Explicit Analytical Solutions Research Articles

Exact solution for heat transfer across the Sakiadis boundary layer

We consider the problem of convective heat transfer across the laminar boundary layer induced by an isothermal moving surface in a Newtonian fluid. In a previous work [Barlow et al., “Exact and explicit analytical solution for the Sakiadis boundary layer,” Phys. Fluids 36, 031703 (2024)], an exact power series solution was provided for the hydrodynamic flow, often referred to as the Sakiadis boundary layer. Here, we utilize this expression to develop an exact solution for the associated thermal boundary layer as characterized by the Prandtl number (Pr) and local Reynolds number along the surface. To extract the location-dependent heat transfer coefficient (expressed in dimensionless form as the Nusselt number), the dimensionless temperature gradient at the wall is required; this gradient is solely a function of Pr and is expressed as an integral of the exact boundary layer flow solution. We find that the exact solution for the temperature gradient is computationally unstable at large Pr, and a large Pr expansion for the temperature gradient is obtained using Laplace's method. A composite solution is obtained, which is accurate to O(10−10). Although divergent, the classical power series solution for the Sakiadis boundary layer—expanded about the wall—may be used to obtain all higher-order corrections in the asymptotic expansion. We show that this result is connected to the physics of large Prandtl number flows where the thickness of the hydrodynamic boundary layer is much larger than that of the thermal boundary layer. The present model is valid for all Prandtl numbers and attractive for ease of use.

Exact solution for heat transfer across the Sakiadis boundary layer

We consider the problem of convective heat transfer across the laminar boundary layer induced by an isothermal moving surface in a Newtonian fluid. In a previous work [Barlow et al., “Exact and explicit analytical solution for the Sakiadis boundary layer,” Phys. Fluids 36, 031703 (2024)], an exact power series solution was provided for the hydrodynamic flow, often referred to as the Sakiadis boundary layer. Here, we utilize this expression to develop an exact solution for the associated thermal boundary layer as characterized by the Prandtl number (Pr) and local Reynolds number along the surface. To extract the location-dependent heat transfer coefficient (expressed in dimensionless form as the Nusselt number), the dimensionless temperature gradient at the wall is required; this gradient is solely a function of Pr and is expressed as an integral of the exact boundary layer flow solution. We find that the exact solution for the temperature gradient is computationally unstable at large Pr, and a large Pr expansion for the temperature gradient is obtained using Laplace's method. A composite solution is obtained, which is accurate to O(10−10). Although divergent, the classical power series solution for the Sakiadis boundary layer—expanded about the wall—may be used to obtain all higher-order corrections in the asymptotic expansion. We show that this result is connected to the physics of large Prandtl number flows where the thickness of the hydrodynamic boundary layer is much larger than that of the thermal boundary layer. The present model is valid for all Prandtl numbers and attractive for ease of use.

Явные решения в проблеме звукоизоляции с использованием многослойных структур при нормальном прохождении волн

The problem of sound isolation at ultra-low frequencies (30–200 Hz) by using multilayered structures is con-sidered, taking into account the normal wave propagation and an arbitrary number of layers. It is assumed that the multilayered structure consists of parallel acoustic layers, connected to each other along vertical boundary lines; acoustic half-spaces (air) are located to the left and right of the structure. For the problem, we define the acoustic parameters of the structure. The wave equations of the harmonic process are described for the left and right half-spaces of the structure, using the Helmholtz equation and a system of linear algebraic equations with given boundary conditions. Besides, explicit analytical solutions are obtained to find the transmission coefficient (transparency coefficient), with an arbitrary number of layers. Examples of solution for low-frequency sound absorption for building structures with different numbers of layers of alternating building materials and air spaces are considered. The effectiveness of using mixed structures, in which air layers are sandwiched between high impedance layers, for low-frequency sound absorption has been confirmed by assessing the level of noise reduction.

