Perturbation Theory Solutions Research Articles

Entropy analysis on the bioconvective peristaltic flow of gyrotactic microbes in Eyring-Powell nanofluid through an asymmetric channel

The educational value of nanofluids in several industrial and biological sectors, particularly in fluid movement systems known as peristalsis, has piqued researchers' interest in studying the peristaltic movement of nanofluids. Additionally, nanoparticles have crucial roles in many engineering and manufacturing processes, including those involving heat exchangers, cooling systems, boilers, MEMS, chemical engineering, laser diode arrays, and cool automotive engines. Various studies have been conducted on this subject. This is done by looking at how migratory gyrotactic microorganisms migrate through an artery that is anisotropically narrowing in a blood-based nanofluid that is non-Newtonian. To comprehend, the Powell-Eyring fluid model is used how the blood's rheology differs from that of a Newtonian fluid. Both Newtonian fluid characteristics and non-Newtonian traits can be seen in this fluid pattern. Equations for continuity, temperature, motile microbes, momentum, and concentration are used to create the mathematical formulation. The series solutions, which are produced using perturbation theory solutions are discussed using graphs for all dominant parameters. Discussion also includes the distribution of temperature, velocity, and swimming microorganisms. Additionally, the effects of wall shear stress, the Nusselt and Sherwood numbers, as well as the phenomena of trapping, are all examined in detail and shown in the graphs. Entropy generation analyses have also been undertaken. The investigation also reveals a crucial behaviour in the use of the heart-lung engine for extracorporeal blood circulation in medicine that may have an impact on the damage of red blood cells as a result of the large fluctuation in wall shear stress. When liquids are transported using arthro pumps and roller pumps in living organs, the results are likewise of significant use. The results are very helpful for executing particle movements in cardiac surgery and may be applicable to the fluid peristaltic pump used in haemodialysis.

Peculiarities of a helium interaction with hydrogen and free electrons from the point of view of the exchange perturbation theory solutions

We consider an adiabatic potential of a Hydrogen-Helium interaction by using a new approach based on the exchange perturbation theory (EPT). This method allows to present results analytically. We find the solution as an expansion in an antisymmetric basis thanks to the proved completeness property for a nonorthogonal set of antisymmetric wave functions. We use the developed invariant form with respect to interatomic electron permutations for both the Hamiltonian describing an unperturbed system and the perturbation operator. The invariant form for both operators allows one to obtain properly antisymmetric corrections to the antisymmetric wave function of the zero approximation. The probability of electron scattering with electron capture by the helium atom is discussed. We use an EPT formalism developed for the time-dependent perturbations. A formalism of the Time-dependent EPT has a form of an invariant perturbation theory that accounts for intercentre electron permutations among the overlapping states. The expressions for the matrix elements of an S-matrix and a T-matrix account for interatomic electron permutations. We use the obtained differential cross section formulas for the description of the electron capture during the scattering processes. The differential cross section of the electron capture by Helium atom has a complex shape of peak for the scattering energy around 0.7 atomic units, associated with the adiabatic creation of the negative ion He(-1).

The formal Kuranishi parameterization via the universal homological perturbation theory solution of the deformation equation

Abstract Using homological perturbation theory, we develop a formal version of the miniversal deformation associated with a deformation problem controlled by a differential graded Lie algebra over a field of characteristic zero. Our approach includes a formal version of the Kuranishi method in the theory of deformations of complex manifolds.

Thermal Hawking radiation of black hole with supertranslation field

Using the analytical solution for the Schwarzschild metric containing supertranslation field, we consider two main ingredients of calculation of the thermal Hawking black hole radiation: solution for eigenmodes of the d’Alambertian and solution of the geodesic equations for null geodesics. For calculation of Hawking radiation it is essential to determine the behavior of both the eigenmodes and geodesics in the vicinity of horizon. The equation for the eigenmodes is solved, first, perturbatively in the ratio O(C)/M of the supertranslation field to the mass of black hole, and, next, non-perturbatively in the near- horizon region. It is shown that in any order of perturbation theory solution for the eigenmodes in the metric containing supertranslation field differs from solution in the pure Schwarzschild metric by terms of order L1/2 = (1 − 2M/r)1/2. In the non-perturbative approach, solution for the eigenmodes differs from solution in the Schwarzschild metric by terms of order L1/2 which vanish on horizon. Using the simplified form of geodesic equations in vicinity of horizon, it is shown that in vicinity of horizon the null geodesics have the same behavior as in the Schwarzschild metric. As a result, the density matrices of thermal radiation in both cases are the same.

