Analytical solutions of a class of optimum orbit modifications
Analytical solutions to optimum orbit modifications including atmospheric maneuver
Marchal, C.↗
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An analytic solution for the J2 perturbed equatorial orbit
An analytic solution for the J2 perturbed equatorial orbit is obtained in terms of elliptic functions and integrals. The necessary equations for computing the position and elocity vectors, and the time are given in terms of known functions. The perturbed periapsis and apoapsis distances are determined from the roots of a characteristic cubic.
Jezewski, D. J.↗
Spherical indentation of a freestanding circular membrane revisited: Analytical solutions and experiments
Because of the development of novel micro-fabrication techniques to produce ultra-thin materials and increasing interest in thin biological membranes, in recent years, the mechanical characterization of thin films has received a significant amount of attention. To provide a more accurate solution for the relationship among contact radius, load and deflection, the fundamental and widely applicable problem of spherical indentation of a freestanding circular membrane have been revisited. The work presented here significantly extends the previous contributions by providing an exact analytical solution to the governing equations of Föppl–Hecky membrane indented by a frictionless spherical indenter. In this study, experiments of spherical indentation has been performed, and the exact analytical solution presented in this article is compared against experimental data from existing literature as well as our own experimental results.
36 MATERIALS SCIENCE↗
Analytical solutions of the Dirac equation under Hellmann–Frost–Musulin potential
The approximate analytical solutions of the Dirac equation with Hellmann–Frost–Musulin potential have been studied by using the generalized parametric Nikiforov–Uvarov (NU) method for arbitrary spin–orbit quantum number k under the spin and pseudospin symmetries. The Hellmann–Frost–Musulin potential is a superposition potential that consists of Yukawa potential, Coulomb potential, and Frost–Musulin potential. As a particular case, we found the energy levels of the non-relativistic limit of the spin symmetry. The energy equation of Yukawa potential, Coulomb potential, Hellmann potential and Frost–Musulin potential are obtained. Energy values are generated for some diatomic molecules.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
An analytical solution of the solution-diffusion-electromigration equations reproduces trends in ion rejections during nanofiltration of mixed electrolytes
Here, this paper develops an analytical solution to the equations governing the solution-diffusion-electromigration transport of the ions from two salts that contain a common ion. The analytical expressions, which rely on constant ion permeances, readily enable simple spreadsheet computations of the permeate ion concentrations in nanofiltration (NF). Despite the model simplicity, calculations of ion rejections as a function of transmembrane volume fluxes, feed-solution compositions, and permeance values reproduce experimental trends, with ion permeances as the only adjustable parameters. For example, with solutions containing dissolved MA and M 2 B, when the membrane permeance to B 2– is much lower than the permeances to A – and M + , plots of A – rejection versus transmembrane volume flux show the negative minimum that is often characteristic of NF. Moreover, B 2– rejection increases with increasing MA in the feed solution, reflecting a decreasing electric field in the membrane. The analytical solution also enables spreadsheet computations of electric fields in membrane separation layers and effectively models experimental data for intrinsic rejections as a function of transmembrane volume flow. Interestingly, the model suggests that with sufficiently high permeances of A – (relative to M + ), at certain feed compositions the A – /B 2– selectivity may increase without bound as a function of volume flow. Otherwise, this selectivity does not vary greatly with feed composition and quickly reaches a limit as the volume flow increases. Despite the simplifying approximation of constant ion permeances, using only a few adjustable parameters this model readily reveals trends in NF ion rejections and may help to identify the desired ion permeances or solution compositions for specific separations.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Comparison of the Amanzi Model against Analytical Solutions and the FEHM Model
This file documents Amanzi code verification of flow simulations for three selected cases: (1) an infinite strip aquifer embedded between two other aquifers, (2) a heterogeneous cylinder (or disc in 2D case) embedded in a background aquifer, and (3) uniform aquifers in bounded domains with fixed heads at the upper- and down-stream directions and no-flow at lateral boundaries. The original FORTRAN codes that solve the analytical solutions for cases (1) and (2) were obtained from Greg Ruskauf. Two modifications were made for compilation on Linux systems: (a) The keyword ‘mode’ in OPEN statements was removed because it is not standard, and (b) variables in COMMON blocks were re-arranged to eliminate a warning message (padding of additional 4 bytes). These codes were compiled using the gfortran compiler on the Linux environment. Note that these analytical solutions were derived for two-dimensional problems only, and therefore in numerical simulations, we used three-dimensional domains with unit thickness for comparison.
54 ENVIRONMENTAL SCIENCES↗
Code associated with Publication “Analytic Solutions and Field-Scale Application for Verification of Coupled Thermo-Hydro-Mechanical Processes in Subsurface Fractured Media”
As part of a submitted paper, which is collection of previously published analytical solutions to coupled thermo-hydro-mechanical problems in subsurface flow and transport, we have prepared a collection of python scripts to compute and plot those analytic solutions. All code to be released implements existing methods; there are no novel algorithms nor any major innovations to existing software.