Loss modeling with the size-biased lognormal mixture and the entropy regularized EM algorithm

The Erlang mixture with a common scale parameter is one of many popular models for modeling insurance losses. However, the actuarial literature recognizes and discusses some limitations of aforementioned model in approximate heavy-tailed distributions. In this paper, a size-biased left-truncated Lognormal (SB-ltLN) mixture is proposed as a robust alternative to the Erlang mixture for modeling left-truncated insurance losses with a heavy tail. The weak denseness property of the weighted Lognormal mixture is studied along with the tail behavior. Explicit analytical solutions are derived for moments and Tail Value at Risk based on the proposed model. An extension of the regularized expectation–maximization (REM) algorithm with Shannon's entropy weights (ewREM) is introduced for parameter estimation and variability assessment. The Operational Riskdata eXchange's left-truncated internal fraud loss data set is used to illustrate applications of the proposed model. Finally, the results of a simulation study show promising performance of the proposed SB-ltLN mixture in different simulation settings.

Dynamically driven correlations in elastic net models reveal sequence of events and causality in proteins.

An explicit analytic solution is given for the Langevin equation applied to the Gaussian Network Model of a protein subjected to both a random and a deterministic periodic force. Synchronous and asynchronous components of time correlation functions are derived and an expression for phase differences in the time correlations of residue pairs is obtained. The synchronous component enables the determination of dynamic communities within the protein structure. The asynchronous component reveals causality, where the time correlation function between residues i and j differs depending on whether i is observed before j or vice versa, resulting in directional information flow. Driver and driven residues in the allosteric process of cyclophilin A and human NAD-dependent isocitrate dehydrogenase are determined by a perturbation-scanning technique. Factors affecting phase differences between fluctuations of residues, such as network topology, connectivity, and residue centrality, are identified. Within the constraints of the isotropic Gaussian Network Model, our results show that asynchronicity increases with viscosity and distance between residues, decreases with increasing connectivity, and decreases with increasing levels of eigenvector centrality.

Analytical solution of the Poiseuille flow of a De Kee viscoplastic fluid

We provide an explicit analytical solution of the planar Poiseuille flow of a viscoplastic fluid governed by the constitutive equation proposed by De Kee and Turcotte (1980). Formulae for the velocity and the flow rate are derived, making use of the Lambert W function. It is shown that a solution does not always exist because the flow curve is bounded from above and hence the rheological model can accommodate stresses only up to a certain limit. In fact, the flow curve reaches a peak at a critical shear rate, beyond which it exhibits a negative slope, giving rise to unstable solutions.

Analytical Solution for the Steady Seepage Field of a Foundation Pit in an Anisotropic Layer

The soil piping and drift sand phenomenon is one of the catastrophic failure forms in foundation pit excavations in coastal buildings. Presently, there is a deficiency in the theoretical research regarding the seepage fields around foundation pits, primarily due to the complexity of theoretical solutions given the difficulty in accurately describing the distribution of the groundwater’s hydraulic head in a seepage field. This study proposes an explicit analytical solution for the steady-state seepage field surrounding a foundation pit under anisotropic conditions. A numerical model, constructed with FLAC3D 7.0 software, was utilized to validate the solution presented in this study. The effects of the foundation pit’s width, the distance between the retaining wall and the impervious layer, and anisotropic seepage conditions on the total head are studied through parameter research. The study shows that the flow behavior of a foundation pit is sensitive to parameters such as the anisotropy of the soil layer and the width of the foundation pit. Further, the study also analyzes the influence of the above parameters on the exit gradient and proposes a simplified algorithm for the exit hydraulic gradient at the base of a foundation pit, which can control the error within 5%. This method makes a certain contribution to improving seepage calculations for foundation pits and is applicable to the seepage problem of anisotropic soil layers.

Nonlinear static solution of suspension bridge formulated from a surrogate model with secant stiffness of suspension cables

Explicit analytical solution may offer great convenience for the design and parametric study of suspension bridges. However, the nonlinear behavior of suspension cable exacerbates the difficulty of such theoretical derivation. Adopting secant stiffness is an effective way to deal with the nonlinearity of the cable, but the crux is how to find its deformed state under external loading. The strategy of this study is primarily targeted at a typically arranged three-span suspension bridge. It starts from formulating the basic form of nonlinear stiffness of suspension cables by setting the elastic stiffness and geometric stiffness in series, then a surrogate model is accordingly established with the nonlinear springs to replace the mid-span and side-span cables. The deformed state of the surrogate model subjected to the specified external load cases will be found by restricting and releasing its tower tops, and thus the analytical formula of the secant stiffness can be obtained. Consequently, the nonlinear static solution of suspension bridge, including force and deformation changes in the cables and towers, can be explicitly expressed with the secant stiffness. The general solution procedure can also be extended to multi-span suspension bridges. Finally, the nonlinear static solution is verified by numerical examples of a three-span suspension bridge and a four-span suspension bridge. The parametric analysis reveals that the nonlinearity of suspension bridge will become significant as the side-span to mid-span ratio increases, or when the side-span cables are hung with hangers.

