Asymptotic Analytical Solutions Research Articles

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.

A Semiclassical Approach to the Nonlocal Nonlinear Schrödinger Equation with a Non-Hermitian Term

The nonlinear Schrödinger equation (NLSE) with a non-Hermitian term is the model for various phenomena in nonlinear open quantum systems. We deal with the Cauchy problem for the nonlocal generalization of multidimensional NLSE with a non-Hermitian term. Using the ideas of the Maslov method, we propose the method of constructing asymptotic solutions to this equation within the framework of semiclassically concentrated states. The semiclassical nonlinear evolution operator and symmetry operators for the leading term of asymptotics are derived. Our approach is based on the solutions of the auxiliary dynamical system that effectively linearizes the problem under certain algebraic conditions. The formalism proposed is illustrated with the specific example of the NLSE with a non-Hermitian term that is the model of an atom laser. The analytical asymptotic solution to the Cauchy problem is obtained explicitly for this example.

Thermal effects on chemically induced Marangoni convection around A + B → C reaction fronts.

Chemical reactions can induce Marangoni flows by changing the surface tension of a solution open to the air, either by changing the composition and/or by modifying the temperature. We consider the case of a simple A + B → C reaction front propagating in a thin horizontal system open to air. The effect of the three chemical species on the surface tension of the aqueous solution is quantified by three solutal Marangoni numbers, while the effect of temperature changes is determined by the thermal Marangoni number. By integrating numerically the incompressible Navier-Stokes equations coupled to reaction-diffusion-convection equations for the chemical concentrations and temperature taking into account the Lewis number (ratio between heat and mass diffusivities), we emphasize the importance of thermal changes occurring due to the heat of reaction on the dynamics of chemically induced Marangoni convection. Based on the reaction-diffusion profiles of concentrations and temperature, asymptotic analytical solutions for the surface tension profiles are obtained and classified as a function of the Marangoni numbers and the Lewis number. This new classification allows for the prediction of the convective patterns in thermo-solutal Marangoni flows. The analytical predictions are further confirmed by numerical results and additional extrema in surface tension profiles induced by the thermal effects are found to affect the nonlinear dynamics.

Coupled models for propagation of explosive shock waves in cylindrical and spherical geometries

The propagation of shock waves in different geometries is crucial in engineering and scientific applications. A comprehensive model is developed to elucidate the hydrodynamic growth and decay of shock waves in cylindrical and spherical geometries by using the strong shock wave assumption. This model takes into consideration the conservation equations governing mass, momentum, and energy, thereby allowing for an accurate description of the coupled behavior between the piston and shock wave propagation. In contrast to the localized analysis employed in previous self-similar methods, this model incorporates the finite sound wave velocity to introduce the concept of retarded pressure on the piston surface. Consequently, the proposed model offers a multitude of advantages by providing a complete set of dynamic information concerning the trajectories, velocities, and accelerations of both the piston and shock wave. Furthermore, an asymptotic analytical solution is derived to describe the decay of shock waves in cylindrical and spherical geometries. To validate the theoretical analysis and illustrate the propagation characteristics of shock waves in these specific geometries, thorough comparisons are conducted. These findings contribute to the advancement of our understanding of shock wave dynamics in various physical systems, particularly in the field of plasma physics.

Two-dimensional vector solitons in Bose-Einstein-condensate mixtures

We derive two decoupled KP-I equations from the system of two-dimensional (2D) Gross-Pitaevskii equations for a two-component Bose-Einstein condensate (BEC), using the multiple-scale expansion method. We produce asymptotic analytical vector-soliton solutions, viz., dark-dark (DD) and dark-antidark (DAD) one-soliton and two-soliton states, by tuning coupling constants and norms of species, and address their evolution numerically under the action of the harmonic-oscillator (HO) trap, in the local-density approximation. We find that shallow single-line DD and DAD solitons are stable, while single-lump DAD solutions (weakly localized truly-2D states) split and lead to nucleation of two half-vortices. We also find that the BEC mixture placed in the HO trap admits stable asymptotic multi-soliton solutions, e.g., two-line DD and DAD solitons and two-lump DD ones, which were not reported before. In particular, the two-lump solutions describe inelastic collisions between the lumps. The analysis developed in this work may also be applied to systems with spin-orbit coupling and gauge fields, which have been realized in atomic BEC.

