Fractional Dispersion,Lévy Motion,and the MADE Tracer Tests

The macrodispersion experiments (MADE) at the Columbus Air Force Base in Mississippi were conducted in a highly heterogeneous aquifer that violates the basic assumptions of local second-order theories. A governing equation that describes particles that undergo Lévy motion, rather than Brownian motion, readily describes the highly skewed and heavy-tailed plume development at the MADE site. The new governing equation is based on a fractional, rather than integer, order of differentiation. This order (), based on MADE plume measurements, is approximately 1.1. The hydraulic conductivity (K) increments also follow a power law of order =1.1. We conjecture that the heavy-tailed K distribution gives rise to a heavy-tailed velocity field that directly implies the fractional-order governing equation derived herein. Simple arguments lead to accurate estimates of the velocity and dispersion constants based only on the aquifer hydraulic properties. This supports the idea that the correct governing equation can be accurately determined before, or after, a contamination event. While the traditional ADE fails to model a conservative tracer in the MADE aquifer, the fractional equation predicts tritium concentration profiles with remarkable accuracy over all spatial and temporal scales.     

Comprehensive mathematical model of microcirculatory dynamics (II)—Calculation and the results

The mathematical model described in Part I was solved using “influence line method” combining analytical method and finite element method. Many important aspects of microcirculatory dynamics were analyzed and discussed. It show that interstitial fluid pressure changes its sign twice within one arteriolar vasomotion period and it is therefore not important that interstitial fluid pressure is a little higher or lower than atmospheric pressure; arteriolar vasomotion can periodically result in lymph formation and interstitial total pressure plays an important role in this procedure; local regulation of microcirculation can meet metabolic need some extent in the form of dynamic equilibrium. The property of arteriole as a “resistant vessel” and the efficiency of microvascular network as heat exchanger are also shown. These results show that the comprehensive mathematical model developed in Part I is physiologically resonable. Foundation item: the Natural Science Foundation of Sichuan Province, P R China Biography: Guo Zhongsan (1947-)     

Application of the extended Fourier amplitude sensitivity testing(FAST) method to infated,axial stretched,and residually stressed cylinders

This paper is dedicated to applying the Fourier amplitude sensitivity test(FAST) method to the problem of mixed extension and inflation of a circular cylindrical tube in the presence of residual stresses. The metafunctions and the Ishigami function are considered in the sensitivity analysis(SA). The effects of the input variables on the output variables are investigated, and the most important parameters of the system under the applied pressure and axial force such as the axial stretch and the a...     

Group Classification of Equations of the Form y″ = f(x,y)

The problem of classification of ordinary differential equations of the form y = f(x,y) by admissible local Lie groups of transformations is solved. Standard equations are listed on the basis of the equivalence concept. The classes of equations admitting a oneparameter group and obtained from the standard equations by invariant extension are described.     

On the “bead,hoop and spring” (BHS) dynamical system

Here, we make the theoretical and numerical analysis of the non-linear equation describing the evolution of the “bead, hoop and spring” (BHS) dynamical system derived by Ochoa and Clavijo in (Eur. J. Phys. 27:1277–1288, 2006). In particular, we solve by standard techniques of non-linear physics an approximation of their equation neglecting the centrifugal effect before giving a more mathematical and exact treatment. The analogy with phase transitions is underlined. We point out the existence of finite-time singularities in the phase-space and we derive a criterion for possible oscillations.     

Representations of ℂ, biharmonic vector fields,and the equilibrium equation of planar elasticity

The equilibrium equation for an elastic body subjected to surface forces asserts the linear dependence of the Laplacian and the gradient of the divergence of the vector field which gives the displacement at each point. James Clerk Maxwell (1831–1879) was the first to point out that the component functions of such a field are biharmonic, i.e., their Laplacians are harmonic functions. Using only algebraic tools familiar to advanced undergraduates we show that the usual complex variable representation of two-variable biharmonic functions falls naturally out of a power series construction based on matrix representations of . Under the assumption of linear stress and strain components, this construction is then used to describe the solutions to the planar equilibrium equation in terms of the geometry of the Moebius plane.     

Strong L p -Solutions of the T α-Type Navier–Stokes Equation and the Regularizing-Decay Rate Estimation

In the present paper, we investigate t ^α-type Navier–Stokes equations introduced in a previous paper of Tu–Zhai. The existence and uniqueness results are given in L ^p space. Moreover, the regularizing decay rate estimates and high order approximation are derived for these solutions.     