Fast computation of the observer motion effects induced on monopole frequency spectra for tabulated functions

Context. Various methods have been studied to compute the boosting effects produced by the observer peculiar motion, which modifies and transfers the isotropic monopole frequency spectrum of the cosmic background radiation to higher multipoles. Explicit analytical solutions for the spherical harmonic expansion coefficients were already presented and applied to different types of background spectrum, strongly alleviating the computational effort needed for accurate theoretical predictions. The frequency spectra at higher multipoles are inherently led by higher-order derivatives of the monopole spectrum. Provided that it can be well described by analytic or semi-analytic functions, the computation of its transfer is not affected by numerical instabilities when evaluated at the required level of numerical accuracy. Instead, monopole frequency spectra described by tabulated functions are computed with a relatively poor frequency resolution in comparison with the Doppler shift, which necessitates interpolation of the tabular representation. The spectra are also affected by uncertainties related to intrinsic inaccuracies in the modelling or in the related observational data as well as to limited accuracy in their numerical computation. These uncertainties propagate and increase with the derivative order, possibly preventing the trustworthy computation of the transfer to higher multipoles and of the observed monopole. Aims. We study methods to filter the original function or its derivatives and the multipole spectra, to mitigate numerical instabilities, and to derive reliable predictions of the harmonic coefficients for different cosmic background models. Methods. From the analytical solutions, and assuming that the monopole spectrum can be expanded in Taylor’s series, we derive explicit expressions for the harmonic coefficients up to the multipole ℓmax = 6 in terms of monopole spectrum derivatives. We then consider different low-pass filters: prefiltering in Fourier space of the tabular representation; filtering in both real and Fourier space of the numerical derivatives; interpolation approaches; and a dedicated method based on amplification and deamplification of the boosted signal. We study the quality of these methods when applied to suitable analytical approximations of the tabulated functions, possibly polluted with simulated noise. These methods are then applied to the tabulations. Results. We consider two very different types of monopole spectra superimposed to the cosmic microwave background: the (smooth) extragalactic source microwave background signal from radio-loud active galactic nuclei and the (feature-rich) redshifted 21 cm line, and present our results in terms of spherical harmonic coefficients. The direct prediction of these coefficients can be noisy at ℓ > 1 or, depending on the uncertainty level, even at ℓ ≤ 1. Without assuming a functional form for the extragalactic background spectrum, the Gaussian prefiltering coupled to the sequential real-space filtering of derivatives allows us to derive accurate predictions up to ℓ ∼ 6, while a log–log polynomial representation, which is appropriate over several decades, gives accurate solutions at any ℓ. Instead, it is difficult to characterise the 21 cm line model variety, and so it is relevant to work without assumptions about the underlying function. Typically, the prefiltering provides accurate predictions up to ℓ ≃ 3 or 4, while the further sequential filtering of the derivatives or the boosting amplification and deamplification method improves the results up to ℓ = 4, while also allowing reasonable estimations of the spectrum at higher ℓ. Conclusions. The proposed methods can significantly extend the range of realistic cosmic background models manageable with a fast computation, beyond the cases characterised a priori by analytical or semi-analytical functions. These methods require only an affordable increase in computation time compared to the direct calculation via simple interpolation.

