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Nanofluids have captured the attention of scientists due to their superior thermophysical properties compared to ordinary liquids. Consequently, nanofluids serve as suitable cooling agents applicable to systems requiring swift response to thermal changes, such as vehicle engines. Therefore, the current article scrutinizes the characteristics of heat transfer on the pressure-driven flow of magnetized nanofluid between two curved corrugated surfaces in the presence of heat generation/absorption. Copper oxide nanoparticles (Cu) have been combined with pure water to form a nanofluid called Cu/H 2 O. The geometry of the channel is represented mathematically in an orthogonal curvilinear coordinate system. The corrugation grooves are described by sinusoidal functions with phase differences between the corrugated curved walls. The boundary perturbation method is used to find the analytical solution for the velocity field, temperature field, and volumetric flow rate taking the corrugation amplitude as the perturbation parameter. The impact of dissimilar parameters such as the curvature parameter (0.5 ≤ k ≤ 10) , wave number (1 ≤ α ≤ 3) , magnetic parameter (1 ≤ M ≤ 5) , and wave amplitude (0.1 ≤ ε ≤ 0.8) on the flow fields are analyzed through graphs and discussed in detail. The results show that the peak of the velocity increases with the radius of curvature and the width of the channel for a constant pressure gradient. If we increase the magnetic parameter from 1 to 4, the velocity profile at the specified point decreases by 30 %. If we increase the heat source/sink parameter from 2 to 5, the temperature profile at the specified point increases by approximately 17 %. The flow rate is increased by the corrugations for any phase difference between the corrugated curved walls depending on the corrugation wavenumber and the channel radius of curvature. [ABSTRACT FROM AUTHOR]

• Contact vibration model between a rigid punch and a half-space with hysteretic damping considering couple stress is studied. • The size effect has a pronounced impact on the peak effect of the dynamic contact stiffness and reciprocal receptance. • The real parts of the reciprocal receptance and dynamic contact stiffness are sensitive to the internal friction. This paper investigates the contact vibration between a rigid spherical indenter and a half-space with hysteretic damping within the framework of the couple stress theory. Firstly, the static contact pressure is determined by using the integral least square approach. Subsequently, the dynamic contact pressure is obtained through the perturbation technology, and the dynamic contact displacement is obtained by employing the Hankel transform. Finally, the receptance is obtained and the dynamic contact stiffness is deduced by adopting approximate dynamic displacement boundary conditions under the assumption of a sufficiently small oscillation force. Parametric studies are presented to clarify the effects of the characteristic material length, Poisson's ratio, shear modulus, and internal friction on the reciprocal receptance and dynamic contact stiffness. The results obtained reveal that the size effect has a pronounced effect on the peak values of the dynamic contact stiffness and reciprocal receptance. Furthermore, the present work provides a useful theoretical framework for applying contact resonance microscopy techniques to micro-structured materials. [ABSTRACT FROM AUTHOR]

Nanofluids have captured the attention of scientists due to their superior thermophysical properties compared to ordinary liquids. Consequently, nanofluids serve as suitable cooling agents applicable to systems requiring swift response to thermal changes, such as vehicle engines. Therefore, the current article scrutinizes the characteristics of heat transfer on the pressure-driven flow of magnetized nanofluid between two curved corrugated surfaces in the presence of heat generation/absorption. Copper oxide nanoparticles (Cu) have been combined with pure water to form a nanofluid called Cu/H2O. The geometry of the channel is represented mathematically in an orthogonal curvilinear coordinate system. The corrugation grooves are described by sinusoidal functions with phase differences between the corrugated curved walls. The boundary perturbation method is used to find the analytical solution for the velocity field, temperature field, and volumetric flow rate taking the corrugation amplitude as the perturbation parameter. The impact of dissimilar parameters such as the curvature parameter (0.5≤k≤10), wave number (1≤α≤3), magnetic parameter (1≤M≤5), and wave amplitude (0.1≤ε≤0.8) on the flow fields are analyzed through graphs and discussed in detail. The results show that the peak of the velocity increases with the radius of curvature and the width of the channel for a constant pressure gradient. If we increase the magnetic parameter from 1 to 4, the velocity profile at the specified point decreases by 30 %. If we increase the heat source/sink parameter from 2 to 5, the temperature profile at the specified point increases by approximately 17 %. The flow rate is increased by the corrugations for any phase difference between the corrugated curved walls depending on the corrugation wavenumber and the channel radius of curvature.

We delve into the ‘effects of hall currents on the dynamics of unsteady magnetohydrodynamic flow and heat transfer within a two-fluid system of ionized gases confined within a horizontal channel bounded by parallel conducting plates. Employing a regular perturbation technique, we solve the governing partial differential equations to unveil the distributions of velocity and temperature, alongside profiles depicting heat transfer coefficients. Through a systematic parametric analysis, we explore the interplay among variables such as the Hartmann number, Hall parameter, and ratios involving viscosities, heights, electrical conductivities, and thermal conductivities. The results highlight the profound influence of these parameters on the dynamics of unsteady magnetohydrodynamic (MHD) heat transfer within a flow regime characterized by a dual-ionized fluid’. This influence is particularly pronounced when the lateral plates of the channel are conductive. Significantly, elevated Hartmann numbers and Hall parameters are associated with augmented heat transfer coefficients at both plates, holding other variables constant.

