Method Of Asymptotic Expansions Research Articles

Near-Field to Far-Field Approximation Formulas Based on the Asymptotic Expansion Method for Electromagnetic Wave Propagation Across Air–Sea Boundary Due to a Vertical Magnetic Dipole

Near-Field to Far-Field Approximation Formulas Based on the Asymptotic Expansion Method for Electromagnetic Wave Propagation Across Air–Sea Boundary Due to a Vertical Magnetic Dipole

Instability of a weakly viscoelastic film flow on an oscillating inclined plane

The long- and finite-wavelength instabilities of weakly viscoelastic film on an oscillating inclined plane are investigated. By using the Chebyshev series solution with the Floquet theory, the combined effects of viscoelasticity and forcing amplitude on instability are described when the inclined plane oscillates in streamwise and wall-normal directions. For long-wavelength instability, the solution to the eigenvalue problem is obtained analytically by the asymptotic expansion method. Results show that the Floquet exponent is independent of the wall-normal oscillation amplitude. The effects of inclined angle, gravity and surface tension on the stability of viscoelastic liquid film are also discussed. For finite-wavelength instability, numerical results corresponding to the wall-normal oscillation disclose that with the increase of viscoelasticity, the unstable gravitational boundary moves to a higher wavenumber, and the critical amplitudes of subharmonic and harmonic instabilities are reduced. The neutral curve of gravity instability for streamwise oscillatory flow is divided into two parts, and a stable bandwidth is formed for a large value of the viscoelastic parameter. Besides, a new oscillatory mode is identified at small angles of inclination, which will be enhanced if the effect of viscoelasticity is incorporated.

Flexural-gravity wave interaction with undulating bottom topography in the presence of uniform current: An asymptotic approach

Using an asymptotic method, this article deals with flexural-gravity wave scattering with undulating bottom topography, including the effect of uniform currents. The interest in this problem lies in developing second-order solutions using the Fourier transform, which minimises the error gap between first and second-order solutions. The present method allows the physical processes involved in the sea-bed topography, uniform current, plate-covered surface, and wave interaction to be studied. Specifically, we observe Bragg resonance between the flexural-gravity waves and the bottom ripples, which are associated with the reflection of incident wave energy. We examine the effects of wave current and emphasise how crucial the asymptotic expansion method is to the emergence of the current response. We demonstrate that bottom topography dominates the effects of Bragg resonance for depth Froude numbers valued at 0.8 or less. Further, most reflected wave components have their frequencies shifted by the current, and wave action conservation causes reflected wave energy to be enhanced for following currents. Using the Joint North Sea Wave Observation Project spectrum and the discrete Fourier transform, the theory derived in the frequency domain is shown in the time domain to analyse wave propagation through the whole system.

Conformal and Non-Minimal Couplings in Fractional Cosmology

Fractional differential calculus is a mathematical tool that has found applications in the study of social and physical behaviors considered “anomalous”. It is often used when traditional integer derivatives models fail to represent cases where the power law is observed accurately. Fractional calculus must reflect non-local, frequency- and history-dependent properties of power-law phenomena. This tool has various important applications, such as fractional mass conservation, electrochemical analysis, groundwater flow problems, and fractional spatiotemporal diffusion equations. It can also be used in cosmology to explain late-time cosmic acceleration without the need for dark energy. We review some models using fractional differential equations. We look at the Einstein–Hilbert action, which is based on a fractional derivative action, and add a scalar field, ϕ, to create a non-minimal interaction theory with the coupling, ξRϕ2, between gravity and the scalar field, where ξ is the interaction constant. By employing various mathematical approaches, we can offer precise schemes to find analytical and numerical approximations of the solutions. Moreover, we comprehensively study the modified cosmological equations and analyze the solution space using the theory of dynamical systems and asymptotic expansion methods. This enables us to provide a qualitative description of cosmologies with a scalar field based on fractional calculus formalism.