Stability, modulation instability and explicit-analytical solutions for the Hamiltonian amplitude equation

Stability, modulation instability and explicit-analytical solutions for the Hamiltonian amplitude equation

A Simple Model for Targeting Industrial Investments with Subsidies and Taxes

We consider an investor, whose capital is divided into an industrial investment xt and cash yt, and satisfy a nonlinear deterministic dynamical system. The investor fixes fractions of capital to be invested, withdrawn, and consumed, and also the production factor parameter. The government fixes a subsidy fraction for industrial investments and a tax fraction for the capital outflow. We study a Stackelberg game, corresponding to the asymptotically stable equilibrium (x∗,y∗) of the mentioned dynamical system. In this game, the government (the leader) uses subsidies to make incentives for the investor (the follower) to maintain the desired level of x∗, and uses taxes to achieve this with the minimal cost. The investor’s aim is to maximize the difference between the consumption and the price of the production factor at equilibrium. We present an explicit analytical solution of the specified Stackelberg game. Based on this solution, we introduce the notion of a fair industrial investment level, which is costless for the government, and show that it can produce realistic results using a case study of water production in Lahore.

Higher order investigation on modulated waves in the Peyrard–Bishop–Dauxois DNA model

Despite the widespread use of transcendental functions in the modeling of the dynamics of DNA, most research efforts are limited in their analytical studies of this enthralling system to cubic order polynomial approximations of the corresponding equations of motion. In this paper, we present an investigation of waves in the Peyrard–Bishop–Dauxois model of DNA; while extending the polynomial approximation of the Morse potential up to the sixth order. We show that, within a generalized version of the reductive perturbation method that we have adopted, the equations governing the envelop consist of the standard cubic nonlinear Schrödinger equation and its non homogeneous linearizations. Exact and explicit analytical solutions that correspond to bright solitary waves are obtained for these coupled amplitude equations. A notable qualitative feature of these solutions is the dependence of their propagation speeds and frequencies on their amplitudes. Our approach additionally unveils that these solutions contain some harmonic terms; which are missed in existing works. A very good agreement is found between our analytical analysis and the numerical simulations of the full discrete nonlinear equation of the lattice which use these solutions as initial conditions.

An isoparametric inclusion model for determining the thermo-elastic fields produced by varying Eigen-temperature gradients

The existence of inclusions or inhomogeneities may disrupt heat flow, resulting in increased stresses and temperature fluctuations at the material interface. Analyzing the thermo-elastic fields surrounding such microstructures is crucial for unveiling the intricate failure mechanisms within materials. Based on the method of Green's function and contour integral, this work presents a complete set of explicit analytical solutions for thermal and elastic fields of an arbitrary polygonal inclusion subjected to linearly varying eigen-temperature gradients or thermal eigenstrains. In contrast to the previous studies on potentials showing analogy with anti-plane elasticity, a mathematical analogy of Green's functions between steady-state heat conduction and thermal inclusion for plane elasticity is accordingly established. By employing the linear interpolation of the nodal eigen-temperature gradients or eigenstrains, the work further extends the traditional isoparametric element idea to the thermos-elastic inclusions, and proposes a novel isoparametric inclusion model to solve arbitrarily shaped inclusions with any distributed eigen-temperature gradients. Unlike the finite element method, which requires meshing for both the matrix and inclusion, the present model exhibits efficiency, flexibility, and versatility with the mesh discretization only conducted inside the inclusion domain. The inclusion based isoparametric method may have potential applications in mechanical engineering and material science for analyzing the thermo-structural behavior of various microstructures.

Analytical solutions of equatorial geodesic motion in Kerr spacetime* *Y. L. is financially supported by the Natural Science Foundation of Shandong Province (ZR2023QA133) and Yantai University (WL22B218). B.S. is Supported by the National Natural Science Foundation of China (12375046) and Beijing University of Agriculture (QJKC-2023032).

The study of Kerr geodesics has a long history, particularly for those occurring within the equatorial plane, which are generally well-understood. However, when compared with the classification introduced by one of the authors [Phys. Rev. D 105, 024075 (2022)], it becomes apparent that certain classes of geodesics, such as trapped orbits, still lack analytical solutions. Thus, in this study, we provide explicit analytical solutions for equatorial timelike geodesics in Kerr spacetime, including solutions of trapped orbits, which capture the characteristics of special geodesics, such as the positions and conserved quantities of circular, bound, and deflecting orbits. Specifically, we determine the precise location at which retrograde orbits undergo a transition from counter-rotating to prograde motion due to the strong gravitational effects near a rotating black hole. Interestingly, the trajectory remains prograde for orbits with negative energy despite the negative angular momentum. Furthermore, we investigate the intriguing phenomenon of deflecting orbits exhibiting an increased number of revolutions around the black hole as the turning point approaches the turning point of the trapped orbit. Additionally, we find that only prograde marginal deflecting geodesics are capable of traversing through the ergoregion. In summary, our findings present explicit solutions for equatorial timelike geodesics and offer insights into the dynamics of particle motion in the vicinity of a rotating black hole.

