Closed-form Analytical Solution Research Articles

Electroosmotic flow and heat transfer in a heterogeneous circular microchannel

Asymmetries in boundary condition are inevitable in practice in microfluidic channels, despite being rarely addressed from theoretical perspectives. Here, by arriving at closed form analytical solutions, we bring out a unique coupling between asymmetries in surface charge and heat transfer in electroosmotically driven microchannel flows. For illustration, we assume that the channel is laterally composed of two parts, each having specified values of the zeta potential and the wall heat flux. Considering low zeta potentials, we obtain analytical solutions in terms of infinite series for the dimensionless forms of the electric potential, the velocity, and the temperature distributions. We demonstrate that, by carefully adjusting the governing parameters, a variety of flow patterns may be achieved, a property that is crucial in applications such as liquid-phase transportation and mixing. Moreover, we show that the average velocity is a linear function of both the zeta potential ratio and the coverage factor. We further show that the average Nusselt number increases when part of the channel having the larger heat flux enlarges and the zeta potential of the part having the smaller surface charge increases. Hence, the maximum heat transfer rates are achieved when the boundary conditions are symmetrical.

Stability buckling and bending of nanobeams including cutouts

This manuscript developed a comprehensive model and numerical studies to illustrate the effect of perforation parameters on critical buckling loads and static bending of thin and thick nanobeams for all boundary conditions, for the first time. Analytical closed-form solutions are presented for buckling loads and static deflections, respectively. Euler–Bernoulli beam theory is exploited for thin beam analysis, and Timoshenko beam theory is proposed to consider a shear effect in case of thick beam analysis. Nonlocal differential form of elasticity theory is included to consider a size scale effect that is missing in case of classical theory and macro-analysis. Geometrical adaptations for perforated beam structures are illustrated in simplest form. Equilibrium equations for local and nonlocal beam are derived in detail. Numerical studies are illustrated to demonstrate influences of long-range atomic interaction, hole perforation size, number of rows of holes and boundary conditions on buckling loads and deflection of perforated nanobeams. The recommended model is helpful in designing nanoresonators and nanoactuators used in NEMS structures and nanotechnology.

Closed-form analytical solutions for options on agricultural futures with seasonality and stochastic convenience yield

Abstract Stochastic commodity price models play a significant role in the pricing and hedging of commodity derivatives and real assets. This paper proposes a commodity pricing model that extends the Ewald and Ouyang two-factor model by adding a time-varying function into the volatility term of convenience yield process to describe seasonality. The stochastic factors in the model are the spot price of the commodity is assumed to follow geometrical Brownian motion process with a stochastic drift, the instantaneous convenience yield follows a mean-reverting Ornstein-Uhlenbeck stochastic process with time-varying seasonal volatility. The model we propose is analytically tractable and allows us to derive closed-form pricing formulas for futures and options by using the Mellin transform method. In addition, we provide the experiments results to illustrate the important properties of commodity futures options with numerical table and graphs.

An extended separation-of-variable method for the free vibration of orthotropic rectangular thin plates

This paper proposes a method to solve for closed-form analytical solutions of the free vibration problems of orthotropic rectangular thin plates with arbitrary homogeneous boundary conditions. This proposed method is called the extended separation-of-variable method, in which the mode functions are in a separation-of-variable form, and the frequencies in two spatial directions are mathematically independent of each other. The closed-form analytical eigensolutions satisfy the Rayleigh's principle exactly. Unlike the extended Kantorovich–-Krylov method, the proposed method solves all eigenvalue equations of two directions simultaneously, not iteratively. Numerical experiments validate the proposed method. More importantly, the proposed method is applicable to any eigenvalue problems of plates, shells, three-dimensional plates and cubes.