Catenary configuration and geometric stiffness matrix of inextensible cables: Analytical high-order asymptotic solutions for parametric design

The analytical and geometric study of catenary curves is a classic matter of applied mathematical modeling. The paper systematizes a methodological strategy to achieve fully analytical solutions for the mechanical problem of determining the equilibrium configuration assumed by inextensible inclined cables under gravitational loads. By developing a two-step perturbation scheme, the statically indeterminate equilibrium problem is asymptotically formulated, and asymptotic solutions are obtained in the form of convergent polynomial series of terms with increasing orders of smallness. As a major achievement, the asymptotic analytical solutions are explicit functions of the governing parameters and automatically satisfy the integral compatibility condition, which traditionally requires a numerical solution to assess the hyperstatic unknowns. The mathematical conditions for the existence of admissible solutions and asymptotic consistency of the perturbation scheme are provided, while the characteristic properties of the asymptotic series are recognized or demonstrated. Parametric analyses successfully verify the high approximation accuracy achievable by high-order asymptotic series, truncated to a large but finite number of terms. With the aim of extending the methodology to the largest possible variety of cable structure applications, a fully analytical, although asymptotic, expression of the geometric stiffness matrix of inextensible inclined cables is determined. Parametric analyses confirm that high-order asymptotic expressions can accurately approximate the exact stiffness matrix assessed numerically. This achievement opens up viable opportunities for analytically studying and parametrically designing complex cable structures (including collaborations with other structural elements), within the framework of the direct stiffness method. Finally, the feasibility and effectiveness of the asymptotic direct stiffness method are successfully verified by solving in a fully-analytical way a paradigmatic static problem for the catenary cable-stayed beam.

Piecewise approximate analytical solutions of high-order reaction-diffusion singular perturbation problems with boundary and interior layers

<abstract> <p>This work aims to present a reliable algorithm that can effectively generate accurate piecewise approximate analytical solutions for third- and fourth-order reaction-diffusion singular perturbation problems. These problems involve a discontinuous source term and exhibit both interior and boundary layers. The original problem was transformed into a system of coupled differential equations that are weakly interconnected. A zero-order asymptotic approximate solution was then provided, with known asymptotic analytical solutions for the boundary and interior layers, while the outer region solution was obtained analytically using an enhanced residual power series approach. This approach combined the standard residual power series method with the Padé approximation to yield a piecewise approximate analytical solution. It satisfies the continuity and smoothness conditions and offers higher accuracy than the standard residual power series method and other numerical methods like finite difference, finite element, hybrid difference scheme, and Schwarz method. The algorithm also provides error estimates, and numerical examples are included to demonstrate the high accuracy, low computational cost, and effectiveness of the method within a new asymptotic semi-analytical numerical framewor.</p> </abstract>

Lubrication dynamics of a settling plate

If a flat, horizontal, plate settles onto a flat surface, it is known that the gap $h$ decreases with time $t$ as a power law: $h\sim t^{-1/2}$ . We consider what happens if the plate is not initially horizontal, and/or the centre of mass is not symmetrically positioned: does one edge contact the surface in finite time, or does the plate approach the horizontal without making contact? The dynamics of this system is analysed and shown to be remarkably complex. We find that, depending upon the initial position of the plate and the position of the centre of force, the plate might either make contact in finite time or settle progressively without ever making contact. Our results show an excellent agreement between analytical exact solutions, asymptotic solutions and numerical studies of the lubrication equations.

Effects of anisotropic permeability on EMHD nanofluid flow and heat transfer in porous microchannel with wavy rough walls

This theoretical study thoroughly examines the complexities of fluid flow and heat transfer within microfluidic devices by employing anisotropic porous media with wavy walls. The effect of anisotropic permeability and electromagnetohydrodynamics (EMHD) on nanofluid transport is examined in a wavy microchannel under a constant pressure gradient. The nonlinear coupled governing equations for electric double-layer potential, velocity, temperature, and nanoparticle volume fraction are solved numerically using shooting techniques with the Runge–Kutta method. The numerical results are validated with the asymptotic analytical solutions. We explore the impact of parameters like the Hartmann number, Darcy number, and joule heating on nanofluidic flow. The interplay between anisotropic permeability ratio and angle significantly influences flow, temperature profiles, and nanoparticle volume fraction. Reduction in nanoparticle concentration is attributed to constrained entry into the porous medium due to elevated anisotropic permeability. Frictional heating, influenced by the Forchheimer inertial effect, enhances temperature and induces slug flow at low Darcy numbers. The Nusselt number peaks at low Darcy numbers with an anisotropic angle of φ=π/2, experiencing a minimum when φ=0. These findings deepen our understanding of the role of anisotropic permeability in shaping fluid flow and heat transfer in microfluidic systems influenced by EMHD effects.