Stability analysis of premixed flames with downstream heat gain

Pulsating instability is a kind of combustion intrinsic instability, that can affect the flammable limit and may couple with other processes to generate more complex combustion instability phenomena. Pulsating instability was found in the premixed flame of the double-flame partially premixed flame (PPF), in which the flame patterns change from stable to pulsating unstable, and to stable again as the equivalence ratio or stretch rate increases. To further investigate this phenomenon, a simpler model, stretched premixed flames with downstream heat gain, is analyzed by the asymptotic expansion method in this paper. The downstream heat gain inhibits the steady extinction of stretched premixed flames with a large Zeldovich number. The stability of premixed flames changes in the zero-limit mode, two-limits mode and one-limit mode sequentially as the Zeldovich number increases. The zero-limit mode represents the premixed flame is stable for all stretch rates. The two-limit mode and one-limit represent the premixed flame changes in the pattern of stable-pulsating-stable and pulsating-stable as the stretch rate increases, respectively. The downstream heat gain suppresses the pulsating instability to yield the stable region at the large stretch rate and large Zeldovich number. The influence of other factors, including the Lewis number, the downstream temperature and the downstream boundary position are also discussed. The current theoretical method is also applied to the extracted parameters from numerical calculations of hydrogen/air double-flame PPF. A qualitative similarity is found between the theoretical results and numerical results.

Asymptotic Study of Flows Induced by Oscillations of Cylindrical Bodies

Hydrodynamic flows induced by translational oscillations of cylindrical bodies of various cross-sectional shapes are studied. The motion of fluid around oscillating bodies is described using the system of Navier–Stokes equations written in a generalized curvilinear coordinate system. Transition to a given body shape is implemented using a conformal mapping. The problem is solved using the method of successive asymptotic expansions under the assumption that the oscillation amplitudes are small. In each asymptotic approximation, the subproblems are solved numerically using the finite-difference method. Based on the results of the work, estimates of the hydrodynamic effect are obtained, the applicability of the high-frequency asymptotic approximation is estimated, and secondary stationary flows near cylinders are studied, in particular, the occurrence of directed stationary flows near an oscillating asymmetric body is considered with reference to the Joukowski airfoil.

Singular perturbation boundary and interior layers problems with multiple turning points

In the study of singularly perturbed boundary problems with turning points, the solution undergoes sharp changes near these points and exhibits various interior phenomena. We employ the matching asymptotic expansion method to analyze and solve a singularly perturbed boundary and interior layers problem with multiple turning points, resulting in a composite expansion that fits well with the numerical solution. The solution demonstrates a strong association with special functions, which is verified by the theory of differential inequalities.

Soret and Dufour effects on two dimensional steady MHD flow past a semi‐infinite vertical porous plate in the presence of thermal radiation and radiation heat absorption with constant heat and mass flux

AbstractThe problem of a two‐dimensional steady, MHD convective, incompressible, viscous flow past a uniformly moving semi‐infinite vertical porous plate embedded in a porous medium in the presence of radiation heat absorption, thermal radiation, thermal diffusion, and diffusion thermo effects with constant heat and mass flux are studied. In the fluid region, a transversely oriented, homogeneous magnetic field of intensity is applied. The non‐dimensional governing equations are solved analytically adopting asymptotic series expansion method. The impacts of several physical quantities including Grashof number, Schmidt number, Prandtl number, Soret number, radiation absorption, radiation parameter, magnetic field within the flow domain and transport characteristic have been explored graphically. It has been found that fluid velocity and skin friction fall with enhancing the values of magnetic parameter. Further, temperature and concentration fields rise with an increase in the Soret effect.

