Full Numerical Solution Research Articles

Constructive A Priori Error Estimates for a Full Discrete Approximation of the Heat Equation

In this paper, we consider the constructive a priori error estimates for a full discrete numerical solution of the heat equation. Our method is based on the finite element Galerkin method with an interpolation in time that uses the fundamental solution for semidiscretization in space. The present estimates play an essential role in the numerical verification method of exact solutions for the nonlinear parabolic equations. This implies that by utilizing the present results we could get the guaranteed a posteriori error estimates for various kinds of nonlinear evolutional problems. Our results can also be considered as an explicit optimal estimate with the limited regularity of solutions, which should be unknown up to the present.

Thermal elastohydrodynamic lubrication of infinite line contact rough surfaces considering pressure and shear flow factors

A full numerical solution of a thermal elastohydrodynamic lubrication of infinite line contact rough surfaces considering pressure and shear flow factors is obtained. The modified average Reynolds equation, elasticity equation and energy equations are solved simultaneously using a modified Houpert and Hamrock fast approach. The present study demonstrates an efficient computational scheme and the combined effects of the stochastic surface roughness with thermal actions on the bearing characteristic with a wide range of load, rolling speed and roughness parameters.

Coalescence of liquid drops: Different models versus experiment

The process of coalescence of two identical liquid drops is simulated numerically in the framework of two essentially different mathematical models, and the results are compared with experimental data on the very early stages of the coalescence process reported recently. The first model tested is the “conventional” one, where it is assumed that coalescence as the formation of a single body of fluid occurs by an instant appearance of a liquid bridge smoothly connecting the two drops, and the subsequent process is the evolution of this single body of fluid driven by capillary forces. The second model under investigation considers coalescence as a process where a section of the free surface becomes trapped between the bulk phases as the drops are pressed against each other, and it is the gradual disappearance of this “internal interface” that leads to the formation of a single body of fluid and the conventional model taking over. Using the full numerical solution of the problem in the framework of each of the two models, we show that the recently reported electrical measurements probing the very early stages of the process are better described by the interface formation/disappearance model. New theory-guided experiments are suggested that would help to further elucidate the details of the coalescence phenomenon. As a by-product of our research, the range of validity of different “scaling laws” advanced as approximate solutions to the problem formulated using the conventional model is established.

Study of a degenerate dipolar Fermi gas of 161Dy atoms

We study the properties of a single-component (spin polarized) degenerate dipolar Fermi gas of 161Dy atoms using a hydrodynamic description. Under axially-symmetric trapping we suggest reduced one- (1D) and two-dimensional (2D) descriptions for the cigar and disc shapes, respectively. In addition to a complete numerical solution of the hydrodynamic model we also consider a variational approximation. For a trapped system under appropriate conditions, the variational approximation as well as the reduced 1D and 2D models are found to yield results for the shape, size and chemical potential of the system in agreement with the full numerical solution of the three-dimensional (3D) model. For the uniform system we consider anisotropic sound propagation in 3D. An analytical result for anisotropic sound propagation in a uniform dipolar degenerate Fermi gas is found to be in agreement with the results of numerical simulation in 3D.

A Refinement of Asymptotic Predictions and Full Numerical Solution of Helicopter Rotor Noise in the Far Field

The asymptotic analysis of Parry and Crighton [1] for propeller noise in the far field, which is based on Hanson's formulation [2] of the FW-H equation, is refined to second order by Laplace's method [3] for evaluating integrals, accounting for second order contributions near the blade tip for loading and thickness noise. The full numerical solution of Hanson's integrals for both thickness and loading noise is also presented. In particular, the theory is applied to a four-bladed helicopter rotor with tip Mach numbers ranging between 0.5 and 0.7. The aerodynamic loading in this case is obtained using a 3D compressible code based on finite volume method with intensified grid density near the blade tip. The far field angular SPL noise distributions of a helicopter rotor in hover show that the present second order asymptotic formula is in better agreement with full numerical computations than that of the first order formula, especially for thickness noise.

Viscous flow in a curved tube filled with a porous medium

Viscous flow in a tube is basic in fluid mechanics. However, there are cases where the tube is filled with a porous medium, such as those in filters, catalytic reactors or matrix-filled biological pores. In these cases the appropriate governing equation is the DarcyBrinkman equation, where both the resistance of the matrix and the bounding walls are taken into account [1, 2]. There are many articles on the flow in a straight tube filled with a porous medium, notably Refs. [3– 10]. This paper considers the flow in a curved tube filled with a porous medium. Previous literature include the works of Avramenko and Kuznetsov [11, 12] who solved analytically for the flow in a curved channel and a curved rectangular tube. Full numerical solution was also done for a helically coiled tube [13]. We shall introduce a Ritz method to treat the flow through a porous medium in a general curved tube, then apply the method to a curved tube of elliptic cross section. The Ritz method and its convergence have been known in elasticity (e.g. [14]), but its applications

Shear viscosity and spin diffusion in a two-dimensional Fermi gas

We investigate the temperature dependence of the shear viscosity and spin diffusion in a two-dimensional Fermi gas with contact interactions, as realized in ultra-cold atomic gases. We describe the transport coefficients in terms of a Boltzmann equation and present a full numerical solution for the degenerate gas. In contrast to previous works we take the medium effects due to finite density fully into account. This effect reduces the viscosity to entropy ratio, $\eta/s$, by a factor of three, and similarly for spin diffusion. The trap averaged viscosity agrees well with recent measurements by Vogt et al. [Phys. Rev. Lett. 108, 070404 (2012)].

