Analytical Solutions for Testing of Baroclinic and Wind-Driven Three-Dimensional Hydrodynamic Models

It is always a challenge for a model developer to verify a three-dimensional hydrodynamic model, especially for the baroclinic term over variable topography, due to a lack of observational data sets or suitable analytical solutions. In this paper, exact solutions for the periodic forcing by surface heat flux and wind stress are given by solving the linearized equations of motion neglecting the rotation, advection and horizontal diffusion terms. The temperature at the bottom is set to a prescribed periodic value and a slip condition on flow is enforced at the bottom. The geometry of the quarter annulus, which has been extensively studied for two- and three-dimensional analytical solutions of unstratified water bodies, is used with a general power law variation of the bottom slope in the radial direction and is constant in the azimuthal direction. The analytical solutions are derived in a cylindrical coordinate system, which describes the three-dimensional fluid field in a Cartesian coordinate system. The results presented in this paper should provide a foundation for studying and verifying the baroclinic term over a varied topography in a three-dimensional numerical model.

Analytical wave solutions of an electronically and biologically important model via two efficient schemes

We search for analytical wave solutions of an electronically and biologically important model named as the Fitzhugh–Nagumo model with truncated M-fractional derivative, in which the expafunction and extended sinh-Gordon equation expansion(ESh GEE) schemes are utilized. The solutions obtained include dark, bright, dark-bright, periodic and other kinds of solitons. These analytical wave solutions are gained and verified with the use of Mathematica software. These solutions do not exist in literature. Some of the solutions are demonstrated by 2D, 3D and contour graphs. This model is mostly used in circuit theory, transmission of nerve impulses, and population genetics. Finally, both the schemes are more applicable, reliable and significant to deal with the fractional nonlinear partial differential equations.

AN ANALYTICAL SOLUTION FOR THE MODEL OF DRUG DISTRIBUTION AND ABSORPTION IN SMALL INTESTINE

According to the physiological and anatomical characteristics of small intestine,neglecting the effect of its motility on the distribution and absorption of drug and nutrient,Y.Miyamoto et al.proposed a model of two-dimensional laminar flow in a circular porous tube with permeable wall and calculated the concentration profile of drugby numerical analysis.In this paper,we give a steady-state analytical solution of the above model including deactivationterm.The obtained results are in agreement with the results of their numerical analysis. Moreover the analytical solution presented in this paper reveals the relation among the physiological parameters of the model and describes the basic absorption rule of drug and nutrient through the intestinal wall and hence pro- vides a theoretical basis for determining the permeability and reflection coefficient through in situ experiments.

THE EXACT ANALYTICAL SOLUTIONS TO THE NEW MODEL OF RESERVOIR FILTRATION PROBLEM

The present study has obtained the new model of the reservoir filtration problem by taking into account the effect of wellbore storage and skin and by making use of the coupled equations of doubled porous media filtration and consequently has got, through various forms of limits, the exact analytical solutions of the three common reservoirs (fissure, homogeneous and the two-layered) pressure distribution under the conditions of three boundaries, i.e., infinite boundary, sealed finite boundary and the finite boundary at constant pressures.

Exact analytical solutions to the mean-field model depicting microcavity containing semiconductor quantum wells

By using a two-mode mean-field approximation, we study the dynamics of the microcavities containing semiconductor quantum wells. The exact analytical solutions are obtained in this study. Based on these solutions, we show that the emission from the microcavity manifests periodic oscillation behaviour and the oscillation can be suppressed under a certain condition.

Two dimensional analytical solution for a partially vegetated compound channel flow

The theory of an eddy viscosity model is applied to the study of the flow in a compound channel which is partially vegetated. The governing equation is constituted by analyzing the longitudinal forces acting on the unit volume where the effect of the vegetation on the flow is considered as a drag force item, The compound channel is divided into 3 sub-regions in the transverse direction, and the coefficients in every region＇s differential equations were solved simultaneously. Thus, the analytical solution of the transverse distribution of the depth-averaged velocity for uniform flow in a partially vegetated compound channel was obtained. The results can be used to predict the transverse distribution of bed shear stress, which has an important effect on the transportation of sediment. By comparing the analytical results with the measured data, the analytical solution in this paper is shown to be sufficiently accurate to predict most hydraulic features for engineering design purposes.