Hyman, Jeffrey↗
Creation of the CMB spectrum: precise analytic solutions for the blackbody photosphere
The blackbody spectrum of CMB was created in the blackbody photosphere at redshifts z∼>2 × 10{sup 6}. At these early times, the Universe was dense and hot enough that complete thermal equilibrium between baryonic matter (electrons and ions) and photons could be established on time scales much shorter than the age of the Universe. Any perturbation away from the blackbody spectrum was suppressed exponentially. New physics, for example annihilation and decay of dark matter, can add energy and photons to CMB at redshifts z∼>10{sup 5} and result in a Bose-Einstein spectrum with a non-zero chemical potential (μ). Precise evolution of the CMB spectrum around the critical redshift of z ≅ 2 × 10{sup 6} is required in order to calculate the μ-type spectral distortion and constrain the underlying new physics. Although numerical calculation of important processes involved (double Compton process, comptonization and bremsstrahlung) is not difficult with present day computers, analytic solutions are much faster and easier to calculate and provide valuable physical insights. We provide precise (better than 1%) analytic solutions for the decay of μ, created at an earlier epoch, including all three processes, double Compton, Compton scattering on thermal electrons and bremsstrahlung in the limit of small distortions. This is a significant improvement over the existing solutions with accuracy ∼ 10% or worse. We also give a census of important sources of energy injection into CMB in standard cosmology. In particular, calculations of distortions from electron-positron annihilation and primordial nucleosynthesis illustrate in a dramatic way the strength of the equilibrium restoring processes in the early Universe. Finally, we point out the triple degeneracy in standard cosmology, i.e., the μ and y distortions from adiabatic cooling of baryons and electrons, Silk damping and annihilation of thermally produced WIMP dark matter are of similar order of magnitude ( ∼ 10{sup −8}–10{sup −10})
79 ASTRONOMY AND ASTROPHYSICS↗
Analytical solutions for transient and steady state beam loading in arbitrary traveling wave accelerating structures
Analytical solutions are derived for both transient and steady state gradient distributions in the traveling wave (TW) accelerating structures with arbitrary variation of parameters over the structure length. The results of the unloaded and beam loaded cases are presented. Finally, the exact analytical shape of the rf pulse waveform was found in order to apply the transient beam loading compensation scheme during the structure filling time. The obtained theoretical formulas were cross-checked by direct numerical simulations on the CLIC main linac accelerating structure and demonstrated a good agreement. The proposed methods provide a fast and reliable tool for the initial stage of the TW structure analysis.
43 PARTICLE ACCELERATORS↗
Revisiting the Analytical Solutions of Heat Transport in Fractured Reservoirs Using a Generalized Multirate Memory Function
Numerous analytical solutions have been developed for modeling thermal perturbations to underground formations caused by deep-well injection of fluids. Each solution has been derived for a specific boundary value problem and a simplified flow network with one set of parallel fractures. In this paper, new generalized solutions G*( x, s ) are developed using (existing) global transfer functions G$^*_0(x,s)$ and a new memory function g*( s ), where x and s are the space and Laplace variable. The memory function represents the solutions of conductive heat exchange between fractures and matrix blocks and between fractured aquifers and unfractured aquitards. The memory function is developed to account for multirate exchange induced by different shapes, sizes, properties, and volumetric fractions of matrix blocks bounded by multiple sets of orthogonal fractures with different spacing. The global transfer functions represent the fundamental solutions to convective, convective-conductive, and convective-dispersive heat transport in fractures (or aquifers) without exchange and are available for various (1-D linear, 1-D radial, 2-D dipole, and single-well injection-withdrawal) flow fields. The new solutions with exchange are developed using $G^*(x,s) = B^*(s)G^*_0{x, s[1 + Θg^*(s)]}$, thereby greatly simplifying solution development in a novel way, where Θ and B*( s ) are a fracture-matrix scaling factor and the boundary condition function. As a result, the new solutions are applied to several example problems, showing that heat transport in fractured aquifers is significantly impacted by (1) thermal dispersion in fractures that is rarely considered, (2) multirate heat exchange with a wide range of size and anisotropy of rectangular matrix blocks, and (3) heat exchange between aquifers and aquitards.
58 GEOSCIENCES↗
An analytic solution to the steady-state double adiabatic equations
A set of 20 generalized moment-transport equations applicable to low-beta (low random energy density/magnetic energy density) plasmas of thermal origin in rotating planetary magnetospheres. An analytic solution is obtained for a set of simplified equations characterizing the steady-state transport of collisionless plasma; the novel element of this analytic solution is a closed-form solution for the parallel-flow velocity variation along magnetic field lines. It is shown that only T(perpendicular)/B remains constant along the field line, while density, parallel temperature, and parallel Mach number vary.