A unified relaxed approach easing the practical application of a paradox in curved beams

The paper deals with an arising paradox in curved beams subjected to bending moment and normal force. This paradox consists in the fact that by laterally removing material from section zones close to the neutral axis, not only an obvious reduction of the beam mass can be obtained, but also an unexpected, though technically negligible, reduction of the bending stress. It has recently been shown that the relaxation of the demanding achievement of a concurrent mass and stress reduction may practically lead to interesting results, yet solvable numerically. In this paper we show that, under some mild assumptions, a remarkable simplification of the intrados stress functional is obtained. Hence, a unified approximate mathematical approach based on linearization is developed for the derivation of analytical closed-form solutions for the lateral grooved zones. A practical example of the application of the relaxed paradox to optimize a crane hook subjected to bending and normal force is illustrated and compared to finite element forecasts.

Design and Analysis of Collective Pulse Oscillators

Collective pulse oscillators (CPOs) are novel designs constructed using pulse regenerative amplifiers that exhibit time-variant gate delay based on the residual charge from past state. This property makes it possible to achieve precise phase resolutions smaller than a pulse gate delay and/or provide identical phase taps at multiple physical locations. CPOs exhibit temporal phase error correction that results in an improvement in frequency stability ∝ -10 log p for power ∝ p across all timescales beyond the correction settling time. While CPOs result in device noise-based figure of merits (FoMs) comparable to that of ring oscillators, they are more resilient to power-coupled and impulse noise. This article presents a systematic time-domain analysis of the properties of CPOs based on an abstract model that captures the time-variant delay of pulse gates. Closedform analytic solutions for CPOs disturbed by impulse noise are derived, and higher order CPOs with continuous noise injection are analyzed using behavioral simulations and characterized using Allan deviation. Hspice simulation results are presented to validate the model and compare CPOs with ring oscillators. Allan deviation and phase noise measurements on CPOs of 8 and 40 gates fabricated in GFUS8RF (130-nm) technology corroborate the theory and simulation results.

Laser powder bed fusion manufacturing of aluminum honeycomb structures: Theory and testing

This paper addresses the theoretical and experimental assessment of thin-walled hexagonal structures manufactured by laser powder bed fusion which is a layerwise powder bed based additive manufacturing process. The lightweight potential and mechanical properties of such cellular structures are governed by their relative density for which the present literature only gives approximate values. Thus, a closed-form analytical solution is given which considers the cell wall overlaps that are neglected by the current models. Effective inplane elastic properties, i.e. two moduli of elasticity and two Poisson's ratios, are treated in the framework of a new closed-form analytical method that uses a homogenization approach applied to a representative volume element and is based on Timoshenko's beam theory by using the principle of the minimum of total complimentary elastic potential in conjunction with Castigliano's 2nd theorem. The general manufacturability of such cellular structures by laser powder bed fusion is investigated by establishing overhang restrictions and by manufacturing trial specimens for the assessment of maximum cell sizes and orientations in the build space of a laser powder bed fusion system. Extensive parameter studies are performed concerning the influence of laser power, scan speed and hatch distance on the resultant hexagon geometries, the surface quality, the occurrence of imperfections and the general limits of manufacturability which enables the targeted manufacturing of cellular structures with clearly defined properties. The resultant material density is determined by Archimedean density measurements, and the study of micrographs reveals degrees of porosity and the quality of the individual melt tracks. The closed-form analytical method for the determination of the effective elastic properties is finally validated by performing compression tests on selected specimens.

Uniformly accelerated traveler in an FLRW universe

This paper provides an analytical treatment of accelerated and geodesic motion within the framework of the Friedmann -Lemaitre-Robertson-Walker (FLRW) spacetime. By employing conformal time transformations we manage to convert second order differential equations of motion in FLRW spacetime to first order equations in the conformally transformed spacetime. This allows us to derive a general analytical solution in closed-form for accelerated motion in spatially curved FLRW spacetime. We provide few examples of this general solution for the spatially flat cases. The last part of our work focuses on the return journey for a traveler exploring a FLRW universe. We derive certain condition for de Sitter universe that must be satisfied in order to have an actual return journey.