Characterizing suspended particle dispersion in wetland flows: Impact of settling velocity and vegetation factor

Predicting the dispersion process of suspended particles with settling velocity in wetland flows holds significant implications for various ecological and environmental applications. This study analytically investigates the dispersion process of fine settling particles in wetland flows due to an instantaneous release source through the asymptotic expansion method. The effect of high-order terms is incorporated. The impact of vegetation factor and settling velocity on characteristic coefficients (including mass retained in the flow, advection velocity, longitudinal dispersion coefficient, skewness, and kurtosis), vertical mean, and two-dimensional concentration distribution are analyzed. Analytical solution is validated by numerical result through random displacement method. Results demonstrate that the vegetation factor does not influence the vertical mass distribution, and a larger settling velocity results in a higher concentration of mass in the bed wall layer. The longitudinal dispersion coefficient does not exhibit a monotonic relationship with the settling velocity. The position of mass centroid of the vertical mean concentration is biased more to the upstream with the larger settling velocity. At larger times, the vertical mean concentration approximates a normal distribution, with skewness and kurtosis nearing zero. Under the influence of settling velocity, the bed wall layer exhibits a high concentration zone in the two-dimensional concentration distribution. These results can help the understanding of sediment dynamics, nutrient cycling, pollutant transport associated with the wetland flows.

A quasi-static model of a thermoelastic body reinforced by a thin thermoelastic inclusion

The problem of description of quasi-static behavior is studied for a planar thermoelastic body incorporating an inhomogeneity, which geometrically is a strip with a small cross-section. This problem contains a small positive parameter δ describing the thickness of the inhomogeneity, i.e., the size of the cross-section. Relying on the variational formulation of the problem, we investigate the behavior of solutions as δ tends to zero. As the result, by the version of the method of formal asymptotic expansions, we derive a closed limit model in which the inhomogeneity is thin (of zero width). After this, using the Galerkin method and the classical techniques of derivation of energy estimates, we prove existence, uniqueness, and stability of a weak solution to this model.

Reynolds Number Effects on the Drag Reduction With a Spanwise Traveling Wave of Blowing and Suction in Turbulent Channel Flows

Turbulent channel flows with Reτ = 180 and Reτ = 550 are controlled to reduce the drag with a spanwise traveling wave of the blowing and suction method. An oscillatory spanwise motion is generated with a periodically reversing propagation direction of the traveling wave, similarly as the wall oscillation. Direct numerical simulation (DNS) results show that this kind of blowing and suction control can achieve a drag reduction rate of 24.5% with Reτ = 180, and 7.5% with Reτ = 550. The reasons for the deterioration in drag reduction rates are thought to be the lift-up mechanism by the actuation through an asymptotic expansion method, and the controlled inner regions and small-scale structures having less significance when the Reynolds number is high.

Bragg scattering of gravity waves by a sea bed of varying depth in the presence of uniform current covered by a floating membrane

We present an asymptotic approach to solving the water wave scattering by undulating bottom topography in the presence of uniform currents where a flexible membrane covers the upper surface. The interest in this problem lies in the development of asymptotic solutions using the Fourier transform under the action of uniform currents. The method allows the physical processes involved in the sea-bed topography and membrane-covered surface and wave interactions to be studied. In particular, we identify the existence of Bragg resonance between gravity waves and the bottom ripples, which are associated with the reflection of incident wave energy. We consider the impacts of uniform current, and we highlight the central role of the asymptotic expansion method in the evolution of the response of current. For depth Froude number in the range of 0.4–0.7, the effects of bottom topography on Bragg resonance dominate. The current shifts the frequency of the most reflected wave components, and wave action conservation results in amplified reflected wave energies for the following currents. The theory developed in the frequency domain is illustrated in the time domain using discrete Fourier transform with the Joint North Sea Wave Observation Project (JONSWAP) spectrum [Hasselmann et al., J. Phys. Oceanogr. 10, 1264 (1980)] to analyze wave propagation through the whole system.