Pressure‐driven movement is a fundamental concept with numerous applications in various industries, scientific disciplines, and fields of engineering. Its proper execution is vital for promoting revolutionary innovations and providing solutions in numerous sectors. Therefore, this article scrutinizes the pressure‐driven flow of magnetized Casson fluid between two curved corrugated walls. The geometry of the channel is represented mathematically in an orthogonal curvilinear coordinate system. The corrugation grooves are described by sinusoidal functions with phase differences between the corrugated curved walls. The boundary perturbation method is used to find the analytical solution for the velocity field and volumetric flow rate, taking the corrugation amplitude as the perturbation parameter. The results show that the peak of the velocity increases with the radius of curvature and the width of the channel for a constant pressure gradient. The velocity exhibited a declining trend due to an increase in the Casson fluid parameter. For a sufficiently large corrugation wavenumber, the flow rate decreases, and the phase difference becomes irrelevant. However, the reduction in flow can be minimized by decreasing the channel radius of curvature. In general, a smooth curved channel will give the maximum flow rate for a large corrugation wavenumber. The model can be used to simulate blood flow in arteries with varying geometries and magnetic fields, aiding in the study of cardiovascular diseases and the design of medical devices like stents. [ABSTRACT FROM AUTHOR]

This article considers the problem of aeroelastic oscillations in the channel wall having a suspension with hardening cubic nonlinearity, which were induced by the vibration of the channel foundation. The narrow flat channel formed by two parallel rigid walls and filled with pulsating viscous gas was examined. The bottom wall was stationary, while the opposite one had a nonlinear elastic suspension. The aeroelasticity problem was formulated for the isothermal state of the gas and channel walls. Considering the narrowness of the channel, the equations of dynamics were derived for a thin layer of the viscous gas, and the asymptotic analysis of the problem was performed by the perturbation method. Using the method of iterations, the law of viscous gas pressure distribution in the channel was determined, and the equation of aeroelastic oscillations in the channel wall was obtained as a generalization of the Duffing equation. This equation was solved by the harmonic balance method. The primary nonlinear aeroelastic response of the channel wall and the nonlinear phase shift were expressed as implicit functions. These characteristics were studied numerically to evaluate the influence of the nonlinear elastic suspension of the channel wall and the viscous gas inertia and compressibility on the nonlinear oscillations in the channel wall.

We consider a regularization of the Lagrange multiplier rule (LMR) in the nondifferential form in a convex constrained optimization problem with an operator equality constraint in a Hilbert space and a finite number of functional inequality constraints. The objective functional of the problem is assumed to be strongly convex, and the convex closed set of its admissible elements also belongs to a Hilbert space. The constraints of the problem contain additively included parameters, which enables using the so-called perturbation method to study it. The main purpose of the regularized LMR is the stable generation of generalized minimizing sequences (GMSs), which approximate the exact solution of the problem using extremals of the regular Lagrange functional. The regularized LMR itself can be interpreted as a GMS-generating (regularizing) operator, which assigns to each set of input data of the constrained optimization problem the extremal of its corresponding regular Lagrange functional, in which the dual variable is generated following one or another procedure for stabilizing the dual problem. The main attention is paid to (1) studying the connection between the dual regularization procedure and the subdifferential properties of the value function of the original problem; (2) proving the convergence of this procedure in the case of solvability of the dual problem; (3) an appropriate update of the regularized LMR; (4) obtaining the classical LMR as a limiting version of its regularized analog. [ABSTRACT FROM AUTHOR]

Motile cilia are hair-like, tiny structures that originate in different types of biological processes. Yeleswarapu fluid model is approximated as non-Newtonian mucus, which is viscous and slippery fluid with key roles in lubrication, protection, and particle trapping in the respiratory and digestive systems. The properties of Yeleswarapu fluid with heat transfer in ciliated channels are investigated in this study. The fluid flow is characterized as laminar (low Reynold number) and incompressible (constant density) in a two-dimensional complex wavy channel generated via metachronal waves of cilia. A static frame of reference is first transformed into a dynamic frame of reference. Afterward, the system of governing equations is transformed into a non-dimensional form by using scaling factors. The half-width of the channel is substantially smaller than the wavelength of metachronal waves. Hence, the constitutive equations are simplified by the long wavelength hypothesis. The perturbation approach is employed to solve the simplified system of differential equations. This research investigates how different boundary walls and rheology affect fluid flow. Additionally, graphs of the streamlines, velocity field, temperature, heat transfer coefficient, and pressure rise are included. [ABSTRACT FROM AUTHOR]

In this paper, we study the local dynamical behavior of a Brusselator-type model with state-dependent delay. First, we investigate the local stability of the positive equilibrium and the existence of the Hopf bifurcation. Secondly, the asymptotic of the solutions near the equilibria are constructed by using the perturbation method which further determines the properties of Hopf bifurcation. Finally, the numerical simulations are carried out to support the theoretical findings. [ABSTRACT FROM AUTHOR]

Most space shuttle fuel tanks use a center column to hold the Propellant Management Device (PMD). This paper analyzes the gas–liquid interface state in the tanks with a central column during microgravity experiments conducted in the Chinese Space Station. It launches an extended study to investigate the gas–liquid interface state under different gravity conditions. Using the perturbation method and boundary layer theory, we numerically calculated the morphology of the gas–liquid interface under varying gravity conditions based on the Young–Laplace equation. The results were then compared to those obtained from existing commercial software and were found to be consistent. Based on this, the study develops two types of calculation procedures. The first procedure generates the corresponding shape of the liquid surface by inputting the height of the liquid surface endpoints and the gravity level. The second procedure is based on the targeting method and generates the corresponding liquid surface by inputting the volume of the liquid in the storage tank and the gravity level. The procedures were used to analyze the variation of gas–liquid interface properties under different gravity conditions. This study offers theoretical support for liquid management in aerospace engineering fuel tanks. [ABSTRACT FROM AUTHOR]

