Closed-form Analytical Solution Research Articles

Energy transition and climate change abatement: A macroeconomic analysis

The paper integrates the characteristics of regenerative energies into a dynamic macroeconomic model with climate change. Learning and economies of scale in new energy moderate the cost of emissions reductions and increase the speed of decarbonization. I provide closed-form analytical solutions for the development of regenerative energies, emissions, consumption, and population. The elasticity of substitution between clean and dirty energy inputs, stringency of climate policy, and potential raw material scarcity constitute critical conditions for reaching carbon neutrality by 2050. I find that a timely carbon phase-out requires sufficient substitution in the energy sector, continued learning and scale effects in regenerative energies, and active climate policy, which is indispensable even with enormous cost degression of regenerative energies. Raw material scarcity induced by regenerative energy use slows down the transition but can be overcompensated by more stringent climate policy at a moderate economic cost.

Pure bending of curved beam with non-uniform cross section

The purpose of this paper is to study the state of stress and displacement field of a curved beam with non-uniform cross section. The material of the beam is non-homogeneous isotropic linearly elastic. The curved beam is subjected to couple moment at the ends of geometry. Utilizing analytical closed form solutions are obtained for the radial and circumferential stresses and for the displacements and cross sectional rotation. A numerical example illustrates the applications of the derived formulae.

Thick and Thin Film Solar Cells: New Formulation

Solar cells rely on photogeneration of charge carriers in p-n junctions and their transport and subsequent recombination in the quasineutral regions. Several basic issues concerning the physics of the operation of solar cells remain obscure. This paper discusses some of those unsolved basic problems. In conventional solar cells, recombination of photogenerated charge carriers plays a major limiting role in the cell efficiency. High quality thin-film solar cells may overcome this limit if the minority diffusion lengths become large as compared to the cell dimensions, but, strikingly, the conventional model fails to describe the cell electric behavior under these conditions. A new formulation of the basic equations describing charge carrier transport in the cell along with a set of boundary conditions is presented. An analytical closed-form solution is obtained under the linear approximation. It is shown that the calculation of the open-circuit voltage of the solar cell diode does not lead to unphysical results in the new given framework.

Application of Yang homotopy perturbation transform approach for solving multi-dimensional diffusion problems with time-fractional derivatives

In this paper, we aim to present a powerful approach for the approximate results of multi-dimensional diffusion problems with time-fractional derivatives. The fractional order is considered in the view of the Caputo fractional derivative. In this analysis, we develop the idea of the Yang homotopy perturbation transform method (YHPTM), which is the combination of the Yang transform (YT) and the homotopy perturbation method (HPM). This robust scheme generates the solution in a series form that converges to the exact results after a few iterations. We show the graphical visuals in two-dimensional and three-dimensional to provide the accuracy of our developed scheme. Furthermore, we compute the graphical error to demonstrate the close-form analytical solution in the comparison of the exact solution. The obtained findings are promising and suitable for the solution of multi-dimensional diffusion problems with time-fractional derivatives. The main advantage is that our developed scheme does not require assumptions or restrictions on variables that ruin the actual problem. This scheme plays a significant role in finding the solution and overcoming the restriction of variables that may cause difficulty in modeling the problem.

Bending theory of composite pressure vessels: A closed-form analytical approach

Composite pressure vessels are strategic for the present and future of the aerospace, transportation, and energy industries. These shell structures are utilized to store and transport liquids and gases and are principally composed of a cylindrical shell and two elliptical heads. The current rules for the structural design of composite pressure vessels make use of the membrane theory that excludes the bending of the structure. Nevertheless, the cylindrical shell and the elliptical heads present different curvature radii. These changes generate a localized bending deformation that determines an increase in the stress state as an effect of a local bending moment and shear force distributions that arise in the geometric transition zone. The stresses related to the bending effects die out rapidly, moving away from the junction plane, but they are sensibly higher than the membrane stress and, therefore, cannot be neglected in the mechanical assessment. Here, a closed-form analytical solution for the bending theory of composite shells is proposed. It can assist in the design of composite pressure vessels and drive the design towards reliable configurations for structural integrity. Using the Donnell–Mustari–Vlasov theory for thin-walled composite shells, the governing fourth-order differential equation is deduced and solved to obtain the displacements and stresses acting in the junction. The analytical results are successfully compared with those of reference finite element models for validation; their accuracy is proved considering different thicknesses, lay-ups, and flattening factors of the vessel