Perturbation theory for the refractive index mismatch between the inclusion and the surrounding tissues

Tissue refractive index is one optical contrast mechanism with diagnostic potential, it is very important to investigate the effect of the refractive index mismatch on light propagation through diffusive regions. Here, we present a new analytical solution of perturbation theory for the refractive index mismatch between the small spherical inclusion and the surrounding tissues. The solution has been used to implement fitting procedures in order to obtain the optical properties of a heterogeneous sphere in semi-infinite medium from measurements of diffuse reflectance. Finally, perturbation theory has been validated by comparisons with the results of Monte Carlo simulation. The new perturbation theory would provide a basis for allowing early disease diagnosis and automatic screening.

Case study of the transformation of swells propagating into Sendai Bay

A high-resolution two-dimensional map of swell wavelength and propagating direction was retrieved from a phased array type L-band synthetic aperture radar scene observed on 24 September 2006 over Sendai Bay in the Tohoku region of Japan. Features of the wavelength map were compared with a wavelength map calculated from in situ measurements of the swell period using infinitesimal amplitude surface wave theory. The maps corresponded well in most areas, although large differences appeared in the near-shore area (area I) and an area of complicated bottom topography (area II). Because the spatial resolution of swell parameters was too coarse to investigate the surface waves in area I, wavelet transform (WT) was applied along a swell ray to improve the spatial resolution. In the resulting WT wavelength map, the large difference in area I disappeared. For area II, perturbation theory was introduced for surface waves propagating obliquely on a gentle slope. The large wavelength difference in area II was well explained by the second-order solution of perturbation theory.

Entrainment by a rotating magnetic field of a ferrofluid contained in a cylinder

Entrainment by a rotating magnetic field of a ferrofluid contained in a cylinder is studied on the basis of spin-diffusion theory. The equations for flow velocity and spin velocity, coupled to Maxwell's equations of magnetostatics, are solved in first-harmonic approximation under the assumption that the magnetic field is small compared to the saturation magnetization. The solution leads to a coupled set of nonlinear integral equations, which can be solved numerically by iteration in a recursive scheme by use of the analytic lowest order perturbation theory solution as the initial state. At a critical applied field, the recursive scheme shows bifurcation. At sufficiently high field, the solution with the lower rate of dissipation shows flow in the direction opposite to the rotating applied field.

Mode expansions in the quantum electrodynamics of photonic media with disorder

AbstractWe address two issues in the quantum electrodynamical description of photonic media with some disorder, neglecting material dispersion. When choosing a gauge in which the static potential vanishes, the normal modes of the medium with disorder satisfy a different transversality condition than the modes of the ideal medium. Our first result is an integral equation for optical modes such that all perturbation-theory solutions by construction satisfy the desired transversality condition. Secondly, when expanding the vector potential for the medium with disorder in terms of the normal modes of the ideal structure, we find the gauge transformation that conveniently makes the static potential zero, thereby generalizing work by Glauber and Lewenstein (1991) [15]. Our results are relevant for the quantum optics of disordered photonic crystals.

Getting Wahlquist's metric from a perturbation theory solution

Perturbation theory is a very convenient and powerful tool to study solutions of Einstein's equations, and its use is extensive nowadays. Nevertheless from the formal point of view, the fact of working with an infinite dimensional manifold results in that it can not be guaranteed in general that a particular perturbation solution corresponds to the Taylor expansion of a supposed exact solution with the desired properties. Here we will present a result obtained from a solution of the perturbed Einstein's system following the CMMR scheme for a perfect fluid region with equation of state μ + (1 − n)p = μ0 being μ density and p pressure. Then we will show how it can be reached from a Post-Minkowskian and slow-rotation approximation of the Wahlquist metric. Additionally we show how conditions we got for it to be Petrov type D are automatically satisfied when we try to mimic Wahlquist's solution, and also identify the Wahlquist parameter r0 in terms of physical quantities.

The bar-halo interaction - I. From fundamental dynamics to revised N-body requirements