The effect of time-periodic magnetic field modulation on the onset of ferroconvection in a magnetic fluid saturated densely packed porous medium is examined. In many systems, such as charges in electrostatic field and ferromagnetic resonance, modulation of a suitable parameter can have marked effects on the motion and can result in increased stability of the system. Making use of isothermal boundary conditions, the subsequent eigenvalue problem is attacked by means of the regular perturbation method with the assumption of small amplitude of modulation. The onset criteria are derived based on the assumption that the principle of exchange of stabilities holds good. The thermal Rayleigh number shift is determined as a function of magnetic force, porous parameters, and magnetic field modulation frequency. The influence of various physical factors is perceived to be significant at moderate values of magnetic field modulation frequency. It is found that subcritical instability is possible for low frequency magnetic field modulation. The effect of magnetic mechanism is shown to be attenuating the stabilizing influence of magnetic field modulation for moderate and large values of the frequency of modulation. However, the stabilizing effect of magnetic field modulation gets amplified due to an increase in the values of Vadasz number. Further, it is delineated that the normalized porosity and magnetic field modulation work in tandem in destabilizing the system for low frequency modulation. The study reveals that the effect of magnetic field modulation could be exploited to control ferroconvective instability in a porous medium saturated with magnetic fluids. [ABSTRACT FROM AUTHOR]

This paper presents a numerical method to solve the non-linear vibration problem of composite structures made of many unidirectional fibers/matrix layers. To take into account damping properties of structures, fractional derivative Zener model for complex moduli is used in each layer. The frequency dependence of these models leads to a non-linear vibration problem. To solve it, a High Order Newton (HON) solver based on homotopy and perturbation techniques is proposed. Thus, the initial non-linear problem turns into a set of linear systems, easier to solve. Numerical results obtained by the HON solver are compared with experimental and numerical results of literature. [ABSTRACT FROM AUTHOR]

It remains a challenge for existing financial models to accurately capture the shapes of implied volatility (IV) of options with all maturities simultaneously. Inspired by Chen et al.'s empirical finding, 'the shorter the option expiry, the higher the IV is', we propose a novel and simple approach to solve this challenge, by introducing a term-structure-based correction (i.e. an exponential increase function of the options expiry) to the volatility of volatility (vol–vol) term of the classical Heston stochastic volatility model. We derive an approximate formula for the IV under the corrected model with the perturbation method and further apply the formula to predict the IVs of options written on the Shanghai Stock Exchange 50 ETF. Numerical experiments and empirical results show that the introduction of a term-structure-based correction function surely overcomes the deficiency of the classical Heston model in capturing the short-term IVs, thus improving notably its performance of IV forecasting. [ABSTRACT FROM AUTHOR]

The stability properties of the Hill equation are discussed, especially those of the Mathieu equation that characterize ion motion in electrodynamic traps. The solutions of the Mathieu-Hill equation for a trapped ion are characterized by employing the Floquet theory and Hill's method solution, which yields an infinite system of linear and homogeneous equations whose coefficients are recursively determined. Stability is discussed for parameters a and q that are real. Characteristic curves are introduced naturally by the Sturm–Liouville problem for the well-known even and odd Mathieu equations c e m (z , q) and s e m (z , q) . In the case of a Paul trap, the stable solution corresponds to a superposition of harmonic motions. The maximum amplitude of stable oscillations for ideal conditions (taken into consideration) is derived. We illustrate the stability diagram for a combined (Paul and Penning) trap and represent the frontiers of the stability domains for both axial and radial motion, where the former is described by the canonical Mathieu equation. Anharmonic corrections for nonlinear Paul traps are discussed within the frame of perturbation theory, while the frontiers of the modified stability domains are determined as a function of the chosen perturbation parameter and we demonstrate they are shifted towards negative values of the a parameter. The applications of the results include but are not restricted to 2D and 3D ion traps used for different applications such as mass spectrometry (including nanoparticles), high resolution atomic spectroscopy and quantum engineering applications, among which we mention optical atomic clocks and quantum frequency metrology. [ABSTRACT FROM AUTHOR]

In present study, the response of a cemented granular and horizontal layer is investigated under one-dimensional harmonic vibrations applied at its base. The modeling was performed considering an infinite horizontal layer with displacements occurred in one direction with uniform shear stress and strain distributions on horizontal planes. It is considered that only shear displacements occur when the soil layer is subjected to seismic excitation at base. The nonlinear behavior due to cyclic loading can be determined using dynamic characteristics of soil like shear modulus and damping ratio. These dynamic characteristics are dependent to different parameters like confining pressure and cement content. In present study, an empirical model was applied for determination of dynamic characteristics of cemented and uncemented soil. By deriving the one degree of freedom equation of motion, an approximate solution was suggested using perturbation method. Finally, the resonance phenomenon was studied for cemented granular layer and the amplitudes were predicted with a precise approximation. Based on the results, the suggested method was able to predict the response of soil layer with good consistency comparing to the results of numerical methods like Runge-Kutta. [ABSTRACT FROM AUTHOR]

Abstract This research contributes to the comprehension of nanofluid behaviour through a wavy channel, emphasizing the significance of considering diverse influences in the modelling process. The study explores the collective influence of pressure gradient variation, magnetic field, porosity, channel waviness, nanoparticle concentration, and heat transfer on nano-blood flow in a two-dimensional wavy channel. In contrast to prior research assuming a constant pulsatile pressure gradient during channel waviness, this innovative study introduces a variable pressure gradient, significantly influencing several associated parameters. The mathematical model characterizing nano-blood flow in a horizontally wavy channel is solved using the perturbation technique. Analytical solutions for fundamental variables such as stream function, velocity, wall shear stress, pressure gradient, and temperature are visually depicted across different physical parameters values. The findings obtained for differing parameter values in the given problem demonstrate a significant influence of the amplitude ratio parameter of channel waviness, Hartmann number of the magnetic field, permeability parameter of the porous medium, volume fraction of nanoparticles, radiation parameter, Prandtl number, and the suction/injection parameter on the flow dynamics. The simulations provide valuable insights into the decrease in velocity with increasing magnetic field and its increase with higher permeability. Additionally, the temperature is observed to escalate with a rising nanoparticle volume fraction and radiation parameter, while it declines with increasing Prandtl number.