Singular asymptotic expansion and Legendre collocation method for two-term weakly singular Volterra integral equation of the second kind

This paper develops a singularity-separation Legendre collocation method for solving two-term linear Volterra integral equation with algebraic and logarithmic singular kernels. The psi-series solution at the origin is obtained by an efficient Picard iteration, which is very accurate near the origin. By splitting the interval into a singular subinterval and a regular one, the equation is converted to one with a sufficiently smooth solution on the regular interval, which is discretized by the Legendre collocation method. The efficient computational process is discussed in detail, and the convergence under the square norm is proved. Finally, three numerical examples, including a Caputo fractional differential equation and a Basset equation with initial conditions, are provided to illustrate the efficiency and accuracy of the proposed singularity-separation method.

Initial layer of the incompressible Oldroyd-B model for weakly viscoelastic fluids at the low Weissenberg number

This paper is concerned with the Newtonian limit of the incompressible Oldroyd-B model for weakly viscoelastic fluids when the Weissenberg number tends to zero. The convergence rates and the thickness of initial layer are obtained. An effective dynamics is constructed to capture the behavior of initial layer via the method of asymptotic expansions.

An analytical solution for convective heat transfer in conical gaps with either cone or disk rotating

This study is devoted to solving two problems of laminar fluid flow in a conical gap with small conicity angles up to 4°: cone rotation with a fixed disk, and disk rotation with a fixed cone. A new improved asymptotic expansion method for energy equation was used to obtain an approximate analytical solution to the convective heat transfer equation. The characteristic Reynolds number ranged from 0.001 to 1.0, the Prandtl number took values 0.71, 1, 5, and 10, and the exponent n* in the power-law for the disk temperature was 0 (constant disk temperature) or 2 (strongly radially increasing disk temperature). A novel model for the asymptotic expansion of the temperature profile and a novel expansion parameter Sv = Re2Pr, which is a new dimensionless number proposed for the first time in the known scientific literature, was developed. For the first time, new approximate analytical solutions were obtained for temperature profiles and Nusselt numbers on the disk and cone for both problems that agree well with the self-similar solution, if the Re and Pr numbers do not exceed threshold values. These analytical solutions are advantageous in analysis of experimental data and further development of one-dimensional models for gases, water, and aqueous solutions (Pr = 0.71–10).

Instability of a viscoelastic film with insoluble surfactants on an oscillating plane

The linear instability of viscoelastic film with insoluble surfactants on an oscillating plane for disturbances with arbitrary wavenumbers is investigated. The combined effects of viscoelastic and insoluble surfactants on the instability are described using Floquet theory. For long-wavelength instability, the solution in the limit of long wave perturbations is obtained by the asymptotic expansion method. The results show that the presence of viscoelastic film shifts the stability boundaries to the low-frequency region in the absence of gravity when the imposed frequency is less than 6. The U-shaped neutral curves with separation bandwidth appear in the presence of gravity. The finite-wavelength instability is solved numerically based on the Chebyshev spectral collocation method. Different from the previous results, a new branch point with special structure of a neutral curve is detected for clean-surface film. Results show that the presence of the surfactants will decrease the unstable frequency bandwidth and increase the critical Reynolds number. Both the travelling-wave mode and standing-wave mode are found due to the existence of surface surfactants. For high-frequency oscillation, the viscoelastic parameter may significantly destabilize the flow and the instability is determined by the finite-wavelength mode over a relatively large frequency range.

Instability of liquid film with odd viscosity and slip effect under the action of external electric field

The influence of odd viscosity on the instability of a liquid film flowing along a porous inclined plane under a normal electric field is investigated. It is assumed that the flow at the porous inclined plane satisfies the Beavers–Joseph slip boundary condition. By utilizing the long-wave approximation and employing the method of systematic asymptotic expansion, a nonlinear evolution equation for the film thickness under the influence of the electric field is derived. The stability analysis of this evolution equation reveals that the odd viscosity of the film has a stabilizing effect, while the electric field has a destabilizing effect. Additionally, the permeability of the porous inclined plane enhances the instability of the liquid film flow. Numerical simulations are conducted using a fast Fourier transform algorithm to solve the nonlinear evolution equations. The numerical results demonstrate that, within the stable region and with all parameters fixed, the wave amplitude decreases as the evolution time increases, indicating a gradual stabilization of the liquid film flow. Conversely, in the unstable region, the opposite behavior is observed. As the evolution time increases, the fluctuation amplitude grows larger, resulting in a gradual destabilization of the liquid film flow. Furthermore, when the evolution time is kept constant and the odd viscosity coefficient is nonzero, the film exhibits greater stability. The amplitude of the wave increases with the electrical parameter E. In the unstable region, an increase in the permeability β of the porous medium leads to a tendency for the film flow to stabilize.