Flow Characterization in a Partially Liquefied Vitreous Humor

The purpose of this study is to systematically examine the basic fluid dynamics associated with a fully liquid region within a porous material. This work has come about as a result of our investigation on the ocular fluid dynamics and transport process in a partially liquefied vitreous humor. The liquid is modeled as a sphere with Stokes flow while the surrounding infinite porous region is described by Brinkman flow. The development here provides basic three-dimensional axisymmetric results on flow characterization and also serves to evaluate the limits of validity of Darcy flow analysis for the same geometry. In the Darcy flow model, the liquid region is also treated as a porous region with a much higher permeability. Therefore, both liquid and porous regions are modeled by Darcy’s law. Besides the analytical results from Brinkman–Stokes model, the simpler case of Darcy–Darcy flow for the same geometry has been provided. The results of both cases are compared and the differences between the two sets of results provide the range of validity of our computational model (Khoobyar et al. in J Heat Transf 144:031208, 2022). Some interesting fluid-dynamical aspects of the system are observed through the analysis. For the Darcy–Darcy system, the liquid region velocity is uniform throughout, as expected for potential flow. With the Brinkman–Stokes model, the liquid region has a paraboloidal profile with the maximum possible peak value of six times the far-field velocity in the porous medium. With the liquid region having a lower resistance, the flow tends to converge there for both models as it seeks the path of least resistance. As for the validation of the Darcy–Darcy model, it is a good approximation as far as the exterior flow is concerned. However, the liquid region flow profiles for the two models are different as noted. The current Brinkman–Stokes model has led to explicit analytical solutions for the flow field for both regions. This has permitted an asymptotic analysis giving deeper insight into the flow characterization.

Tension buckling and postbuckling of nanocomposite laminated plates with in-plane negative Poisson’s ratio

Abstract Mechanical metamaterials with negative Poisson’s ratio (NPR) have emerged as a novel class of engineering material, and have attracted increasing attention in various engineering sectors. Most studies available on the buckling problem of laminated plates with positive or NPR are those under uniaxial compression. Here, we report that the buckling phenomenon may occur for auxetic nanocomposite laminated plates under uniaxial tension when the unloaded edges of the plates are immovable. Two types of nanocomposites are considered, including graphene/Cu and carbon nanotube/Cu composites. Governing equations of the auxetic nanocomposite laminated plates are formulated based on the framework of Reddy’s higher-order shear deformation theory. In modeling, the von Kármán nonlinear strain–displacement relationship, temperature-dependent material properties, thermal effects, and the plate–substrate interaction are considered. The explicit analytical solutions for postbuckling of auxetic nanocomposite laminated plates subjected to uniaxial tension are obtained for the first time by employing a two-step perturbation approach. Numerical investigations are performed for tension buckling and postbuckling behaviors of auxetic nanocomposite laminated rectangular plates with in-plane NPR rested on an elastic substrate under temperature environments.

Elastic foundation-introduced defective phononic crystals for tunable energy harvesting

Defective phononic crystals offer the advantage of concentrating elastic waves, thereby enhancing the potential for piezoelectric energy harvesting (PEH). However, a key limitation is their reliance on a fixed operating frequency, rendering them susceptible to the prevailing vibration environment. To surmount this constraint, this study introduces a novel approach involving a tunable elastic foundation system for defective phononic crystal structures. The newly developed phononic crystal is fashioned by integrating a periodically elastic foundation beneath a uniform beam. A defect is induced by selectively removing specific elastic foundations and integrating piezoelectric components. Explicit analytical solutions are established through the transfer matrix method and the spectral element method, which are subsequently corroborated via comparison with finite element results. The findings underscore that the periodic elastic foundations impart bandgaps in the elastic wave band structure. The absence of specific elastic foundations results in the emergence of distinct defect modes. Additionally, frequency response analysis exposes the potential for energy enhancement, albeit with inherent variations. Noteworthy is the revelation that manipulating the stiffness of the elastic foundation triggers shifts in the resonant frequency of the output voltage. Therefore, the proposed tunable elastic foundation system exhibits promising potential to engender versatile and adaptive phononic crystal configurations, thereby advancing the domain of PEH.