Gombosi, Tamas I.↗
Direct analytical solutions to non-uniform beam problems
The direct analytical solution to the vibration of non-uniform beams with and without discontinuities and with various boundary conditions is presented. Results are compared to results from the exact solution for certain cases where the exact solution has been obtained. It is shown that the direct solution converges to the exact solution, in fact, with 'indefinite accuracy' just as Hamilton stated that it would.
Bailey, C. D.↗
Steady hydromagnetic flows in open magnetic fields. I - A class of analytic solutions
In the case of an establishment of theoretical models of the hydromagnetic solar wind, the inclusion of the effects of the magnetic field in the solar wind makes it extremely dificult to solve the mathematical problem. This paper has the objective to present a set of particular analytic solutions. The general formulation of Tsinganos (1982) is used to identify a class of analytic solutions to the equations of steady hydromagnetic flows in spherical coordinates. Flow in an open magnetic field are studied, taking into account the problem in dimensionless form, the special case of radial flows with alpha = 0, general radial flows, illustrative examples for flows in which alpha is not equal to 0, a parametric study of nonradial flows in which alpha is not equal to zero, variations in the parameter nu, and variations in the initial speed eta.
Low, B. C.↗
Analytical Solutions for Seismic Fluid-Structure Interaction of Head-Supported Cylindrical Tanks
Fluid-structure interaction (FSI) of earthquake shaking is a design consideration in the civil, nuclear, chemical, and mechanical industries. Preliminary sizing of structures and components in these industries will often use analytical solutions because there is insufficient information to warrant the analysis of numerical models. Here, this study extends prior analytical work on seismic FSI of base-supported tanks in the 1980s to accommodate head-supported tanks with geometries similar to those proposed for new-build, liquid (fluid)-filled, advanced reactors. Analytical solutions are derived for flexible, head-supported, cylindrical tanks subjected to small-amplitude, unidirectional, horizontal seismic motion, and address frequencies of lateral motions of the tank, hydrodynamic pressures, frequencies of waves, wave heights, and reactions at the head support. Solutions are presented for a range of tank dimensions and could be used for a) preliminary design of head-supported cylindrical tanks and their supporting structures with idealized seismic input, and b) verification of numerical models to be used for final seismic analysis, design, and probabilistic risk assessment.
42 ENGINEERING↗
Analytic solutions of the DGLAP evolution and theoretical uncertainties
The energy dependence for the singlet sector of Parton Distributions Functions (PDFs) is described by an entangled pair of ordinary linear differential equations. Although there are no exact analytic solutions, it is possible to provide approximated results depending on the assumptions and the methodology adopted. These results differ in their sub-leading, neglected terms and ultimately they are associated with different treatments of the theoretical uncertainties. In this work, a novel analytic approach in Mellin space is presented and a new methodology for obtaining closed and exponentiated analytic solutions is devised. Different results for the DGLAP evolution at Next-Leading-Order are compared, discussing advantages and disadvantages for each solution. The generalizations to higher orders are addressed.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
Convenient analytical solution for vibrational distribution function of molecules colliding with a wall
We study formation of the Vibrational Distribution Function (VDF) in a molecular gas at low pressure, when vibrational levels are excited by electron impact and deactivated in collisions with walls and show that this problem has a convenient analytical solution that can be used to obtain VDF and its dependence on external parameters. The VDF is determined by excitation of vibrational levels by an external source and deactivation in collisions with the wall. Deactivation in wall collisions is little known process. However, we found that the VDF is weakly dependent on the functional form of the actual form of probability ${\gamma }_{\upsilon ^{\prime} \to \upsilon }$ for a vibrational number $\upsilon ^{\prime} $ to transfer into a lower level $\upsilon $ at the wall. Because for a given excitation source of vibrational states, the problem is linear the solution for VDF involves solving linear matrix equation. The matrix equation can be easily solved if we approximate probability, in the form: ${\gamma }_{\upsilon ^{\prime} \to \upsilon }=(1/\upsilon ^{\prime} )\theta (\upsilon ^{\prime} -\upsilon ).$ In this case, the steady-state solution for VDF($\upsilon $) simply involves a sum of source rates for levels above $\upsilon ,$ with a factor of $1/(\upsilon +1).$ As an example of application, we study the vibrational kinetics in a hydrogen gas and verify the analytical solution by comparing with a full model.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
ANALYTIC SOLUTION FOR ATMOSPHERIC DENSITY FROM SATELLITE MEASUREMENTS OF STELLAR REFRACTION
Analytic solution for atmospheric density from satellite observations of stellar refraction
STAR↗
Analytical solution of Euler-Lagrange equations for optimum coast trajectories
Analytical solution of Euler-Lagrange equations for coast trajectory optimization
TRAJECTORY OPTIMIZATION↗