An Analytical Solution for One-Dimensional Water Infiltration and Redistribution in Unsaturated Soil

Soil infiltration and redistribution are important processes in field water cycle, and it is necessary to develop a simple model to describe the processes. In this study, an algebraic solution for one-dimensional water infiltration and redistribution without evaporation in unsaturated soil was developed based on Richards equation. The algebraic solution had three parameters, namely, the saturated water conductivity, the comprehensive shape coefficient of the soil water content distribution, and the soil suction allocation coefficient. To analyze the physical features of these parameters, a relationship between the Green-Ampt model and the algebraic solution was established. The three parameters were estimated based on experimental observations, whereas the soil water content and the water infiltration duration were calculated using the algebraic solution. The calculated soil water content and infiltration duration were compared with the experimental observations, and the results indicated that the algebraic solution accurately described the unsaturated soil water flow processes.

A Review and Update of Analytical and Numerical Solutions of the Terzaghi One-Dimensional Consolidation Equation

Practical resolution of consolidation problems that we often face requires an extensive and solid knowledge of the different parameters highlighted by the Terzaghi one-dimensional consolidation theory. This theory, with its assumptions, leads to a partial differential equation of second order in space and first order in time of pore water pressure. Analytical and numerical resolutions of this equation allow determining the water pressure variation before and after the application of a charge. Numerical modeling has enabled the simulation of the whole results obtained by the two methods of resolution (pressure, degree of consolidation, time factor, among others) to have a physical analysis and a lawful observation that lead to a suitable understanding of the phenomenon of Terzaghi one-dimensional consolidation.

ANALYTICAL SOLUTIONS FOR SOME NONLINEAR EVOLUTION EQUATIONS

The following partial differential equations are studied: generalized fifth-order KdV equation,water wave equation, Kupershmidt equation, couples KdV equation. The analytical solutions to these problems via using various ansatzes by introducing a second-order ordinary differential equation are found out.

Exact analytical solution to three-dimensional phase change heat transfer problems in biological tissues subject to freezing

Analytically solving a three-dimensional （3-D） bioheat transfer problem with phase change during a freezing process is extremely difficult but theoretically important. The moving heat source model and the Green function method are introduced to deal with the cryopreservation process of in vitro biomaterials. Exact solutions for the 3-D temperature transients of tissues under various boundary conditions, such as totally convective cooling, totally fixed temperature cooling and a hybrid between them on tissue surfaces, are obtained. Furthermore, the cryosurgical process in living tissues subject to freezing by a single or multiple cryoprobes is also analytically solved. A closed-form analytical solution to the bioheat phase change process is derived by considering contributions from blood perfusion heat transfer, metabolic heat generation, and heat sink of a cryoprobe. The present method is expected to have significant value for analytically solving complex bioheat transfer problems with phase change.

An Analytic Approximate Solution of the SIR Model

The SIR(D) epidemiological model is defined through a system of transcendental equations, not solvable by elementary functions. In the present paper those equations are successfully replaced by approximate ones, whose solutions are given explicitly in terms of elementary functions, originating, piece-wisely, from generalized logistic functions: they ensure exact (in the numerical sense) asymptotic values, besides to be quite practical to use, for example with fit to data algorithms;moreover they unveil a useful feature, that in fact, at least with very strict approximation, is also owned by the (numerical) solutions of the exact equations. The novelties in the work are: the way the approximate equations are obtained, using simple, analytic geometry considerations;the easy and practical formulation of the final approximate solutions;the mentioned useful feature, never disclosed before. The work’s method and result prove to be robust over a range of values of the well known non-dimensional parameter called basic reproduction ratio, that covers at least all the known epidemic cases, from influenza to measles: this is a point which doesn’t appear much discussed in analogous works.