Back-reaction of perturbation wave packets on gray solitons

Within the Bogoliubov-de Gennes linearization theory of quantum or classical perturbations around a background solution to the one-dimensional nonlinear Schr\"odinger equation, we study the back-reaction of wave packet perturbations on a gray soliton background. From our recently published exact solutions, we determine that a wave packet effectively jumps ahead as it passes through a soliton, emerging with a wavelength-dependent forward translation in comparison to its motion in absence of the soliton. From this and from the full theory's exact momentum conservation, we deduce that post-Bogoliubov back-reaction must include a commensurate forward advance by the soliton itself. We quantify this effect with a simple theory, and confirm that it agrees with full numerical solution of the classical nonlinear Schr\"odinger equation. We briefly discuss the implications of this effect for quantum behavior of solitons in quasi-condensed dilute gases at finite temperature.

Approximate Scheme for Calculating van der Waals Interactions between Finite Cylindrical Volume Elements

A successful approach to calculating van der Waals (vdW) forces between irregular bodies is to divide the bodies into small cylindrical volume elements and integrate the vdW interactions between opposing elements. In this context it has been common to use Hamaker's expression for parallel plates to approximate the vdW interactions between the opposing elements. This present study shows that Hamaker's vdW expression for parallel plates does not accurately describe the vdW interactions for co-axial cylinders having a ratio of cylinder radius to separation distance (R/D) of 10 or less. This restricts the systems that can be simulated using this technique and explicitly excludes consideration of topographical or compositional variations at the nanoscale for surfaces that are in contact or within a few nm of contact. To address this limitation, approximate analytical expressions for nonretarded vdW forces between finite cylinders in different orientations are derived and are shown to produce a high level of agreement with forces calculated using full numerical solutions of the corresponding Hamaker's equations. The expressions developed here allow accurate calculation of vdW forces in systems where particles are in contact or within a few nm of contact with surfaces and the particles and/or surfaces have heterogeneous nanoscale morphology or composition. These calculations can be performed at comparatively low computational cost compared to the full numerical solution of Hamaker's equations.

All bent out of shape: buckling of sheared fluid layers

AbstractBuckling instabilities of thin sheets or plates of viscous fluid occur in situations ranging from food and polymer processing to geology. Slim, Teichman & Mahadevan (J. Fluid Mech., this issue, vol. 694, 2012, pp. 5–28) study numerically the buckling of a sheared viscous plate floating on a denser fluid using three approaches: a classical ‘thin viscous plate’ model; full numerical solution of the three-dimensional Stokes equations; and a novel ‘advection-augmented’ thin-plate model that accounts (in an asymptotically inconsistent way) for the advection of perturbations by the background shear flow. The advection-augmented thin-plate model is markedly superior to the classical one in its ability to reproduce the predictions of the Stokes solution, illustrating the utility of judicious violations of asymptotic consistency in fluid-mechanical models.

Framework for discrete-time quantum walks and a symmetric walk on a binary tree

We formulate a framework for discrete-time quantum walks, motivated by classical random walks with memory. We present a specific representation of the classical walk with memory 2, on which this is based. The framework has no need for coin spaces, it imposes no constraints on the evolution operator other than unitarity, and is unifying of other approaches. As an example we construct a symmetric discrete-time quantum walk on the semi-infinite binary tree. The generating function of the amplitude at the root is computed in closed form, as a function of time and the initial level $n$ in the tree, and we find the asymptotic and a full numerical solution for the amplitude. It exhibits a sharp interference peak and a power-law tail, as opposed to the exponentially decaying tail of a broadly peaked distribution of the classical symmetric random walk on a binary tree. The probability peak is orders of magnitude larger than it is for the classical walk (already at small $n$). The quantum walk shows a polynomial algorithmic speedup in $n$ over the classical walk, which we conjecture to be of the order $2/3$, based on strong trends in data.