Viscous-slip,thermal-slip,and temperature-jump coefficients based on the linearized Boltzmann equation (and five kinetic models) with the Cercignani–Lampis boundary condition

A polynomial expansion procedure and the ADO (analytical discrete-ordinates) method are used to compute the viscous-slip coefficient, the thermal-slip coefficient, and the temperature-jump coefficient from the linearized Boltzmann equation (LBE) for rigid-sphere interactions and the Cercignani–Lampis (CL) boundary condition. These same quantities are also computed from five kinetic models, with the CL condition, and compared to the LBE result. Equivalent results for the LBE and the kinetic models, all based on the usual Maxwell boundary condition, are also reported.     

Application of the “proportionate partitioning” method suggested by Poulovassilis and Kargas (2000) for determination of the domain distribution function

The soil water hysteresis model proposed by Poulovassilis and Kargas (Soil Sci. Soc. Am. J. 64:1947–1950, 2000) is considered in the present study. According to this model, the bivariate domain density distribution function f can be derived by partitioning the slopes of either of two main curves proportionally to the slopes of another. Accordingly, there are two possible ways of deriving function f. The basic claim of Poulovassilis and Kargas is that both possibilities lead to the same resultant function f, which can be evaluated using integral equation presented by them. The present study shows that the above two ways of determining function f actually lead to two incompatible partitioning models yielding different domain density distribution functions. Moreover, none of these two partitioning models can reproduce the measured hysteresis loop used for calibration. Whether the partitioning of the main wetting curve slopes proportionally to the main drying curve slopes or vice versa is applied, most of the predicted primary scanning curves deviate considerably from the measured ones, cross out the measured boundary loop and do not converge at an appropriate edge of the loop. The present study reveals that the above-mentioned integral equation, presented by Poulovassilis and Kargas, appears to be at variance with both partitioning models. It is shown herein that this integral equation unambiguously follows from Mualem (Water Resour. Res. 9:1324–1331, 1973) similarity hypothesis and, accordingly, the correspondent domain density distribution function derived as the unique analytical solution of this equation is evidently identical to that obtained by Mualem (1973). The predicted curves presented by Poulovassilis and Kargas are not obtained when any of the two partitioning models is applied, but when using the integral equation of Mualem’s (1973) model.     

Simulation of the Emission Spectrum of SiH (A2Δ → X2Π) and Measurement of the Rotational Temperature of the A2Δ State in an Electron-Beam Plasma

The 0–0 band emission spectrum of the A² X² transition of the SiH molecule was modeled numerically. The results obtained agree well with known calculated and experimental data. The rotational temperature of the A² state of SiH in a free stream of pure monosilane (SiH₄) and in a mixture with helium (He+SiH₄) activated by an electron beam is determined by comparing calculated and experimental spectra. The assumption that the emission of SiH results from dissociative excitation of SiH₄ by electron impact is confirmed. Rotational temperatures for various monosilane concentrations and distances from the nozzle are given. The spectra obtained exhibit the emission of silicon ions at wavelengths of 412.807 and 413.089 nm.     

Solutions of the Dirac–Fock Equations and the Energy of the Electron-Positron Field

We consider atoms with closed shells, i.e. the electron number N is 2, 8, 10,..., and weak electron-electron interaction. Then there exists a unique solution γ of the Dirac–Fock equations with the additional property that γ is the orthogonal projector onto the first N positive eigenvalues of the Dirac–Fock operator . Moreover, γ minimizes the energy of the relativistic electron-positron field in Hartree–Fock approximation, if the splitting of into electron and positron subspace is chosen self-consistently, i.e. the projection onto the electron-subspace is given by the positive spectral projection of. For fixed electron-nucleus coupling constant g:=α Z we give quantitative estimates on the maximal value of the fine structure constant α for which the existence can be guaranteed.     

Combined slit and plate–plate magnetorheometry of a magnetorheological fluid (MRF) and parameterization using the Casson model

We describe a magneto-slit die of 0.34 mm height and 4.25 mm width attached to a commercial piston capillary rheometer, enabling the measurement of apparent flow curves of a magnetorheological fluid (MRF) in the high shear rate regime (apparent shear rates 276 up to 20,700 s^???1, magnetic flux density up to 300 mT). The pressure gradient in the magnetized slit is measured via two pressure holes. While the flux density versus coil current without MRF could directly be measured by means of a Hall probe, the flux density with MRF was investigated by finite element simulations using Maxwell® 2D. The true shear stress versus shear rate is obtained by means of the Weissenberg–Rabinowitsch correction. The slit die results are compared to plate–plate measurements performed in a shear rate regime of 0.46 up to 210 s^???1. It is shown that the Casson model yields a pertinent fit of the true shear stress versus shear rate data from plate–plate geometry. Finally, a joint fit of the slit and plate–plate data covering a shear rate range of 1 up to 50,000 s^???1 is presented, again using the Casson model. The parameterization of the MRF behavior over the full shear rate regime investigated is of relevance for the design of MR devices, like, e.g., automotive dampers. In the Appendix, we demonstrate the drawbacks of the Bingham model in describing the same data. We also show the parameterization of the flow curves by applying the Herschel–Bulkley model.     