Slow Rotation of Coaxial Slip Colloidal Spheres about Their Axis

The flow field around a straight chain of multiple slip spherical particles rotating steadily in an incompressible Newtonian fluid about their line of centers is analyzed at low Reynolds numbers. The particles may vary in radius, slip coefficient, and angular velocity, and they are permitted to be unevenly spaced. Through the use of a boundary collocation method, the Stokes equation governing the fluid flow is solved semi-analytically. The interaction effects among the particles are found to be noteworthy under appropriate conditions. For the rotation of two spheres, our collocation results for their hydrodynamic torques are in good agreement with the analytical asymptotic solution in the literature obtained by using a method of twin multipole expansions. For the rotation of three spheres, the particle interaction effect indicates that the existence of the third particle can influence the torques exerted on the other two particles noticeably. The interaction effect is stronger on the smaller or less slippery particles than on the larger or more slippery ones. Torque results for the rotation of chains of many particles visibly show the shielding effect among the particles.

Asymptotic analysis of the angular motion of a projectile based on Lyapunov artificial-small-parameter perturbation

In the traditional analysis of the angular motion of a projectile, the coefficient freezing method is often used to simplify the differential equation of angular motion. However, this method gives a poor approximation for long time scales. In this paper, based on Lyapunov artificial-small-parameter perturbation theory, an asymptotic analytical solution for the angular motion of a projectile is derived by constructing artificial small parameters for the variable coefficient terms in the original equation. In an example, the angular motion of a single-channel rotating projectile with two-dimensional trajectory correction was analyzed. The method proposed in this paper was compared with the coefficient freezing method. The results show that the proposed method can give a good approximation over a long time scale. This study may be significant for the solution of differential equations with variable coefficients and may be useful for certain engineering applications.

Dissipation instability of Couette-like adiabatic flows in a plane channel

The mixed convection flow in a plane channel with adiabatic boundaries is examined. The boundaries have an externally prescribed relative velocity defining a Couette-like setup for the flow. A stationary flow regime is maintained with a constant velocity difference between the boundaries, considered as thermally insulated. The effect of viscous dissipation induces a heat source in the flow domain and, hence, a temperature gradient. The nonuniform temperature distribution causes, in turn, a buoyancy force and a combined forced and free flow regime. Dual mixed convection flows occur for a given velocity difference. Their structure is analysed where, in general, only one branch of the dual flows is compatible with the Oberbeck–Boussinesq approximation, for realistic values of the Gebhart number. A linear stability analysis of the basic stationary flows with viscous dissipation is carried out. The stability eigenvalue problem is solved numerically, leading to the determination of the neutral stability curves and the critical values of the Péclet number, for different Gebhart numbers. An analytical asymptotic solution in the special case of perturbations with infinite wavelength is also developed.

Thermophoresis migration of an aerosol spherical particle embedded in a Brinkman medium at small non-zero Péclet numbers

The method of matched asymptotic expansions is used to investigate the problem of thermophoresis migration of an aerosol spherical particle immersed in a porous medium saturated by a viscous fluid at a small non-zero Péclet number Pe. A uniform temperature gradient is imposed on the system parallel to the diameter of the particle in the opposite direction of z axis. It is assumed that the Knudsen number is in the range of the slip fluid flow through the pores of the porous medium and is compatible with the assumption of the continuum model. The porous medium is modeled by the Brinkman equation and is assumed to be homogenous and isotropic, and the solid matrix is in thermal equilibrium with the fluid through the voids of the medium. In the analysis of motion, the thermal stress slip is considered in addition to the temperature jump, the thermal creep, and the frictional slip. The thermophoretic velocity of the particle is obtained in the closed form up to order Pe3 as a function of the thermal properties of the system and the permeability of the porous medium. The present asymptotic analytical solutions can be viewed as an effective method for checking the numerical schemes for future work on arbitrary values of the Péclet number. The limiting case of the thermophoretic velocity for the Stokes clear fluid is recovered.

Boundary Element Method for Tangential Contact of a Coated Elastic Half-Space

We present a formulation of the boundary element method (BEM) for simulating the tangential contact with an elastic half-space coated with an elastic layer with different elastic properties. We use the fast Fourier-transform-based formulation of BEM, while the fundamental solution is determined directly in the Fourier space. Numerical tests are validated by comparison with available asymptotic analytical solutions for a very thin and a very thick layer, as well as with FEM calculations for layers with finite thickness.

On Boussinesq's Problem for a Power-Law Graded Elastic Half-Space on Elliptical and General Contact Domains.

The indentation of a power-law graded elastic half-space by a rigid counter body is considered in the framework of linear elasticity. Poisson's ratio is assumed to be constant over the half-space. For indenters with an ellipsoidal power-law shape, an exact contact solution is derived, based on the generalizations of Galin's theorem and Barber's extremal principle for the inhomogeneous half-space. As a special case, the elliptical Hertzian contact is revisited. Generally, elastic grading with a positive grading exponent reduces the contact eccentricity. Fabrikant's approximation for the pressure distribution under a flat punch of arbitrary planform is generalized for power-law graded elastic media and compared with rigorous numerical calculations based on the boundary element method (BEM). Very good agreement between the analytical asymptotic solution and the numerical simulation is obtained for the contact stiffness and the contact pressure distribution. A recently published approximate analytic solution for the indentation of a homogeneous half-space by a counter body, whose shape slightly deviates from axial symmetry but is otherwise arbitrary, is generalized for the power-law graded half-space. The approximate procedure for the elliptical Hertzian contact exhibits the same asymptotic behavior as the exact solution. The approximate analytic solution for the indentation by a pyramid with square planform is in very good agreement with a BEM-based numerical solution of the same problem.