In this paper, we analyze a perturbed formulation of bilevel optimization problems, which we refer to as δ-perturbed formulation. The δ-perturbed formulation allows to handle the lower level optimization problem efficiently when there are multiple lower level optimal solutions. By using an appropriate perturbation strategy for the optimistic or pessimistic formulation, one can ensure that the optimization problem at the lower level contains only a single (approximate) optimal solution for any given decision at the upper level. The optimistic or the pessimistic bilevel optimal solution can then be efficiently searched for by algorithms that rely on solving the lower level optimization problem multiple times during the solution search procedure. The δ-perturbed formulation is arrived at by adding the upper level objective function to the lower level objective function after multiplying the upper level objective by a small positive/negative δ. We provide a proof that the δ-perturbed formulation is approximately equivalent to the original optimistic or pessimistic formulation and give an error bound for the approximation. We apply this scheme to a class of algorithms that attempts to solve optimistic and pessimistic variants of bilevel optimization problems by repeatedly solving the lower level optimization problem.

The thermal-hydraulic dynamics in containment are governed by a system of stiff ordinary differential equations (ODEs). A fully implicit discretization scheme is adopted to discretize these ODEs in order to mitigate the effects of stiffness. In comparison with explicit or semi-implicit discretization schemes that are subject to Courant limits on time steps, the fully implicit discretization scheme is more suitable for a containment analysis code that focuses on predicting both shortterm and long-term thermal-hydraulic parameters after an accident. This study introduces a general-purpose ODE solver for the containment analysis code. The outline of the solver is as follows: The fully implicit discrete equations lead to a large set of nonlinear equations that need to be solved using Newton's iterative method. The partial derivative components in the Jacobi matrix are calculated by the perturbation method using finite difference approximation, which avoids the complicated derivation of partial derivatives. The scaling modification technique is incorporated into this ODE solver to deal with significant differences in unknown variable magnitudes, and the line search method is introduced to address the difficulty of obtaining an accurate root estimate with Newton's method when the initial guess is far from the actual root. This proposed ODE solver was applied to two typical stiff ODE problems to test its stiffness-suppressed ability and to demonstrate that this proposed solver can perform calculations with a very large time step. Then, the CASSIA code, a containment analysis code developed by China Nuclear Power Technology Research Institute Co., Ltd (CNPRI), equipped with this ODE solver, was applied to the CSNI (Committee on the Safety of Nuclear Installations) benchmark problem and the Carolinas Virginia Tube Reactor (CVTR) test 3 problem to preliminarily demonstrate that the proposed ODE solver can perform containment thermal-hydraulic analysis correctly. This study could provide references for the development of a home-made containment analysis code. [ABSTRACT FROM AUTHOR]

A novel analysis of the pulsatile nano-blood flow through a sinusoidal wavy channel, emphasizing the significance of diverse influences in the modelling, is investigated in this paper. This study examines the collective effects of slip boundary conditions, magnetic field, porosity, channel waviness, nanoparticle concentration, and heat source on nano-blood flow in a two-dimensional wavy channel. In contrast to prior research that assumed a constant pulsatile pressure gradient during channel waviness, this innovative study introduces a variable pressure gradient that significantly influences several associated parameters. The mathematical model characterising nano-blood flow in a horizontally wavy channel is solved using the perturbation technique. Analytical solutions for fundamental variables such as stream function, velocity, wall shear stress, pressure gradient, and temperature are visually depicted across different physical parameter values. The findings obtained for various parameter values in the given problem demonstrate a significant influence of the amplitude ratio parameter of channel waviness, Hartmann number of the magnetic field, permeability parameter of the porous medium, Knudsen number due to the slip boundary, volume fraction of nanoparticles, radiation parameter, Prandtl number, and heat source parameters on the flow dynamics. The simulations provide valuable insights into the decrease in velocity with increasing magnetic field and its increase with increasing permeability and slip parameters. Additionally, the temperature increases with increasing nanoparticle volume fraction and radiation parameter, while it decreases with increasing Prandtl number. [ABSTRACT FROM AUTHOR]

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We consider the radially symmetric positive solutions to quasilinear problem \begin{equation*}-\triangle u-u\triangle u^{2}+\lambda u=f(u),\quad{\rm in} \ \mathbb{R}^{N},\end{equation*} having prescribed mass $\int_{\mathbb{R}^{N}}|u|^2 =a^2,$ where a > 0 is a constant, λ appears as a Lagrange multiplier. We focus on the pure L 2-supercritical case and combination case of L 2-subcritical and L 2-supercritical nonlinearities \begin{equation*}f(u)=\tau |u|^{q-2}u+|u|^{p-2}u,\quad \tau \gt 0,\qquad{\rm where}\ \ 2 \lt q \lt 2+\frac{4}{N} \ {\rm and} \quad \ p \gt \bar{p},\end{equation*} where $\bar{p}:=4+\frac{4}{N}$ is the L 2-critical exponent. Our work extends and develops some recent results in the literature. [ABSTRACT FROM AUTHOR]