Explicit analytical solution to the minimum safety thickness of waterproof-resistant slab in front of karst tunnel face

Water and mud inrush disasters are typical disaster for tunnel engineering in karst area, and the rock mass between tunnel face and water-rich karst cave plays a key role in preventing water and mud inrush disasters. In this paper, a modified theoretical model is established and explicit analytical solution to the minimum safety thickness is obtained on the basis of the rigid body limit equilibrium theory and elasticity theory. The theoretical formula is validated through a combination of comparisons with some classical theoretical formulations reported in the literature and numerical simulations. The theoretical formula shows that the minimum safety thickness depends heavily on the hydrostatic pressure in karst cave, the tunnel section size and the rock quality. Parameter sensitivity analysis is conducted and the results illustrate that the minimum safety thickness is positive correlated with karst water pressure and tunnel cross-section size, while negative with the tensile strength of the surrounding rock. In additional, the karst tunnel excavation simulations under four tunnel section shapes are carried out, i.e., actual section, elliptical, circular, and rectangular sections, based on the engineering background of the Pengshui Tunnel in Chongqing Province. Displacement of the monitoring sections and distribution characteristics of the plastic zone are analyzed. The results prove that simplifying the waterproof-resistant slab into an elliptical plate is more accurate than the other models.

Transverse-inertia-induced wave dispersion in plate–shell chains with a variable Poisson ratio

One-dimensional shell chain structures can cause stress wave dispersion, and the dispersion ability is mainly affected by the apparent elastic Poisson ratio of the unit cell composing the chain. In this study, a novel plate–shell chain structure with a variable Poisson ratio is proposed by combining bent plates and arc shells (BPAS). The dispersive wave propagation features in the BPAS chains under striker impact are studied theoretically, numerically, and experimentally. An equivalent continuum model of the unit cell under compression in a linear-elastic deformation regime is established, which contains two material parameters, i.e., the apparent elastic modulus and the apparent elastic Poisson ratio. It is found that the unit cell can reach a relatively large Poisson ratio by adjusting its structural parameters. A continuum-based wave propagation model considering transverse inertia is developed, and an explicit analytical solution is obtained to represent the dispersion wave propagation behavior within a linear-elastic deformation regime. The model can reasonably predict the force and velocity evolution features of the chains and is verified by the numerical/experimental tests. The results show that the BPAS chain with a large Poisson ratio exhibits superior wave dispersion performances. The underlying mechanism is that more impact energy is dispersed by the transverse movement of the cells in the BPAS chain. Increasing the aspect ratio of the unit cell can improve the wave dispersion ability of the BPAS chains, achieving a high attenuation of the stress wave and is helpful for impact mitigation.

Analytical Solution for the Steady Seepage Field of an Anchor Circular Pit in Layered Soil

An analytical study was carried out on an anchored circular pit with a submerged free surface in layered soil. The seepage field around the anchor circular pit was divided into three zones. Separate variable method was used to obtain the graded solution forms of head distribution in the column coordinate system for each of the three regions. Combined with the continuity condition between the regions the Bessel function orthogonality was used to obtain the explicit analytical solution of the seepage field in each region, and the infiltration line was determined. Comparison with the calculation results of Plaxis 2D 8.5 software verified the correctness of the analytical solution. Based on the analytical solution, the influence of the radius of the pit and the distance of the retaining wall from the top surface of the impermeable layer on the total head distribution on both sides of the retaining wall was analyzed. And the variation in the infiltration line was determined with the above parameters. The results show that as the pit radius, r, decreased, the total head on both sides of the retaining wall gradually increased, and the height of the submerged surface drop also increased. As the distance, a, between the retaining wall and the impermeable boundary at the bottom increased, the hydraulic head on the outer side of the retaining wall decreased and the head on the inner side increased. The height of the submerged surface drop increased with decreasing depth of insertion of the retaining wall. The depth of insertion of the retaining wall had a greater influence on the degree of diving surface drop than the pit radius.