Comparison of numerical and exact analyses in a neutron activation and decay chain

A method for solving a set of coupled linear first-order ordinary differential equations as it applies to neutron activation and subsequent radioisotope decay chains was developed that is easy to implement and accurate enough for most preliminary engineering design calculations. The method is applied to a specific radioisotope production scheme to evaluate the yield of the desired isotope as well as production of secondary and tertiary contaminant radioisotopes. An analytical closed-form solution was also derived for the same activation and decay chain to provide a basis for comparison; the analytical solution was benchmarked against several limiting cases to validate the equations. Results of the simulation involving the numerical approach were compared to the exact analytical solution. The numerical solution, while obviously not exact, is found to be close enough for engineering purposes. A discussion of the numerical precision of the result for the particular case are discussed.

Multiscale Modeling of a Chain Comprising Selective Laser Melting and Post-Machining toward Nanoscale Surface Finish.

The generation of rough surfaces is an inherent drawback of selective laser melted (SLM) material that makes post-treatment operation a mandatory process to enhance its surface condition and service performance. However, planning an appropriate and optimized chain to attain the best surface finish needs an integrated simulation framework that includes physics of both additive manufacturing and post-processing. In the present work, an attempt is made to model the alternation of surface roughness which is produced by SLM and post-processed by milling and sequential surface burnishing. The framework includes a series of closed-form analytical solutions of all three processes embedded in a sequence where the output of the preceding operation is input of the sequential one. The results indicated that there is close agreement between the measured and predicted values of arithmetic surface roughness for both SLM material and the post-processed ones. It was also found that a nanoscale surface finish is obtained by finishing milling and single pass rolling at a static force of 1500 N. In addition, the results of the simulation showed that elimination of the milling process in the chain resulted in a six-times-longer production time that requires three times bigger rolling force compared to a chain with an included milling operation.

Influence of non-linear slip dependent zeta potential on the bi-layered electrohydrodynamic flow in an electrically actuated microsystem

In this paper, the electro-osmotic flow (EOF) of two immiscible fluids through an electrically actuated micro-slit with a non-linear slip-dependent zeta potential is considered. The main objective of this work is to enhance the EOF velocity of non-polar fluid with an intensive flow variation in micro-confinements. We present the comparison between the Poisson–Nernst–Planck (P–N–P) and the Poisson–Boltzmann model and illustrate the impact of different parameters, such as Debye–Hückel parameter, slip coefficient, and interfacial potential drop, on the ionic concentration and the induced potential in detail. In this study, the interface between two immiscible fluids is considered to be planar, and a potential drop is observed close to the interface due to the development of back-to-back diffuse layers. At the interface between two layers, the continuity of the velocity and total stresses (Maxwell stress and shear stress) are taken into account to investigate the flow field in the system of immiscible electrolytes. The P–N–P model, linked with the ion transport equation and the Poisson equation, is employed to describe the motion of electrolyte solutions. The non-linear governing equations are numerically computed using a pressure-correction-based finite volume technique based on a staggered grid algorithm. Closed-form analytical solutions are derived for both steady and unsteady EOF field of two immiscible fluids for the low wall zeta potential and non-overlapping electrical double layer. The analytical solutions are well validated with our numerical results under suitable assumptions. It is also observed that the slip-dependent zeta potential successfully provides a considerable enhancement in EOF velocity over a broad range of parameters such as zeta potential, Debye–Hückel parameter, as well as boundary slip parameter. The results for both layers are highly affected by unequal wall potential, and the ion concentrations are highly actuated by the external electric field.

Topological Singularity Pair in Nonreciprocal Non-Hermitian Lossy-dielectric Gyromagnetic Metasurface

The transmission properties of a nonreciprocal non-Hermitian metasurface composed of lossy dielectric and gyromagnetic materials are studied with closed-form analytical solutions. It is found that a pair of opposite topological singularities of isolation ratio exist in a two-dimensional (2D) parameter space, and their evolution with parameter adjustment is preliminarily studied. These results provide clues for the design of optical devices. Further study of the mechanism of this singular behavior may reveal the complex systematic effect of non-Hermiticity and non-reciprocity on optical performance.