In this paper different lumped parameters models are presented for the stability analysis of rotors supported on gas bearings. The analysis is carried out taking into account the damping and stiffness coefficients of journal bearings, calculated with the perturbation method. Lumped parameters models of different complexity are discussed for the description of the main rigid modes of the spindle. In case of symmetric systems, it is shown that the complexity of the model can be reduced by halving the number of the degrees of freedom. The analogy with the single mass approach is demonstrated. The considerations discussed in the paper are preliminary for the stability analysis of more complex systems, such as the ones with non-fixed bushes. [ABSTRACT FROM AUTHOR]

In this study, we have analyzed magnetohydrodynamic (MHD) consequences on the heat and mass transmission within unsteady dissipated liquid film flow. Flow is generated due to stretchable surface accompanied with effects of ohmic heating, chemical reaction and heat absorption. Moreover, the flow governing partial differential equations (PDEs) are further modified into equivalent ordinary differential equations (ODEs) by applying regular perturbation method to get its analytical solution after that we have applied sixth-order Runge–Kutta technique to get its numerical solution. These two solutions are validating each other in the simulations. Figures are plotted to study the changes in physical quantities like skin friction coefficient, concentration, velocity, temperature, Sherwood and Nusselt number with the variations of Prandtl numbers Pr, parameters of chemical reaction β , Eckert numbers Ec, magnetic parameter Ha (also known as Hartman number) Schmidt number and coefficient of heat absorption Δ. [ABSTRACT FROM AUTHOR]

This research contributes to the comprehension of nanofluid behaviour through a wavy channel, emphasizing the significance of considering diverse influences in the modelling process. The study explores the collective influence of pressure gradient variation, magnetic field, porosity, channel waviness, nanoparticle concentration, and heat transfer on nano-blood flow in a two-dimensional wavy channel. In contrast to prior research assuming a constant pulsatile pressure gradient during channel waviness, this innovative study introduces a variable pressure gradient, significantly influencing several associated parameters. The mathematical model characterizing nano-blood flow in a horizontally wavy channel is solved using the perturbation technique. Analytical solutions for fundamental variables such as stream function, velocity, wall shear stress, pressure gradient, and temperature are visually depicted across different physical parameters values. The findings obtained for differing parameter values in the given problem demonstrate a significant influence of the amplitude ratio parameter of channel waviness, Hartmann number of the magnetic field, permeability parameter of the porous medium, volume fraction of nanoparticles, radiation parameter, Prandtl number, and the suction/injection parameter on the flow dynamics. The simulations provide valuable insights into the decrease in velocity with increasing magnetic field and its increase with higher permeability. Additionally, the temperature is observed to escalate with a rising nanoparticle volume fraction and radiation parameter, while it declines with increasing Prandtl number. [ABSTRACT FROM AUTHOR]

In this study, vibrations of stepped nanobeams were investigated according to Eringen's nonlocal elasticity theory. Multi-time scale method, which is one of the perturbation methods, has been applied to solve dimensionless state equations. The solution is considered in two steps. First-order terms obtained from the perturbation expansion formed the linear problem in the first step. In the second step, the solution of the second order of the perturbation expansion was made and nonlinear terms emerged as corrections to the linear problem from this solution. The main issue that the study wants to emphasize is the examination of the mechanical effects of the steps, which are discontinuities encountered at the nanoscale, on the system. For this purpose, while the findings of the research were obtained, various nonlocal parameter values were obtained to capture the nano-scale effect, and frequency-response and nonlinear frequency-amplitude curves corresponding to the 1st Mode values of the beam for different step ratios and step locations were obtained to capture the step effect. One of the important features of the nonlinear system is the formation of internal resonance between the modes of the system. How this situation affects the characteristics of the system has also been examined and results have been given by graphs. The obtained data show that taking into account the nanoscale step is essential for the accuracy and sensitivity of many nanostructures such as sensors, actuators, biostructures, switches, etc. that are likely to be produced at the nanoscale in practice. [ABSTRACT FROM AUTHOR]

We investigate the dynamics of fast two-pulse collisions in linear diffusion-advection systems with weak quadratic loss in spatial dimension 2. We introduce a two-dimensional perturbation method, which generalizes the perturbation method used for studying two-pulse collisions in spatial dimension 1. We then use the generalized perturbation method to show that a fast collision in spatial dimension 2 leads to a change in the pulse shape in the direction transverse to the advection velocity vector. Moreover, we show that in the important case of a separable initial condition, the longitudinal part in the expression for the amplitude shift has a simple universal form, while the transverse part does not. Additionally, we show that anisotropy in the initial condition leads to a complex dependence of the amplitude shift on the orientation angle between the pulses. Our perturbation theory predictions are in very good agreement with results of extensive numerical simulations with the weakly perturbed diffusion-advection model. Thus, our study significantly enhances and generalizes the results of previous works on fast collisions in diffusion-advection systems, which were limited to spatial dimension 1. The results of our study can have applications in multisequence chemical transmission, in reshaping of chemical pulses, and in chemical detection. [ABSTRACT FROM AUTHOR]

Purpose: In the present study, the nonlinear vibration analysis of a nanoscale beam with different boundary conditions named as simply supported, clamped-clamped, clamped-simple and clamped-free are investigated numerically. Methods: Nanoscale beam is considered as Euler-Bernoulli beam model having size-dependent. This non-classical nanobeam model has a size dependent incorporated with the material length scale parameter. The equation of motion of the system and the related boundary conditions are derived using the modified couple stress theory and employing Hamilton's principle. Multiple scale method is used to obtain the approximate analytical solution. Result: Numerical results by considering the effect of the ratio of beam height to the internal material length scale parameter, h/l and with and without the Poisson effect, υ are graphically presented and tabulated. Conclusion: We remark that small size effect and poisson effect have a considerable effect on the linear fundamental frequency and the vibration amplitude. In order to show the accuracy of the results obtained, comparison study is also performed with existing studies in the literature. [ABSTRACT FROM AUTHOR]