Improved asymptotic expansion method for laminar fluid flow and heat transfer in conical gaps with disks rotating

PurposeThe purpose of this paper was to study laminar fluid flow and convective heat transfer in a conical gap at small conicity angles up to 4° for the case of disk rotation with a fixed cone.Design/methodology/approachIn this paper, the improved asymptotic expansion method developed by the author was applied to the self-similar Navier–Stokes equations. The characteristic Reynolds number ranged from 0.001 to 2.0, and the Prandtl numbers ranged from 0.71 to 10.FindingsCompared to previous approaches, the improved asymptotic expansion method has an accuracy like the self-similar solution in a significantly wider range of Reynolds and Prandtl numbers. Including radial thermal conductivity in the energy equation at small conicity angle leads to insignificant deviations of the Nusselt number (maximum 1.23%).Practical implicationsThis problem has applications in rheometry to experimentally determine viscosity of liquids, as well as in bioengineering and medicine, where cone-and-disk devices serve as an incubator for nurturing endothelial cells.Social implicationsThe study can help design more effective devices to nurture endothelial cells, which regulate exchanges between the bloodstream and the surrounding tissues.Originality/valueTo the best of the authors’ knowledge, for the first time, novel approximate analytical solutions were obtained for the radial, tangential and axial velocity components, flow swirl angle on the disk, tangential stresses on both surfaces, as well as static pressure, which varies not only with the Reynolds number but also across the gap. These solutions are in excellent agreement with the self-similar solution.

Lie group analysis of flow and heat transfer of a nanofluid in cone–disk systems with Hall current and radiative heat flux

A study of the rheological and heat transport characteristics in cone–disk systems finds relevance in many applications such as viscometry, conical diffusers, and medical devices. Therefore, a three‐dimensional axisymmetric flow with heat transport of a magnetized nanofluid in a cone–disk system subjected to Hall current and thermal radiation effects is investigated. The simplified Navier–Stokes (NS) equations for the cone–disk system given by Sdougos et al. [18] Journal of Fluid Mechanics, 138, 379–404 are solved by using the asymptotic expansion method for the four different models, such as rotating cone with static disk (Model I), rotating disk with static cone (Model II), co‐rotating cone and disk (Model III), and counter‐rotating cone and disk (Model IV). The Khanafer–Vafai–Lightstone (KVL) model along with experimental data‐based properties of 37 nm Al2O3–H2O nanofluid is considered. To obtain the transformations leading to self‐similar equations from the Navier–Stokes (NS) and energy conservation equations, the Lie group technique is used. The self‐similar nonlinear problem is solved numerically to examine the effects of physical parameters. There are critical values of the power exponent at which no heat transport from the disk surface occurs. Nanoparticles significantly enhance heat transport when both the cone and disk rotate in the same or opposite directions. The centrifugal force and thermal radiation improve the heat transport in cone–disk systems.

Solving Kolmogorov PDEs without the curse of dimensionality via deep learning and asymptotic expansion with Malliavin calculus

This paper proposes a new spatial approximation method without the curse of dimensionality for solving high-dimensional partial differential equations (PDEs) by using an asymptotic expansion method with a deep learning-based algorithm. In particular, the mathematical justification on the spatial approximation is provided. Numerical examples for high-dimensional Kolmogorov PDEs show effectiveness of our method.