Dual solutions for the two-dimension copper oxide with silver (CuO–Αg) and zinc oxide with silver (ΖnO–Αg) hybrid nanofluid flow past a permeable shrinking sheet in a dusty fluid with velocity slip

PurposeThis paper aims to present dual solutions for the two-dimension copper oxide with silver (CuO–Ag) and zinc oxide with silver (ZnO–Ag) hybrid nanofluid flow past a permeable shrinking sheet in a dusty fluid with velocity slip.Design/methodology/approachThe governing partial differential equations for the two dust particle phases are reduced to the pertinent ordinary differential equations using a similarity transformation. Closed-form analytical solutions for the reduced skin friction and reduced Nusselt number, as well as for the velocity and temperature profiles, were presented, both graphically and in tables, under specific non-dimensional physical parameters such as the suction parameter, Prandtl number, slip parameter and shrinking parameter, which are also presented in both figures and tables.FindingsThe results indicate that for the shrinking flow, the wall skin friction is higher in the dusty fluid when compared with the clear (viscous) fluid. In addition, the effect of the fluid–particle interaction parameter to the fluid phase can be seen more clearly in the shrinking flow. Furthermore, multiple (dual, upper and lower branch solutions) are found for the governing similarity equations and the upper branch solution expanded with higher values of the suction parameter. It can be confirmed that the lower branch solution is unstable.Practical implicationsIn practice, the study of the stretching/shrinking flow is crucially important and useful. Both the problems of steady and unsteady flow of a dusty fluid have a wide range of possible applications in practice, such as in the centrifugal separation of particles, sedimentation and underground disposal of radioactive waste materials.Originality/valueEven though the problem of dusty fluid has been broadly investigated, very limited results can be found for a shrinking sheet. Indeed, this paper has succeeded to obtain analytically dual solutions. The stability analysis can be performed by following many published papers on stretching/shrinking sheets. Finally, the critical values and plotting curves for obtaining single or dual solution are successfully presented.

A hyperbolic theory for the analysis of laminated shallow shells with double curvature

In this paper, higher-order closed-form analytical solutions to the static bending and free vibration problems of laminated composite shells with double curvature are obtained using a hyperbolic shear deformation theory. The current theory is a modification of the shape function provided by Soldatos [30] in his well-known hyperbolic theory. The distributions of transverse shear stresses through the thickness of the shell are precisely predicted by the current theory satisfying traction free boundary conditions at the top and the bottom surfaces of the shell. Hamilton's principle serves as the foundation for the development of equations of motion. Navier's method is used for the analysis of simply-supported laminated shells under static and free vibration conditions. Displacements, stresses, and natural frequencies are presented for different shells with double curvature. The results from past investigations are compared to verify the accuracy and efficacy of the present hyperbolic shell theory.

Distributed Transfer Function Formulations for Vibration Analysis of Cracked Beam and Frame Structures

In this paper, based on distributed transfer function method (DTFM), the closed-form analytical solutions for vibrations of Euler–Bernoulli beam and frame structures with arbitrary number of cracks are studied. First, generalized DTFM is employed to characterize the dynamical model for a single cracked beam and its analytical solutions for eigenvalue problem and frequency response are obtained. Then, a new DTFM cracked element that encapsulates one crack of arbitrary location inside the beam is proposed. Using the DTFM cracked element and global dynamic stiffness matrix assembly technique, damaged frame structures of arbitrary form can be modeled for vibration analysis. Previous analytical methods only addressed low-frequency vibration of simple cracked beam structures, the proposed method can yield analytical solutions in the medium- and high-frequency regions, which is critical for the small crack detection in complex frames. Lastly, three numerical examples are given to illustrate the correctness and effectiveness of the DTFM in analyzing natural frequencies, modal shapes and frequency responses for cracked structures. By comparing with the Finite Element Method (FEM) and benchmarks from literatures, we proved that the DTFM has better performances in terms of accuracy and efficiency.