Intrinsic Geometry and Analysis of Diffusion Processes and L ∞-Variational Problems

The aim of this paper is twofold. First, we obtain a better understanding of the intrinsic distance of diffusion processes. Precisely, (a) for all n ≧ 1, the diffusion matrix A is weak upper semicontinuous on Ω if and only if the intrinsic differential and the local intrinsic distance structures coincide; (b) if n = 1, or if n ≧ 2 and A is weak upper semicontinuous on Ω, the intrinsic distance and differential structures always coincide; (c) if n ≧ 2 and A fails to be weak upper semicontinuous on Ω, the (non-)coincidence of the intrinsic distance and differential structures depend on the geometry of the non-weak-upper-semicontinuity set of A. Second, for an arbitrary diffusion matrix A, we show that the intrinsic distance completely determines the absolute minimizer of the corresponding L ^∞-variational problem, and then obtain the existence and uniqueness for given boundary data. We also give an example of a diffusion matrix A for which there is an absolute minimizer that is not of class C ¹. When A is continuous, we also obtain the linear approximation property of the absolute minimizer.     

Resonance and deflection of multi-soliton to the (2+1)-dimensional Kadomtsev–Petviashvili equation

In this work, a modified three-soliton method with a perturbation parameter is proposed, and it is applied to the (2+1)-dimensional Kadomtsev–Petviashvili equation (KP), and new breather multi-soliton solutions are obtained. The dependence of new mechanical structures on the perturbed parameter for multi-soliton including resonance and deflection for KP equation are investigated and exhibited.     

Existence and construction of quadratic invariants for the equation d2dξ2ν = λ(ξ)ddξν + f(ξ,ν)

If the equation ν_ζζ = λ(ξ)ν_ζ + F(ξ,ν) admits a first integral quadratic in the momentum c., the function F(ξ,ν) is defined; a new procedure is introduced to find the first integral. Among the examples is discussed the existence of quadratic invariants for the equation u″ λu′ = Q(u), where λ is a constant, and for the equation ν_ξ,ξ =βξ^σνⁿ. This last equation admits a first integral, quadratic in ν_ξ if σ = 0, or n = −2σ − 3, or n = −σ − 3. Particular eases are exhibited where no invariant quadratic in ν_ξ is found for the equation ν_ξξ = βξ^σνⁿ but it admits a first integral which is a polynomial of degree 4 or 6 in ν_ξ.     

Quantifying the parameter dependent basin of the unsafe regime of asymmetric Lévy-noise-induced critical transitions

In real systems,the unpredictable jump changes of the random environment can induce the critical transitions(CTs)between two non-adjacent states,which are more catastrophic.Taking an asymmetric Lévy-noise-induced tri-stable model with desirable,sub-desirable,and undesirable states as a prototype class of real systems,a prediction of the noise-induced CTs from the desirable state directly to the undesirable one is carried out.We first calculate the region that the current state of the given model is absorbed into the undesirable state based on the escape probability,which is named as the absorbed region.Then,a new concept of the parameter dependent basin of the unsafe regime(PDBUR)under the asymmetric Lévy noise is introduced.It is an efficient tool for approximately quantifying the ranges of the parameters,where the noise-induced CTs from the desirable state directly to the undesirable one may occur.More importantly,it may provide theoretical guidance for us to adopt some measures to avert a noise-induced catastrophic CT.     

Hysteretic Θ(S) Curve Prediction: Comparison of Two Models

The prediction of the first-order wetting and drying scanning curves is attempted by two different methods. These are: Mualem model II and Poulovassilis and Kargas (P–K). Model II by Mualem was chosen deliberately as the most appropriate. Experimental (S) data obtained in the laboratory for a sand mixture and a real soil were used for the comparison. Moreover, data presented originally by Poulovassilis (1970) were also used for the same purpose. It is shown that the P–K method gives better results than Mualems model II. Some remarks on Mualems model are also included.     

Nonlocal symmetry,CRE solvability and soliton–cnoidal solutions of the ($$$$)-dimensional modified KdV-Calogero–Bogoyavlenkskii–Schiff equation

In this paper, a modified KdV-CBS equation is investigated by using the truncated Painlevé expansion and consistent Riccati expansion method, respectively. It is shown that the modified KdV-CBS equation has a nonlocal symmetry related to the residue of its truncated Painlevé expansion. It is also proved that the modified KdV-CBS equation is consistent Riccati expansion solvable. Furthermore, with the help of the consistent Riccati expansion method, the soliton–cnoidal wave interaction solutions are explicitly given.