An analytical analysis for the mechanical performance of fully-grouted rockbolts based on the exponential softening model

This paper presented a novel bond-slip model to better reveal the mechanical behavior of the bolt-grout interface for fully-grouted rockbolts under tensile loads by considering the non-linear response in the softening stage.The exponential decay function is adopted for describing the non-linear response in the softening stage.Based on the improved bond-slip model,the corresponding analytical solutions for the interfacial shear stress and the axial force of the bolt under different loading stages are solved.Then,the validity of this proposed model was verified by comparing with the experimental results.The results show that compared with the linear softening model,the proposed model is more suitable for predicting the mechanical performance of fully-grouted rockbolts.Finally,a series of parametric studies are conducted to explore the effect of model parameters on the mechanical properties of fully-grouted rockbolts.The results indicate that compared with the anchor length,the bolt diameter and the bond strength of the bolt-grout interface have a significance influence on the ultimate load of bolt,especially for the elastic and softening stage.Moreover,it can be found that using the linear softening model maybe overestimates the supporting performance of grouted bolt,resulting in an unsafe design for bolt.

Comparison between Non-Gaussian Puff Model and a Model Based on a Time-Dependent Solution of Advection-Diffusion Equation

A comparison between a non-Gaussian puff model and an advanced time-dependent model to simulate the pollutant dispersion in the Planetary Boundary Layer is presented. The puff model is based on a general technique for solving the K-equation, using the truncated Gram-Charlier expansion (type A) of the concentration field and finite set equations for the corresponding moments. The other model (named ADMM: Analytical Dispersion Multilayers Model) is an semi- analytical solution to the time-dependent two-dimensional advection-diffusion equation based on a discretization of the PBL in N sub-layers;in each sub-layers the advection-diffusion equation is solved by the Laplace transform technique, considering an average value for eddy diffusivity and the wind speed. A preliminary performance evaluation is shown in the case of continuous emission from an elevated source in a variable boundary layer. Both models were able to correctly reproduce the concentration field measured and so to be used as operative air pollution models.

An Analytical Air Pollution Model with Time Dependent Eddy Diffusivity

Air pollution transport and dispersion in the atmospheric boundary layer are modeled by the advection-diffusion equation, that is, essentially, a statement of conservation of the suspended material in an incompressible flow. Many models simulating air pollution dispersion are based upon the solution (numerical or analytical) of the advection-diffusion equation assuming turbulence parameterization for realistic physical scenarios. We present the general time dependent three-dimensional solution of the advection-diffusion equation considering a vertically inhomogeneous atmospheric boundary layer for arbitrary vertical profiles of wind and eddy-diffusion coefficients. Numerical results and comparison with experimental data are shown.

AN ANALYTIC SOLUTION OF DENSE TWO-PHASE FLOW IN A VERTICAL PIPELINE

According to a mathematical model for dense two-phase flows presented in the previous pape[1],a dense two-phase flow in a vertical pipeline is analytically solved, and the analytic expressions of velocity of each continuous phase and dispersed phase are respectively derived. The results show that when the drag force between twophasesdepends linearly on their relative velocity, the relative velocity profile in the pipeline coincides with Darcy's law except for the thin layer region near the pipeline wall, and that the theoretical assumptions in the dense two-phase flow theory mentioned are reasonable.

An improved analytical solution for solving the shield tunnel uplift problem

The problem of shield tunnel uplift is a common issue in tunnel construction.Due to the decrease in shear stiffness at the joints between the rings,uplift is typically observed as bending and dislocation deformation at these joints.Existing modeling methods typically rely on the Euler-Bernoulli beam theory,only considering the bending effect while disregarding shear deformation.Furthermore,the constraints on the shield tail are often neglected in existing models.In this study,an improved theoretical model of tunnel floating is proposed.The constraint effect of the shield machine shell on the tunnel structure is considered using the structural forms of two finite long beams and one semi-infinite long beam.Furthermore,the Timoshenko beam theory is adopted,providing a more accurate description of tunnel deformation,including both the bending effect and shear deformation,than existing models.Meanwhile,the buoyancy force and stratum resistance are calculated in a nonlinear manner.A reliable method for calculating the shear stiffness correction factor is proposed to better determination of the calculation parameters.The proposed theoretical model is validated through five cases using sitemonitored data.Its applicability and effectiveness are demonstrated.Furthermore,the influences of soil type,buried depth,and buoyancy force on the three key indicators of tunnel floating(i.e.the maximum uplift magnitude,the ring position with the fastest uplift race,and the ring position with the maximum uplift magnitude)are analyzed.The results indicate that the proposed model can provide a better understanding of the floating characteristics of the tunnel structure during construction.