Purpose: This paper focuses on the analysis of non-linear forced vibrations of a sandwich beam with a viscoelastic core layer. The analytical formulation considers normal, transverse, and shear deformations in the core layer by utilizing the refined higher-order zig-zag theory (HZZT). Method: The Ritz's method coupled to the modal transformation was employed to derive the nonlinear differential equations of motion. In order to investigate superharmonic resonance cases, the perturbation technique was applied to the equation of motion, and analytical frequency-amplitude relationships were obtained using the multiple scale method. This model is commonly applied to describe materials exhibiting both elastic and viscous characteristics. Results: The frequency response curves, which depict superharmonic resonance behaviour were analyzed and commented upon by varying the material characteristics of the viscoelastic core layer and the geometric parameters of the sandwich beams. Particular attention was given to assessing the influence of the transverse stress on the response curves. [ABSTRACT FROM AUTHOR]

We are concerned with the existence of traveling waves for density‐dependent diffusion reaction nonlinear convection equations with small time delay. We first study the existence and uniqueness of smooth and sharp‐type traveling wavefront solution of the wave speed c ≥ c∗ for equation without time delay, where c∗ is the minimal wave speed. Meanwhile, we construct a variational principle for c∗ from which the upper bound of c∗ is determined, while the lower bound of c∗ is attained by using the phase plane analysis. Then, we obtain the existence of smooth traveling wavefront solution for equation with small time delay for c > c∗ by applying the perturbation method and implicit function theorem. [ABSTRACT FROM AUTHOR]

This paper conducts the analytical investigation on the buckling of cylindrical shells with axially variable elastic modulus subjected to axial compressive load for the first time. First, it proves that the axially distributed elastic modulus can be expressed as the combination of constant and variable component. Then, governing differential equations for buckling analysis are derived and exactly solved by the combined perturbation method and Fourier analysis. Accordingly, the closed analytical solutions for the cylinder with arbitrarily variable elastic modulus are obtained, which reveal the explicit relations among buckling load, shell sizes and elastic modulus functions. Based on the presented analytical formulas, four types of elastic modulus variations for shell material which are uniform, periodic, linear and combined are studied in detail, and the results are also well verified. The derived analytical solutions in this paper can serve as benchmarks for buckling analyses of thin-walled cylinders with elastic modulus variations resulted from design, material manufacturing process, material imperfections and so on. [ABSTRACT FROM AUTHOR]

Spontaneous symmetry breaking occurs when the underlying laws of a physical system are symmetric, but the vacuum state chosen by the system is not. The (3+1)d ϕ 4 theory is relatively simple compared to other more complex theories, making it a good starting point for investigating the origin of non-trivial vacua. The adaptive perturbation method is a technique used to handle strongly coupled systems. The study of strongly correlated systems is useful in testing holography. It has been successful in strongly coupled QM and is being generalized to scalar field theory to analyze the system in the strong-coupling regime. The unperturbed Hamiltonian does not commute with the usual number operator. However, the quantized scalar field admits a plane-wave expansion when acting on the vacuum. While quantizing the scalar field theory, the field can be expanded into plane-wave modes, making the calculations more tractable. However, the Lorentz symmetry, which describes how physical laws remain the same under certain spacetime transformations, might not be manifest in this approach. The proposed elegant resummation of Feynman diagrams aims to restore the Lorentz symmetry in the calculations. The results obtained using this method are compared with numerical solutions for specific values of the coupling constant λ = 1, 2, 4, 8, 16. Finally, we find evidence for quantum triviality, where selfconsistency of the theory in the UV requires λ = 0. This result implies that the ϕ 4 theory alone does not experience SSB, and the ⟨ϕ⟩ = 0 phase is protected under the RG-flow by a boundary of Gaussian fixed-points. [ABSTRACT FROM AUTHOR]

The hybrid stochastic deterministic continuous-energy coarse mesh transport method (COMET) has been recently extended for high-fidelity efficient kinetics calculations in highly heterogeneous reactor cores. The method discretizes the time variable as a series of time grids and solves the resulting set of steady-state neutron transport equations. In this work, a high-order perturbation method is developed to update the COMET unperturbed response function library on the fly for changes in the discretized time step size $$\Delta {t_h}$$ Δ t h . The unperturbed response functions are precomputed with $$1/v\Delta {t_h} = 0$$ 1 / v Δ t h = 0. The perturbation expansion coefficients are also generated during the unperturbed response function library precomputation. The adjoint solution needed by the perturbation method is calculated using the reciprocity relation without solving the corresponding adjoint problems. As a result, the method can be easily implemented into any stochastic code to generate the perturbation expansion coefficients together with the unperturbed response functions. The high-order perturbation method is benchmarked by comparing both the response functions and the time-dependent COMET core solution (fission density) with the corresponding reference solutions. It is found that the response functions generated by the perturbation method at second order are in excellent agreement with those directly computed by the Monte Carlo method. When $$\Delta {t_h}$$ Δ t h changes from 1.0E-4 s to infinity, the average and maximum relative differences in the various response functions were found to be in the range of 0.0000106% to 0.00116% and 0.000011% to 0.00121%, respectively. The fission density as a function of time calculated by COMET using the perturbation method is in excellent agreement with the reference solutions, with an average relative difference of 0.0065% to 0.075%. These comparisons indicate that the perturbation method at second order is highly accurate. [ABSTRACT FROM AUTHOR]