Influence of pore water pressure change on consolidation behavior of saturated poroelastic medium

Abstract. There are many factors causing land subsidence, and groundwater extraction is one of the most important causes of subsidence. A set of coupled partial differential equations are derived in this study by using the poro-elasticity theory and linear stress-strain constitutive relation to describe the one-dimensional consolidation in a saturated porous medium subjected to pore water pressure change due to groundwater table depression. Simultaneously, the closed-form analytical solutions for excess pore water pressure and total settlement are obtained. To illustrate the consolidation behavior of the poroelastic medium, the saturated layer of clay sandwiched between two sand layers is simulated, and the dimensionless pore water pressure changes with depths and the dimensionless total settlement as function of time in the clay layer are examined. The results show that the greater the water level change in the upper and lower sand layers, the greater the pore water pressure change and the total settlement of the clay layer, and the more time it takes to reach the steady state. If the amount of groundwater replenishment is increased, the soil layer will rebound.

Stochastic Optimal Control Models for Online Stores: Indefinite Horizon

In this work we propose and analyze a model which addresses the pulsing behavior of sellers in an online auction (online store). This pulsing behavior is observed when sellers switch between advertising and processing states. We assert that a seller switches her state in order to maximize her profit, and further that this switch can be identified through the seller's reputation. We find that for each seller there is an optimal reputation, i.e. the reputation at which the seller should switch her state in order to maximize her total profit. Relying on techniques of stochastic optimal control, we design a stochastic behavioral model for an online seller. This model incorporates the dynamics of resource allocation and reputation. We optimize the design model by using a stochastic advertising model put forth by Sethi (Optimal Control Applications and Methods, vol. 4, no. 2, 1983, pp. 179-184) and used effectively in Raman's more recent work (Automatica, vol. 42, no. 8, 2006, pp. 1357-1362). This model of reputation is combined with the effect of online reputation on sales price empirically verified by by Mink and Seifert (Group Decision and Negotiation International Conference, June 2006, pp. 253-255), and we derive the resulting Hamilton-Jacobi-Bellman (HJB) differential equation. The solution of this HJB equation relates optimal wealth level to a seller's reputation. We formulate both a full model, as well as a reduced model with fewer parameters, both of which have the same qualitative description of optimal seller behavior. Coincidentally, this reduced model has a closed form analytical solution which we also construct. Although the techniques we use are standard, we believe that the optimal strategy presented both analytically and numerically has implications on the way reputational information can be used to predict the behavior of an individual seller in an online market.

Decentralized optimal coordination of connected and automated vehicles for multiple traffic scenarios

Connected and automated vehicles (CAVs) provide the most intriguing opportunity to optimize energy consumption and travel time. Several approaches have been proposed in the literature that allow CAVs to coordinate in situations where there is a potential conflict, for example, in signalized intersections, merging at roadways and roundabouts, to reduce energy consumption and optimize traffic flow. In this paper, we consider the problem of coordinating CAVs in a corridor consisting of multiple traffic scenarios. We formulate a two-level optimization problem in which we maximize traffic throughput in the upper-level problem, and derive a closed-form analytical solution that yields the optimal control input for each CAV, in terms of fuel consumption, in the low-level problem. We validate the effectiveness of the solution through simulation under 100% CAV penetration rate. Fuel consumption and travel time for the vehicles are significantly reduced compared to a baseline scenario consisting of human-driven vehicles.

An analytical solution of the inverse problem of capillary imbibition

The inverse problem of capillary imbibition involves determination of the capillary geometry from the measurements of the time-varying meniscus position. This inverse problem is known to have multiple solutions, and to ensure a unique solution, measurements of imbibition kinematics in both directions of the capillary are required. We here present a closed-form analytical solution of the inverse problem of determining the axially varying radius of a capillary from experimental data of the meniscus position as a function of time. We demonstrate the applicability of the method for solving the inverse capillary imbibition problem for two cases, wherein the data for imbibition kinematics are obtained (i) using numerical simulations and (ii) from published experimental work. In both cases, the axially varying capillary radius predicted by the analytical solution agrees with the true capillary radius. In contrast to the previously proposed iterative methods for solving the inverse capillary imbibition problem, the analytical method presented here yields a direct solution. This analytical solution of the inverse capillary imbibition problem can be helpful in determining the internal geometry of micro- and nano-porous structures in a non-destructive manner and design of autonomous capillary pumps for microfluidic applications.