边坡植被恢复中考虑微生物矿化和植被作用的Green-Ampt模型

在边坡植被恢复中降雨入渗量的估算是很重要的,为探究石漠化边坡格构内微生物矿化联合植被作用的填土降雨入渗过程,在传统Green-Ampt模型基础上,针对表层矿化的土柱引入稳定入渗率,提出考虑微生物表层矿化(M)作用的改进G-A模型(M-G-A);针对含植株土柱,考虑复合土层的入渗过程,分别提出考虑植被(P)作用的改进G-A模型(P-G-A)以及考虑微生物矿化联合植被作用的改进G-A模型(PM-G-A)。利用改进G-A模型进行累计入渗量计算,并将计算结果与土柱试验实测值、Hydrus数值解进行对比分析。结果表明:M-G-A模型可以较好地描述累计入渗量随时间的变化,且矿化程度越高模型计算偏差越小,在强矿化作用下误差可控制在5%以内;P-G-A模型的计算误差随降雨历时而增加,但相较于Hydrus数值解可以减少误差约10%;PM-G-A模型的计算偏差随降雨历时呈现先增加后减小的趋势,总体上优于Hydrus数值解。改进G-A模型能够针对微生物表层矿化石漠化边坡格构内填土的生态恢复技术进行累计入渗量即蓄水量分析,具有一定的工程实用价值。

Air Pollution Steady-State Advection-Diffusion Equation: The General Three-Dimensional Solution

Atmospheric air pollution turbulent fluxes can be assumed to be proportional to the mean concentration gradient. This assumption, along with the equation of continuity, leads to the advection-diffusion equation. Many models simulating air pollution dispersion are based upon the solution (numerical or analytical) of the advection-diffusion equation as- suming turbulence parameterization for realistic physical scenarios. We present the general steady three-dimensional solution of the advection-diffusion equation considering a vertically inhomogeneous atmospheric boundary layer for arbitrary vertical profiles of wind and eddy-diffusion coefficients. Numerical results and comparison with experimental data are shown.

Analytical and Experimental Studies on Wave Scattering by a Horizontal Perforated Plate at the Still Water Level

This research investigates water-wave scattering via a horizontal perforated plate fixed at the still water level through analytical studies and physical model tests.The velocity potential decomposition method is combined with an efficient iterative algorithm to develop an analytical solution in which the quadratic pressure drop condition is imposed on the horizontal perforated plate.The analytical results are in good agreement with the results of an independently developed iterative boundary element method(BEM)solution.Experimental tests are carried out in a wave flume to measure the reflection coefficient and transmission coefficient of the horizontal perforated plate,and the analytical results agree reasonably well with the experimental data.The influence of various structural parameters of the horizontal perforated plate on the hydrodynamic parameters of reflection coefficient,transmission coefficient,energy-loss coefficient,and wave force are analyzed on the basis of the analytical solution.Useful results for the practical engineering application of horizontal perforated plates are also presented.

Closed-form solution for shock wave propagation in density-graded cellular material under impact

Density-graded cellular materials have tremendous potential in structural applications where impact resistance is required.Cellular materials subjected to high impact loading result in a compaction type deformation,usually modeled using continuum-based shock theory.The resulting governing differential equation of the shock model is nonlinear,and the density gradient further complicates the problem.Earlier studies have employed numerical methods to obtain the solution.In this study,an analytical closed-form solution is proposed to predict the response of density-graded cellular materials subjected to a rigid body impact.Solutions for the velocity of the impinging rigid body mass,energy absorption capacity of the cellular material,and the incident stress are obtained for a single shock propagation.The results obtained are in excellent agreement with the existing numerical solutions found in the literature.The proposed analytical solution can be potentially used for parametric studies and for effectively designing graded structures to mitigate impact.