A mathematical model is constructed in this investigation to examine the effects of heat and mass transfer in tangent‐hyperbolic fluid bounded within horizontal channel. The lower‐wall of channel is taken as heated. The dimensional momentum, energy and concentration equations are determined by defining a stress‐tensor of undertaken fluid (Tangent‐hyperbolic fluid). To calculate the solution, the system of dimensional equations is transformed into nondimensional system by using normalized variables and then solved by applying perturbation technique. The impacts of several dimensionless parameters that is, power‐law index parameter m⌣, pressure gradient parameter P⌣, Reynolds number (Re), time constant parameter (Γ), Peclet number (λ4), viscous heating parameter (λ5), Schmidt number (Sc), thermophoresis parameter (NT), and Brownian motion (NB) parameter are sketched and also physically discussed. The results reveal that power‐law index parameter and pressure gradient cause increment in velocity and temperature fields and decrement in concentration field. Further, the variations in skin‐friction coefficient, Nusselt number and Sherwood number via emerging parameters are also expressed in form of table in this exploration. [ABSTRACT FROM AUTHOR]

In this paper, we study the semilinear subelliptic equation where is the self-adjoint Hörmander operator associated with the vector fields X = (Xl, X2, ..., Xm) satisfying the Hörmander's condition, is a Carathéodory function on Ω × ℝ, and Ω is an open bounded domain in ℝn with smooth boundary. Combining the perturbation from the symmetry method with the approaches involving the eigenvalue estimate and the Morse index in estimating the minimax values, we obtain two kinds of existence results for multiple weak solutions to the problem above. Furthermore, we discuss the difference between the eigenvalue estimate approach and the Morse index approach in degenerate situations. Compared with the classical elliptic cases, both approaches here have their own strengths in the degenerate cases. This new phenomenon implies that the results in general degenerate cases would be quite different from the situations in classical elliptic cases. [ABSTRACT FROM AUTHOR]

In this paper, we apply the stochastic perturbation technique to solve the optimal control problem governed by elliptic partial differential equation with small uncertainty in the random input. We first use finite-dimensional noise assumption and perturbation technique to establish the first-order and second-order deterministic optimality systems, and then discretize the two systems by standard finite-element method. Furthermore, we derive a posteriori error estimators for the finite-element approximation of the state, co-state and control in two different norms, respectively. These error estimators are then used to build our adaptive algorithm. Finally, some numerical examples are presented to verify the effectiveness of the derived estimators. [ABSTRACT FROM AUTHOR]

This study focuses on optimization of pressure gradient on MHD peristaltic nanofluid flow by response surface methodology (RSM). The governing equations, including continuity, motion, nanoparticle, and concentration force are solved by disregarding inertial forces and employing the approximation of long-wavelength. The perturbation method is used to solve the resultant nonlinear coupled partial differential equation analytically. Mathematical and graphical outputs for concentration, temperature, and pressure rise, considering all physical parameters, are presented. Numerical computation is applied to assess expressions for friction forces and pressure rise. Magnetohydrodynamics has various applications in various microchannel designs for efficacious flow control in pumping fluids for both pulsating and non-pulsating continuous flow. Finally, Response Surface Methodology (RSM) is used for performing sensitivity analysis and its optimization. ANOVA tables are generated with the help of MINITAB-19 which is a statistical software. The sensitivity results are displayed in tabular and graphical form and concluded that M is more sensitive than other input parameters for ΔPat the low level and the input parameter Nt is most sensitive among others at middle and high levels. Further, Temperature and pressure of the flow has different responses for various values of magnetic, thermophoresis and Brownian motion parameter.

The primary goal of this study is to achieve isothermal temperature control of the fluid at the outlet of a circular pipe through a gradient-based method. Typically, temperatures are set on the outer wall of the pipe to warm the fluid and achieve the desired uniformly consistent exit temperature. However, maintaining a constant temperature on the outer wall of the pipe is challenging due to the heat exchange between the fluid and the wall, as well as external environmental factors. Thus, in this current investigation, the temperature distribution is designated along the inner pipe wall to heat the fluid, employing an optimization algorithm to deduce the appropriate optimal heating function on the outer wall. This enables the attainment of the specified temperature on the inner wall. To guarantee consistent temperature control during optimal heating procedures across extensive distances or expansive areas, a gradient-based optimization technique is employed. In the context of a smaller pipe diameter scenario, applying the designated floor function temperature distribution produces the most accurate outcomes for exit temperatures at the pipe exit. The assessed heating function exhibits abrupt surges and disjointed behaviors, which can be alleviated by introducing a weight coefficient into the cost function. This adaptation preserves the precision of exit temperature and its uniformity. In the context of a larger pipe diameter scenario, while adhering to the floor function temperature specification, the estimated exit temperature remains precise, and the uniformity reaches a contentedly acceptable level. This implies that a larger pipe diameter could potentially require a lengthened tube to enhance the uniformity of exit temperatures.

Abstract This study advances the understanding of nanofluid behaviour within stenosed arteries, highlighting the importance of considering multifaceted effects in the modelling process. It investigates the combined impact of pressure gradient variation, heat transfer, chemical reactions, and magnetic field effects on nano-blood flow in stenosed arteries. Unlike previous studies that made the assumption that the pulsatile pressure gradient remains constant during channel narrowing, this novel investigation introduces a variable pressure gradient. This, in turn, significantly impacts several associated parameters. The mathematical model describing nano-blood flow in a horizontally stenosed artery is solved using perturbation techniques. Analytical solutions for key variables, including velocity, temperature, concentration, wall shear stress, flow rate, and pressure gradient, are visually presented for various physical parameter values.