A hydromechanical FEM fracturing simulator under the leak-off phenomenon perspective

Carter’s model of fluid loss is used in the oil and gas industry performing the numerical approach of leak-off. Its deduction applies some simplifying assumptions, such as the difference between fluid pressure and constant pore pressure and filtration process approximation to a unidirectional flow, perpendicular to the fracture plane. For this paper, the so-called Chimas is an hydromechanical simulator of oil reservoirs, fully coupled and implicit, in a way that considers the poroelastic effects on the propagation of fractures. The leak-off process is explicitly treated in the fracture boundary conditions using Darcy’s law. The objective of this work is to validate the filtration phenomenon of Chimas hydromechanical simulator comparing the parameters of length, width and injection pressure with those obtained using an iterative method implemented in MatLab and which includes Carter’s analytical equation. Numerical accuracy was validated against asymptotic analytical solutions. The agreement between the iterative method and the analytical solutions was particularly good in relation the length fracture and injection pressure.

Green’s Functions on Various Time Scales for the Time-Fractional Reaction-Diffusion Equation

The time-fractional diffusion equation coupled with a first-order irreversible reaction is investigated by employing integral transforms. We derive Green’s functions for short and long times via approximations of the Mittag-Leffler function. The time value for which the crossover between short- and long-time asymptotic holds is presented in explicit form. Based on the developed Green’s functions, the exact analytic asymptotic solutions of the time-fractional reaction-diffusion equation are obtained. The applicability of the obtained solutions is demonstrated via quantification of the reaction-diffusion kinetics during heterogeneous catalytic chitin conversion to chitosan.

ANALYTIC HEAT AND SOLUTE TRANSPORT OF MHD REACTIVE MIXED CONVECTION FLOW OVER A HORIZONTAL POROUS STRETCHING SHEET WITH MULTIPLE SLIPS

The present paper interprets the unsteady magnetohydrodynamic (MHD) flow with heat and mass transfer over a stretching sheet embedded in a horizontal porous medium. The effects of viscous and Darcy dissipation, heat source, and chemical reaction is elaborately discussed in the presence of multiple slip effects for velocity, temperature, and concentration. Introducing a similarity transformation, the governing partial differential equations are transformed into a system of nonlinear, coupled partial differential equations. The asymptotic analytical solutions are obtained by using double-parameter transformation perturbation techniques and, compared with the numerical results and the verification of the computational code made with the earlier works, serves as the benchmark of reliability of the present study. The important findings reported herein are: porosity acts as an aiding force for the fluid velocity, more dissipative heat leads to higher velocity and temperature, chemical reaction parameter adversely affects the concentration. The main motivation behind this study is to perform the fluid flow, heat, and mass transfer analysis of a Newtonian fluid with mixed MHD flow, which has important applications in many industries, in particular, the process of extrusion/layer of fluid dispersed with solutal particles is extruded over other materials to increase the strength and durability of the final product.

Learning the single-dominant modal system on resonant sloshing in a rectangular tank

A machine learning technique is proposed to derive the damping rate functions in the co-called single-dominant modal system describing the resonant liquid sloshing in a rectangular tank performing harmonic longitudinal motions. Its implementation is demonstrated for the steady-state (periodic) resonance waves when the modal system admits an analytical asymptotic solution depending on the introduced damping rate functions. Recent experiments by Bäuerlein & Avila (2021) are employed to show that the viscous damping of the higher natural sloshing modes matters, and the damping functions depend on the wave amplitude.

Asymptotic analysis of the SIR model and the Gompertz distribution

The SIR (Susceptible–Infected–Removed) is one of the simplest models for epidemic outbreaks. The present paper derives a novel, simple, analytical asymptotic solution for the I-variable, which is valid on the entire real line. Connections with the Gompertz and Gumbel distributions are also demonstrated. The approach is applied to the ongoing coronavirus disease 2019 (COVID-19) pandemic in four European countries — Belgium, Italy, Sweden, and Bulgaria. The reported raw incidence data from the outbreaks in 2020–2021 have been fitted using constrained least squares. It is demonstrated that the asymptotic solution can be used successfully for parametric estimation either in stand-alone mode or as a preliminary step in the parametric estimation using numerical inversion of the exact parametric solution.