Steady-state response of annular wedge-shaped aquifers to arbitrarily-located multiwells with regional flow

Wedge-shaped aquifers are sometimes cut by crossing streams so that the areal geometry can be delineated by an annular wedge of infinite radial extent. Despite their worldwide spread, the problem of groundwater flow in such geometrical entities has not received much attention in analytical studies. This paper presents a closed-form analytical solution for the aquifer head distribution in response to a system of multiwells superimposed on a background regional flow. The aquifer is isotropic, homogeneous and in prefect hydraulic connection with the surrounding streams. A steady-state Green’s function is constructed for this laterally bounded aquifer system. The formulation is general in the sense that the aquifer vertex angle as well as number of wells and their positions can be arbitrary chosen. Closed-form expression for the stream function is obtained by integration of the Cauchy-Riemann equations. Three types of hydrogeological boundary conditions are considered, combining streams of constant and linearly varying head and no-flow barrier. The results are applicable to both confined and unconfined flows. Extremely simple expressions are derived to quantify the amount of water exchange across stream-aquifer interfaces. Flow nets are given for a number of hypothetical test cases, demonstrating different aspects of the generated flow field. Sensitivity maps are prepared for the stream depletion rates due to a single pumping well and the role of aquifer vertex angle on the model response is investigated. Multiple stagnation points in the flow field are identified on a semi-analytical basis. This helps to assess how the position of a single stagnation point is affected by changes in the gradient of regional flow. For relatively high gradients, it is observed that the radial position of the stagnation point coincides with that of the well itself. Finally, the formulation is extended to account for rainfall recharge over an unconfined aquifer having the same areal extent. The aquifer response is found to agree well with other analytical and numerical results.

Exact solutions of Mathieu’s equation

Abstract Mathieu’s equation originally emerged while studying vibrations on an elliptical drumhead, so naturally, being a linear second-order ordinary differential equation with a Cosine periodic potential, it has many useful applications in theoretical and experimental physics. Unfortunately, there exists no closed-form analytic solution of Mathieu’s equation, so that future studies and applications of this equation, as evidenced in the literature, are inevitably fraught by numerical approximation schemes and nonlinear analysis of so-called stability charts. The present research work, therefore, avoids such analyses by making exceptional use of Laurent series expansions and four-term recurrence relations. Unexpectedly, this approach has uncovered two linearly independent solutions to Mathie’s equation, each of which is in closed form. An exact and general analytic solution to Mathieu’s equation, then, follows in the usual way of an appropriate linear combination of the two linearly independent solutions.

An analytical model for the bio-inspired nacreous composites with interlocked “brick-and-mortar” structures

This paper establishes a closed-form analytical solution for the bio-inspired nacreous composites with interlocked “brick and mortar” (B-and-M) structure, with considering the influence of tension zones of the soft matrix, which was neglected in the classical tension-shear chain (TSC) model. It is found that vertical mortar layers have a significant effect on the mechanical properties of B-and-M structure in the state of plane strain. Parametric study shows that the deformation mechanism of the interlocked B-and-M structure is dependent on the length of vertical mortar layers and interlocking angle. On basis of this analytical solution, the influence of the interlocked structures on mechanical properties and deformation modes of nacreous composites is further evaluated, which shows that the mechanical properties of the nacreous composites can be optimized, with tuning of the length of the bricks and the interlocking angle. The results present a deep understanding of the deformation mechanisms of the bio-inspired nacreous composites with interlocked B-and-M structure and potentially provide guideline to the design and optimization of bio-inspired composites with high strength and high ductility.

A closed-form solution for the 3D steady-state thermoporoelastic field in an infinite transversely isotropic rock weakened by an elliptical crack

This paper develops a closed-form analytical solution for the problem of an infinite transversely isotropic thermoporoelastic rock weakened by an elliptic crack lying in the transverse plane. The crack is symmetrically subjected to three pairs of uniform generalized loads, i.e., mechanical pressure, pore pressure and temperature increment, on the upper and lower crack surfaces. Based on the general solution, the three-dimensional (3D) steady-state thermoporoelastic field in the rock are explicitly derived in terms of elemental functions and elliptic integrals by the generalized potential theory method. The crack surface displacements, crack stiffness, normal stress in the cracked plane, and stress intensity factor at the crack tip are obtained as well. In the numerical calculation, the obtained analytical solution is first degenerated into the one for a circular crack for the purpose of verification, and then used to demonstrate the thermoporoelastic field around an elliptic crack and its geometric effect on the stress intensity factor at its tips. The present solution can serve as a convenient benchmark to various numerical codes for simulations of rock fracture.