The computation of Lie point symmetry generators, modulational instability, classification of conserved quantities, and explicit power series solutions of the coupled system

The well-known Chaffee–Infante reaction hierarchy is examined in this article along with its reaction–diffusion coupling. It has numerous variety of applications in modern sciences, such as electromagnetic wave fields, fluid dynamics, high-energy physics, ion-acoustic waves in plasma physics, coastal engineering, and optical fibres. The physical processes of mass transfer and particle diffusion might be expressed in this way. The Lie invariance criteria is taken into consideration while we determine the symmetry generators. The suggested approach produces the six dimensional Lie algebra, where translation symmetries in space and time are associated to mass conservation and conservation of energy respectively, the other symmetries are scaling or dilation. Additionally, similarity reductions are performed, and the optimal system of the sub-algebra should be quantified. There are an enormous number of exact solutions can construct for the traveling waves when the governing system is transformed into ordinary differential equations using the similarity transformation technique. The power series approach is also utilized for ordinary differential equations to obtain closed-form analytical solutions for the proposed diffusive coupled system. The stability of the model under the limitations is ensured by the modulation instability analysis. The reaction diffusion hierarchy’s conserved vectors are calculated using multiplier methods using Lie Backlund symmetries. The acquired results are presented graphically in 2-D and 3-D to demonstrate the wave propagation behavior.

An analytical solution for the filtering effect of piles in two-layer soil

The aim of this work is to investigate the kinematic response of a floating pile (either free- or fixed-head) embedded in a two-layer soil subjected to upward-propagating seismic waves. Closed form analytical solutions are provided for the kinematic interaction factors, based on a Beam-on-Dynamic-Winkler-foundation (BDWF) model properly adjusted for the problem at hand. For the fixed-head pile case, a simplified solution to assess the pile-to-soil acceleration ratio is provided adopting the concept of average soil motion over an effective pile length. The proposed solutions compare well with rigorous elastodynamic Finite-Element analysis. Insight is given into the interpretation of the mechanisms regulating the reduction of design spectra as function of structural period. Finally, as an outcome of a wide parametric study, ready-to-use formulae are proposed for a rapid assessment of the pile-induced spectral reduction.

Newtonian annular Poiseuille and Couette flows with dynamic wall slip

Analytical solutions are derived for the cessation of annular Poiseuille and Couette flows of a Newtonian fluid in the presence of dynamic wall slip. We employ linear dynamic slip law involving a slip relaxation parameter, which results in the appearance of the eigenvalue parameters in the boundary conditions and thus leads to distinct Sturm–Liouville problems that differ from their static slip counterparts. The proper orthogonality condition is derived and closed-form analytical solutions are obtained for both flows of interest. Representative results are then presented and discussed. In agreement with previous reports for other flows with dynamic wall slip, the present solutions show that wall slip slows down flow dynamics and this effect becomes more pronounced as the slip-relaxation parameter is increased.

Analytical closed-form solution of the Navier–Stokes equations for the aerodynamic near-field and sonic boom from axisymmetric bodies

An analytical closed-form solution is presented for the aerodynamic near-field and ground signature from an axisymmetric body with a low thickness ratio. The Navier–Stokes equations are formulated as a boundary value problem that incorporates the incoming ambient flow-field and the aerodynamic properties on the body surface. The shape of the aerodynamic body is defined as a product of generalized functions. A direct solution for the density of the aerodynamic near-field, represented as a function of both space and time, is proposed through the integration of the Navier–Stokes equations in a generalized functional form. Pressure, temperature, velocity, and Mach number are then derived in the near-field. The methodology, being fully nonlinear, surpasses the traditional F-function, impulse, and hypersonic similarity theories originally developed for near-field prediction. The presentation outlines the major steps in deriving the analytical solution and provides predictions from an aerodynamic body in the near-field, along with the associated ground signature. The methodology is focused on aerodynamic bodies operating at high-speeds, ranging from the supersonic to the hypersonic regime. [Work supported by the Defense Advanced Research Project Agency, under Grant No. W911NF-21-1-0342.]