We consider the regularization of classical optimality conditions—the Lagrange principle and the Pontryagin maximum principle—in a convex optimal control problem with an operator equality constraint and functional inequality constraints. The controlled system is specified by a linear functional–operator equation of the second kind of general form in the space , and the main operator on the right-hand side of the equation is assumed to be quasinilpotent. The objective functional of the problem is only convex (perhaps not strongly convex). Obtaining regularized classical optimality conditions is based on the dual regularization method. In this case, two regularization parameters are used, one of which is "responsible" for the regularization of the dual problem, and the other is contained in the strongly convex regularizing Tikhonov addition to the objective functional of the original problem, thereby ensuring the well-posedness of the problem of minimizing the Lagrange function. The main purpose of the regularized Lagrange principle and Pontryagin maximum principle is the stable generation of minimizing approximate solutions in the sense of J. Warga. The regularized classical optimality conditions Are formulated as existence theorems for minimizing approximate solutions in the original problem with a simultaneous constructive representation of these solutions. Are expressed in terms of regular classical Lagrange and Hamilton–Pontryagin functions. "Overcome" the properties of the ill-posedness of the classical optimality conditions and provide regularizing algorithms for solving optimization problems. Based on the perturbation method, an important property of the regularized classical optimality conditions obtained in the work is discussed in sufficient detail; namely, "in the limit" they lead to their classical counterparts. As an application of the general results obtained in the paper, a specific example of an optimal control problem associated with an integro-differential equation of the transport equation type is considered, a special case of which is a certain inverse final observation problem. [ABSTRACT FROM AUTHOR]

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This study investigates the nonlinear vibration motions of the Euler–Bernoulli microbeam on a nonlinear elastic foundation in a uniform magnetic field based on Modified Couple Stress Theory (MCST). The effect of size, foundation, and magnetic field on the nonlinear vibration motion of microbeam has been examined. The governing equations related to the nonlinear vibration motions of the microbeam are obtained by using Hamilton's Principle, and the Multiple Time Scale Method was used to obtain the solutions for the governing equations. The linear natural frequencies of microbeam are presented in the table according to nonlinear parameters and boundary conditions. The linear and nonlinear natural frequency ratio graphs are shown. The present study results are also compared with previous work for validation. It is observed that length scale parameters and magnetic force have a more significant effect on the natural frequency of microbeams. It is seen that when the linear elastic foundation coefficient, the Pasternak foundation and the magnetic force effects increase, the ratio of nonlinear and linear natural frequency decreases. [ABSTRACT FROM AUTHOR]

In this paper, the approximate analytical solutions obtained by using the constrained parameter-splitting-multiple-scales (C-PSMS) method to the primary and 1 / 3 sub-harmonic resonances responses of a cantilever-type energy harvester are presented. The C-PSMS method combines the multiple-scales (MS) method with the harmonic balance (HB) method. Different from the erroneous stability results obtained by using the Floquet theory and the classical HB method, accurate stability results are obtained by using the C-PSMS method. It is found that the correction to the erroneous solution when the HB method and Floquet theory are adopted in the stability analysis of the primary and 1 / 3 sub-harmonic resonances of a largely deflected cantilever-type energy harvester is necessary. On the contrary, the C-PSMS method gives much improved results compared to those obtained by using Floquet theory and HB method when the numbers of terms in each response expression are the same. The frequency response curves of the primary resonance and the 1 / 3 sub-harmonic resonance of the harvester obtained by the C-PSMS method are compared to those obtained by the HB method and verified by those obtained by the fourth-order Runge–Kutta method. Moreover, the basin of attraction based on the fourth-order Runge–Kutta method is presented to confirm the inaccurate stability results obtained by using the HB method and Floquet theory. The convergence examinations on the stability analysis carried out by the HB method and Floquet theory show that enough terms in the response assumption are needed to achieve relatively accurate stability results when studying the stability of the primary and sub-harmonic resonances of a cantilever by using the HB method and the Floquet theory. However, the low-order C-PSMS method is able to give an accurate frequency-amplitude response and accurate stability results of the primary and sub-harmonic resonances of a largely deflected cantilever-type energy harvester. [ABSTRACT FROM AUTHOR]

The key aim of this paper is to provide a new nth generalized order perturbed isogeometric fast multistage technique of boundary elements to compute the propagation of time harmonics in an infinite region. Structural geometry and boundary integral equations are constructed by using non-uniform rational B-splines. The source of system uncertainty is believed to be the incident plane wave number's unpredictability. The actual field, depending on the input random variables, is simulated using the extended nth-order perturbation method. The field and kernel values for boundary integral formulas are generated via the nth-order generalized series of Taylor expansions using perturbation parameters. The fast multipole method (FMM) is utilized to speed up the process. The effectiveness and correctness of the proposed algorithm are verified by Monte Carlo simulations (MCs) with numerical examples. [ABSTRACT FROM AUTHOR]

AbstractIn this article, we present a non-isothermal study of the wire coating process using the Carreau–Yasuda fluid model. Our aim is to explore how the Carreau–Yasuda fluid model parameters are linked to key operational factors in the wire coating process. These factors encompass the radius of the coated wire, the volumetric flow rate, and the force needed to pull the wire. The equations that depict the flow and heat transfer within the die are solved: (a) analytically using the perturbation method to get the analytical expression for velocity, temperature distribution, and relevant engineering quantities; (b) numerically using shooting method with the “Runge–Kutta–Fehlberg algorithm.” The physical effects of the Weissenberg number (We), power law index (n), pressure gradient (A), and Brinkman number (Br) on the velocity profile, temperature distribution, and engineering quantities are discussed with the help of different graphs. Our findings reveal that, in the case of shear thinning (n < 1) with a constant pressure gradient, as compared to the Newtonian scenario, both the thickness of the coated wire and the force required to pull the wire decrease with an increase in the Weissenberg number (We). Moreover, the force required to pull the wire also decreases with an increase in the values of the pressure gradient parameter. [ABSTRACT FROM AUTHOR]