A galaxy remains near equilibrium for most of its history. Only through resonances can nonaxisymmetric features such as spiral arms and bars exert torques over large scales and change the overall structure of the galaxy. In this paper, we descri be the resonant interaction mechanism in detail and derive explicit criteria for the particl e number required to simulate these dynamical processes accurately using N-body simulations, and illustrate them with numerical experiments. To do this, we perform a direct numerical solution of perturbation theory, in short, by solving for each orbit in an ensemble and make detailed comparisons with N-body simulations. The criteria include: sufficient particle cov erage in phase space near the resonance and enough particles to minimise gravitational potential fluctuations that will change the dynamics of the resonant encounter. These criteria are general in concept and can be applied to any dynamical interaction. We use the bar‐halo interaction as our primary example owing to its technical simplicity and astronomical ubiquity. Some of our more surprising findings are as follows. First, th e Inner-Lindblad-like resonance (ILR), responsible for coupling the bar to the central halo cusp, requires more than O(10 8 ) equal mass particles within the virial radius for a Milky-Way-like bar in an NFW profile (Navarro et al. 1997). Second, orbits that linger nea r the resonance receive more angular momentum than orbits that move through the resonance quickly. Small-scale fluctuations present in state-of-the-art particle-particle simulatio ns can knock orbits out of resonance, preventing them from lingering and, thereby, decrease the torque. This particularly affects the ILR. However, noise from orbiting substructure remains at least an order of magnitude too small to be of consequence. The required particle numbers are sufficiently high for scenarios of interest that apparent convergence in particle number is misleading: the convergence is in the noise-dominated regime. State-of-the-art simula tions are not adequate to follow all aspects of secular evolution driven by the bar-halo interac tion. It is not possible to derive particle number requirements that apply to all situations, e.g . more subtle interactions may be even more difficult to simulate. Therefore, we present a proc edure to test the requirements for individual N-body codes to the actual problem of interest.

Weak elastic anisotropy by perturbation theory

We demonstrate the advantages of adopting a wave-vector-based coordinate system (WCS) for the application of perturbation theory to derive and display approximate expressions for qP- and qS-wave polarization vectors, phase velocities, and group velocities in general weakly anisotropic media. The advantages stem from two important properties of the Christoffel equation when expressed in the WCS: (1) Each element of the Christoffel matrix is identical to a specific stiffness component in the WCS, and (2) the Christoffel matrix of an isotropic medium is diagonal in the WCS. Using these properties, one can easily identify the small components of the Christoffel matrix in the WCS for a weakly anisotropic medium. Approximate solutions to the Christoffel equation are then obtained by straightforward algebraic manipulations, which make our perturbation theory solution considerably simpler than previously published methods. We compare and contrast our solutions with those discussed by other workers. Numerical comparisons between the exact, first-order, and zero-order qS-wave polarization vectors illustrate the accuracy of our approximate formulas. The form of the WCS phase-velocity expressions facilitates the derivation of closed-form, first-order expressions for qP- and qS-wave group-velocity vectors, providing explicit formulas for the direction of propagation of seismic energy in general weakly anisotropic media. Numerical evaluation of our group-velocity expressions demonstrates their accuracy. We discuss problems with the approximate qS-wave group velocities and polarizations in neighboring directions of singularities. Standard methods are used to transform our solutions from the WCS to the acquisition coordinates, as illustrated by application to orthorhombic symmetry.

Simulation of sloshing motions in fixed and vertically excited containers using a 2-D inviscid σ-transformed finite difference solver

Nonlinear effects of standing wave motions in fixed and vertically excited tanks are numerically investigated. The present fully nonlinear model simulates two-dimensional waves in stable and unstable regions of the free-surface flow. Numerical solutions of the governing nonlinear potential flow equations are obtained using a finite-difference time-stepping scheme on adaptively mapped grids. A σ-transformation in the vertical direction that stretches directly between the free surface and bed boundary is applied to map the moving free-surface physical domain onto a fixed computational domain. A horizontal linear mapping is also applied, so that the resulting computational domain is rectangular, and consists of unit square cells. Predictions of small-amplitude free-surface motions in fixed and vertically excited tanks compare well with second order small perturbation theory. For stable steep waves in the vertically excited tank, the free surface exhibits nonlinear behaviour. Parametric resonance is evident in the instability zones, as the amplitudes grow infinitely large, even for small forcing amplitudes. For steep initial amplitudes the predictions differ considerably from the small perturbation theory solution, demonstrating the importance of nonlinear effects. The present numerical model provides a simple way of simulating steep nonbreaking waves. It is computationally quick and accurate. The σ-transformation removes the need for free-surface smoothing for the cases considered herein.

Application of radiative renormalization to electromagnetically induced transparency

Radiative renormalization has been used to deduce and describe electromagnetically induced transparency succinctly in four-wave mixing for both one and two strong fields. We find both additive and nested renormalization terms and give a prescription for obtaining renormalized expressions starting from perturbation theory solutions. The advantages of using a second strong field are also discussed.