Transport and evaporation of exhaled respiratory droplets: An analytical model

An important vector for host-to-host infectious disease transmission is given by the transport of tiny pathogen-laden droplets. These are commonly exhaled by individuals while breathing, speaking, coughing, or sneezing. Depending on their size and ambient conditions, they may follow different paths, either settling on surfaces where the pathogen can be further transmitted by contact, or remaining airborne after evaporation where the pathogen can be inhaled. Our understanding of pathogen transmission from the fluid mechanics perspective is still somewhat limited, especially in quantitative terms. In the current work, starting from the fundamental laws of fluid mechanics and diffusion, a detailed analytical model of droplet transport and evaporation in humid air streams is presented and successfully validated against available data in the literature finding remarkable agreement. The model implements closed-form analytical solutions of the equations of transport, evaporation, and energy balance, and an algebraic model to account for the droplet chemical composition. It also features an analytical model of droplet transport within the buoyant exhaled breath cloud based on momentum conservation addressing both jet and puff phases and is able to handle periodic respiratory events. Turbulent dispersion is modeled with a discrete random walk approach. A simple inhalation model is also proposed. Such a model may help in better understanding droplets' fluid dynamic behavior and may be used to assess the risks associated with pathogen transmission under different scenarios for any type of respiratory event. Overall, the computational cost is relatively low, allowing extensive simulation campaigns to be performed easily.

On the gravitational fall of a non-spherical particle in a quiescent fluid medium

Closed-form analytical solutions have been derived for the gravitational fall of a non-spherical particle in a quiescent fluid. The Boussinesq-Basset-Oseen equation accounting for history force and particle slip effects was applied to derive the particle motion equation, using a sphericity-dependent drag law for transitional flow in the range 0.001<Re≤100000 and valid for sphericities >0.50. The derived closed-form solutions were validated with published experimental data and the accuracy was verified using published approximate analytical solutions and numerical results. The closed-form solutions were found to agree with experiments and produced more accurate results than other published approximate analytical solutions. Also, the effects of sphericity, Basset force and particle slip on the particle motion were investigated and discussed. The present analytical solutions are applicable to zero and non-zero initial velocity, steady and unsteady motion, and the corresponding displacements and accelerations. Additionally, new settling velocity formula valid for non-spherical particles was derived.

Low complexity channel tracking algorithms for coordinated and uncoordinated pilot access over high-rate internet of things

Given that high-rate internet-of-things (IoT) applications are supported by ubiquitous massive MIMO services, this paper introduces novel algorithms with significantly reduced complexity to estimate and track the channels of active and inactive devices, which can resolve intra-cell pilot collisions and utilize their information. Then, we propose an optimum coordinated access (O-CA) tracker whose complexity grows squarely with the number of devices instead of cubically. In addition, the error-boundedness and steady-state characteristics of O-CA are demonstrated. Although uncoordinated pilot access eliminates control signaling associated with active IoT devices, it needs the base station (BS) to identify the devices’ pilot transmission patterns (DPTPs). A low-complexity optimal uncoordinated access (O-UA) tracker is intended to contain DPTPs that expand exponentially, which is impractical. To this end, we derive an analytical closed-form solution for a near-optimal uncoordinated access (NO-UA) tracker that minimizes an upper bound on the error between O-UA and the simplified DPTPs. Consequently, the NO-UA results in significantly lower computational costs and superior estimation performance compared to existing trackers. Moreover, we derive the optimal linear uncoordinated access (OL-UA) tracker and its stability. Specifically, two heuristic trackers are introduced to detect DPTPs and track the devices’ channels. The simulation results and complexity analysis demonstrate that the designed trackers significantly reduce complexity compared with conventional trackers.

An analytical closed-form solution for the in-plane elastic modulus of plain-woven textile composites

Engineering application of textile composites needs simple relations for the estimation of their mechanical properties. Many homogenization and simplified models have been already proposed for the analytical characterization of the elastic moduli of this class of materials. Along this route, the current study introduces an analytical solution for the longitudinal elastic modulus of plain-woven textile composites. A hybrid beam-spring model of the representative unit cell (RUC) is postulated consisting of a horizontal beam connected to an inclined beam, two supporting springs beneath the free end of the horizontal beam and the common end of the horizontal and inclined beams, and a torsion spring at the free end of the inclined beam. The mechanical and geometrical properties of the beams and springs are calculated from the weave structure and the symmetry conditions. A reduction technique from the matrix analysis of structures is borrowed to consider the effect of internal degrees of freedom on the in-plane stiffness coefficient and a closed-form formula is extracted for the elastic modulus. The proposed analytical formula is finally verified against available experimental data and a wide range of FE predictions for the GFRP and CFRP plain-woven composites.