternative hypothesis. The results are illustrated via simulations.

Study of the singular term in mixed adhesive joints

In the numerical modelling of mixed adhesive joints, where different materials are disposed along the bond-lines, trimaterial singularities may arise at the adhesive bands transition points, resulting in mesh-dependent models. This work is focused on the study, characterization and treatment of these type of singularities. Firstly, the selection of an intermediate adhesive material for generating a tailored transition between bands is addressed, using analytical solutions and then corroborating by means of a novel technique for determining the influence distance of the singularity with the finite element method as calculation tool. Furthermore, it is proved how a continuum material field transition between bands produces a convergent finite element solution applying the considerations of this framework (analytical asymptotic solutions). Finally, in order to achieve the best numerical convergence rate in the models, the optimum material transition function is obtained and checked through FEM.

Analytical study of electromechanical buckling of a micro spherical elastic film on a compliant substrate part I: Formulation and linear buckling of periodic patterns

This work presents an analytical study of the electromechanical buckling of a micro spherical thin film bonded to a compliant elastic substrate. The spherical film is subjected to electrostatic attraction forces that are induced by applying a voltage difference between an outer spherical elastic film electrode and inner rigid, fixed and grounded spherical electrode. When the applied voltage is small, the film contracts while maintaining its complete spherical shape. However, when the applied voltage reaches a critical value, the film buckles into high-ordered periodic patterns (i.e. elastic surface wrinklings). Motivated by experimental results of other studies, this work examines the critical buckling state of one-dimensional, square checkerboard and hexagonal patterns. As will be shown, the above considered patterns are associated with the same critical state, and therefore, all patterns have equal buckling voltage and critical wavelength. Furthermore, with increasing radius of the film/substrate system the electromechanical buckling response converges to the electromechanical pull-in instability of the well-known two parallel plate electrodes, as will be revealed by the asymptotic analytical solution. Finally, the elastic ripples of the electromechanically buckled film can be generated or removed by a simple On/Off switching of the applied voltage. The ability to generate and remove elastic ripples tremendously increases the potential of such microsystem to be utilized in different applications in the field of Micro and Nano electromechanical systems.

Analytical asymptotic solutions for heat transfer during transient compression filling of spherical and cylindrical pressure vessels in the limit of small Reynolds number

Taking the limit of small Reynolds number for a vessel being filled with compressed gas, the energy equation was found to asymptote to the unsteady heat conduction equation with heat generation and variable density. This equation was solved analytically for cylindrical and spherical geometry. Assuming the density changes linearly with time, a solution is obtained which is identical in form to the constant density solution if the Fourier number is defined using the log-mean density rather than the instantaneous density. At steady state conditions, the Nusselt number based on the diameter for cylinders with aspect ratios larger than 1 rapidly approaches an integer solution of NuD = 8. For cylinders with aspect ratios less than 1, the Nusselt number based on the cylinder length (height) characteristic dimension rapidly approaches NuL = 6. It is shown experimentally and numerically that during compression filling, the heat transfer asymptotically approaches this analytical solution at low Reynolds numbers.

Analytic asymptotic solution to spherical relativistic shock breakout

We investigate the relativistic breakout of a shock wave from the surface of a star. In this process, each fluid shell is endowed with some kinetic and thermal energy by the shock, and then continues to accelerate adiabatically by converting thermal energy into kinetic energy. This problem has been previously studied for a mildly relativistic breakout, where the acceleration ends close to the surface of the star. The current work focuses on the case where the acceleration ends at distances much greater than the radius of the star. We derive an analytic description for the hydrodynamic evolution of the ejecta in this regime and validate it using a numerical simulation. We also provide predictions for the expected light curves and spectra from such an explosion. The relevance to astrophysical explosions is discussed, and it is shown that such events require more energy than is currently believed to result from astrophysical explosions.

Automatic trajectory planning for low-thrust active removal mission in low-earth orbit

In this paper two strategies are proposed to de-orbit up to 10 non-cooperative objects per year from the region within 800 and 1400km altitude in Low Earth Orbit (LEO). The underlying idea is to use a single servicing spacecraft to de-orbit several objects applying two different approaches. The first strategy is analogous to the Traveling Salesman Problem: the servicing spacecraft rendezvous with multiple objects in order to physically attach a de-orbiting kit that reduces the perigee of the orbit. The second strategy is analogous to the Vehicle Routing Problem: the servicing spacecraft rendezvous and docks with an object, spirals it down to a lower altitude orbit, undocks, and then spirals up to the next target.In order to maximise the number of de-orbited objects with minimum propellant consumption, an optimal sequence of targets is identified using a bio-inspired incremental automatic planning and scheduling discrete optimisation algorithm. The optimisation of the resulting sequence is realised using a direct transcription method based on an asymptotic analytical solution of the perturbed Keplerian motion. The analytical model takes into account the perturbations deriving from the J2 gravitational effect and the atmospheric drag.

Three-dimensional poroelastic effects during hydraulic fracturing in permeable rocks

A fully coupled three-dimensional finite-element model for hydraulic fractures in permeable rocks is presented, and used to investigate the ranges of applicability of the classical analytical solutions that are known to be valid in limiting cases. This model simultaneously accounts for fluid flow within the fracture and rock matrix, poroelastic deformation, propagation of the fractures, and fluid leakage into the rock formation. The model is validated against available asymptotic analytical solutions for penny-shaped fractures, in the viscosity-dominated, toughness-dominated, storage-dominated, and leakoff-dominated regimes. However, for intermediate regimes, these analytical solutions cannot be used to predict the key hydraulic fracturing variables, i.e. injection pressure, fracture aperture, and length. For leakoff-dominated cases in permeable rocks, the asymptotic solutions fail to accurately predict the lower-bound for fracture radius and apertures, and the upper-bound for fracture pressure. This is due to the poroelastic effects in the dilated rock matrix, as well as due to the multi-dimensional flow within matrix, which in many simulation codes is idealised as being one-dimensional, normal to the fracture plane.