Phase equilibria of associating fluids of spherical and chain molecules

The development of equations of state for strongly associating fluids and fluid mixtures has proved over the years to be a difficult problem. The first-order perturbation theory solution of a resummed cluster expansion has been used to investigate the effect of molecular associations on the critical and phase coexistence properties of fluids with one and two off-center attractive sites. The individual molecules are represented by hard-sphere repulsive cores with square-well attractive sites. Model systems comprising chains of hard spheres have also been examined. Isothermal-isobaric Monte Carlo simulations of hard-sphere fluids with one and two attractive sites are shown to be in good agreement with the results of the theory. A simple van der Waals mean-field term is also added to account for the dispersion forces. The critical points and phase equilibria of the associating fluids are determined for various values of the strength and range of the attractive site, as well as for different chain lengths. The theory can treat fluids with strong hydrogen-bonding associations such as the carboxylic acids the aliphatic alcohols, hydrogen fluoride, water, etc.

Investigation of equations for coefficient functions of the Yang-Mills field and their solutions

Coefficient functions in Euclidean four-dimensional space-time are used to study the theory of Yang-Mills fields. Equations and formal solutions that are identical to the perturbation-theory solutions are constructed for them. To eliminate the ultraviolet divergences of the formal solutions, the equations are considered in a pair of function spaces of translationally invariant functions; a projection space is introduced. Renormalized solutions defined in the pair of function spaces are constructed by a projection and iteration method.

Excitation of Neutron Flux Waves in Reactor Core Transients

An analysis of the excitation of neutron flux waves in reactor core transients has been performed. A perturbation theory solution has been developed for the time-dependent thermal diffusion equatio...

WT4 Millimeter Waveguide: TE01Transmission in Waveguide with Axial Curvature

In this paper we examine the relationship between the loss of the TE 01 mode in a multimode circular waveguide and the waveguide geometry. First-order perturbation theory solutions of the coupled line equations were used to predict the loss from measurements of the waveguide geometry. The predicted loss disagreed with the measured loss for the 14 km long waveguide line in the WT4 field evaluation test. An analysis of the coupled line equations which considers the effects of both first- and second-order mode conversion is described. The appropriate coupling coefficients are derived and discussed. Second-order perturbation theory solutions of the coupled line equations are developed and examined. Losses predicted from the measured waveguide curvature and the second-order theory agree well with measured loss data. The results indicate that axial curvature leads to significant mode conversion-reconversion effects between the TE 01 –TM 11 , TM 01 –TE 12 , TM 11 –TM 21 , TE 12 –TM 21 modes in the installed waveguide of the WT4 field evaluation test. The TM 21 conversion is due to curvature fluctuations with a characteristic wavelength of 10 to 30 m.

Comparison of Numerical and Perturbation Solutions of Two-Dimensional Nonlinear Water-Wave Problems

Numerical solutions of the nonlinear problem of the steady two-dimensional potential flow past a submerged line vortex are obtained using the finite-difference iterative technique previously presented by the authors. These solutions are compared in detail with third-order perturbation theory solutions. It is found that very good agreement is obtained for cases of positive circulation of the vortex with strength large enough to produce downstream waves whose steepness is within 15 percent of the maximum possible steepness of irrotational free waves. These computed waves are as steep as the steepest waves obtained in a certain experiment involving the flow past a two-dimensional hydrofoil. For negative circulation, there is substantial difference between the numerical results and third-order perturbation theory. The failure of the perturbation theory is discussed. Details of the far-downstream wave system obtained by the numerical method are compared with other numerical solutions and very high-order perturbation theory solutions of the free-wave problem. Very good agreement is obtained in most cases.

Large-ω Asymptotic Expansions and Padé Approximants for the Harmonic Oscillator

An approximate solution to the Schrödinger equation for a particle in various perturbed harmonic-oscillator potentials is obtained by utilizing large-ω (ω is the natural frequency of unperturbed oscillator) asymptotic expansion theory. Both the basic asymptotic and perturbation theory solutions are obtained and are shown to be related. Explicit high-order perturbation expansions for the energy and mean-square displacement of the symmetric anharmonic oscillator are obtained and are seen to be divergent but asymptotic. It is shown that by rearranging these expansions as Padé approximants, accurate numerical values are obtained for these observables, even for nonperturbative values of the coupling constant.

AB Initio Calculations for Helium-4 of Thermodynamic and Transport Properties of the Gas and the Ground State Energy of the Liquid

A potential energy curve for the 4He diatom obtained from a perturbation theory solution of the Schr�dinger equation is used to calculate the viscosity, thermal conductivity, self-diffusion coefficient, and second virial coefficient of gaseous 4He. In addition, the ground state energy of the zero-temperature liquid is obtained by a variation procedure using a trial wavefunction in conjunction with the Monte Carlo method. The results are in good agreement with experiment.