Effects of Transpiration on the Unsteady Separated Stagnation-Point Flow and Heat Transfer Over a Moving Porous Plate

Abstract The combined influence of normal transpiration and tangential movement of a porous surface on the unsteady separated stagnation-point flow and heat transfer of a viscous fluid is studied. This study is based on five physical parameters, namely (i) flow strength parameter a, (ii) suction/injection parameter d, (iii) plate velocity parameter λ, (iv) unsteadiness parameter β, and (v) Prandtl number Pr. This analysis shows an interesting relation β = 2a which allows us to derive some closed-form analytic solutions depending upon the unlikely values of d. For suction d > 0, two attached flow solutions (AFS) without point of inflection are found in the range (−ad2/4−3<λ<−3), whereas for λ > −3 only one solution of the same type is found for any given value of d. Besides them, the numerical computations reveal two types of AFS—one without and the other with a point of inflection in the range (−1.24658 ≤λ≤ −1.07) when d = β = 0. The present analysis confirms the nonexistence of the second attached flow solution after a certain value of suction d depending upon the choice of the values of λ in this range. A reverse flow solution (RFS) along with the above two solutions is found for a negative value of β which continues even for large rate of suction d. The asymptotic solutions of this flow problem have also been derived for large values of d which provide with the exact results after a certain value of d depending upon the values of the other parameters.

A closed-form solution for stress analysis of hollow circular cylinder structure under non-uniform external load and its engineering application

Hollow circular cylinder structures are widely used in industry for their high bearing capacity. In some engineering cases, these structures are always subjected to complicated non-uniform external loads. For example, casings used for oil production are subjected to non-uniform ground stresses. In this study, a generalized closed-form analytical solution for stress analysis of hollow circular cylinder under non-uniform external load was derived. The common non-uniform external load was decomposed by Fourier series under the principle of superposition by theory of elasticity. Analytical solutions for stress results of sine or cosine series external load problems were obtained by the semi-inverse method. A baseline analysis of a casing under non-uniform ground stress was presented using the proposed analytical method and the finite element method to validate the accuracy of the proposed analytical model. A parametric analysis was conducted finally to discuss the effects of non-uniform coefficients on the stress results. Results show that, the hollow circular cylinder structure’s anti-collapse capacity will be strongly weakened, when the non-uniform coefficient increases. This proposed analytical model can be referenced in strength verification of hollow circular cylinder structures in engineering practice.

Analytical Solutions for Ultimate Limit State Design of Thermal Piles

AbstractThis work aimed at providing analytical closed-form solutions for the design of thermal piles. To this end, a model in which a cylindrical pile is attached along the shaft to a series of di...

Method of Green’s Function for Characterization of SH Waves in Porous-Piezo Composite Structure with a Point Source

An approach of Green’s function is adopted to solve the inhomogeneous linear differential equations representing wave equations in piezo-composite materials. In particular, transference of horizontally polarised shear (SH) waves is considered in bedded structure comprising of porous-piezo electric layer lying over a heterogeneous half-space. Propagation of SH-waves is considered to be influenced by point source, situated in the heterogeneous substrate. A closed form analytical solution is obtained to establish the dispersion relation. Remarkable influence of different parameters (like elastic constant, piezoelectric constant, heterogeneity parameter, initial stress and layers thickness) on the phase and group velocity are shown graphically. Moreover, a special case of present study is shown by replacing the porous piezoelectric material with piezoelectric material. Some numerical examples are illustrated by taking the material constants of Lead Zirconate Titanate (PZT-1, PZT-5H and PZT-7) for the porous piezoelectric layer where the phase velocity of SH waves is high rather than that of piezoelectric layer.