Natural ventilation driven by periodic gusting of wind

We investigate the mixing of a warm enclosed space by a series of discrete gusts of cold air from a high-level opening. Initially we examine the case of a series of gusts of identical size each modelled as a turbulent buoyant thermal. We develop a model of the filling box-like flow which develops in the space and identify the key parameter in the system as the ratio between the initial gust size and the product of the height of the room and the entrainment coefficient. We find an approximate analytic solution for the evolution of the density profile within the space which is in good agreement with a full numerical solution of the governing equations. We successfully test the predictions of the model with a series of new laboratory experiments. The experiments combined with the model also provide a new independent estimate for the entrainment coefficient of a thermal, ε = 0.37 ± 0.02, based on the propagation speed of a filling box front. We then examine the mixing produced by a series of thermals of non-identical size which we characterize in terms of a mean size and coefficient of variation. We find that as the coefficient of variation increases, the density profile becomes progressively more stratified owing to the asymmetry of dilution through entrainment of large and small thermals. We discuss the implications of these results for the ventilation of a building subject to gusts of wind.

Asymptotic analysis of stresses near a crack tip in a two-dimensional colloidal packing saturated with liquid

The consolidation of colloidal particles in drying colloidal dispersions is influenced by various factors such as particle size and shape, and interparticle potential. The capillary pressure induced by the menisci, formed between the top layer of particles in the packed bed, compresses the bed of particles while the constraints imposed by the boundaries result in tensile stresses in the packing. Presence of flaws or defects in the bed determines its ultimate strength under such circumstances. In this study, we determine the asymptotic stress distribution around a flaw in a two-dimensional colloidal packing saturated with liquid and compare the results with those obtained from the full numerical solution of the problem. Using the Griffith's criterion for equilibrium cracks, we relate the critical capillary pressure at equilibrium to the crack size and the mechanical properties of the packed bed. The analysis also gives the maximum allowable flaw size for obtaining a crack-free packing.

Spin-triplet supercurrent through inhomogeneous ferromagnetic trilayers

Motivated by a recent experiment [J. W. A. Robinson, J. D. S. Witt and M. G. Blamire, Science, \textbf{329}, 5987 (2010)], we here study the possibility of establishing a long-range spin-triplet supercurrent through an inhomogeneous ferromagnetic region consisting of a Ho$\mid$Co$\mid$Ho trilayer sandwiched between two conventional s-wave superconductors. We utilize a full numerical solution in the diffusive regime of transport and study the behavior of the supercurrent for various experimentally relevant configurations of the ferromagnetic trilayer. We obtain qualitatively very good agreement with experimental data regarding the behavior of the supercurrent as a function of the width of the Co-layer, $L_\text{Co}$. Moreover, we find a synthesis of 0-$\pi$ oscillations with superimposed rapid oscillations when varying the width of the Ho-layer which pertain specifically to the spiral magnetization texture in Ho. We are not able to reproduce the anomalous peaks in the supercurrent observed experimentally in this regime, but note that the results obtained are quite sensitive to the exact magnetization profile in the Ho-layers, which could be the reason for the discrepancy between our model and the experimental reported data for this particular aspect. We also investigate the supercurrent in a system where the intrinsically inhomogeneous Ho ferromagnets are replaced with domain-wall ferromagnets, and find similar behavior as in the Ho$\mid$Co$\mid$Ho case. Furthermore, we propose a novel type of magnetic Josephson junction including only a domain-wall ferromagnet and a homogeneous ferromagnetic layer, which in addition to simplicity regarding the magnetization profile also offers a tunable long-range spin-triplet supercurrent. Finally, we discuss some experimental aspects of our findings.

Axion cosmology revisited

The misalignment mechanism for axion production depends on the temperature-dependent axion mass. The latter has recently been determined within the interacting instanton liquid model, and provides for the first time a well-motivated axion mass for all temperatures. We reexamine the constraints placed on the axion parameter space in the light of this new mass function. Taking this mass at face value, we find an accurate and updated constraint ${f}_{a}\ensuremath{\le}2.8(\ifmmode\pm\else\textpm\fi{}2)\ifmmode\times\else\texttimes\fi{}{10}^{11}\text{ }\text{ }\mathrm{GeV}$ or ${m}_{a}\ensuremath{\ge}21(\ifmmode\pm\else\textpm\fi{}2)\text{ }\text{ }\ensuremath{\mu}\mathrm{eV}$ from the misalignment mechanism in the classic axion window (thermal scenario). However, this is superseded by axion string radiation which leads to ${f}_{a}\ensuremath{\lesssim}{3.2}_{\ensuremath{-}2}^{+4}\ifmmode\times\else\texttimes\fi{}{10}^{10}\text{ }\text{ }\mathrm{GeV}$ or ${m}_{a}\ensuremath{\gtrsim}{0.20}_{\ensuremath{-}0.1}^{+0.2}\text{ }\text{ }\mathrm{meV}$. In this analysis, we take care to precisely compute the effective degrees of freedom and, to fill a gap in the literature, we present accurate fitting formulas. We solve the evolution equations exactly, and find that analytic results used to date generally underestimate the full numerical solution by a factor 2--3. In the inflationary scenario, axions induce isocurvature fluctuations and constrain the allowed inflationary scale ${H}_{I}$. Taking anharmonic effects into account, we show that these bounds are actually weaker than previously computed. Considering the fine-tuning issue of the misalignment angle in the whole of the anthropic window, we derive new bounds which open up the inflationary window near ${\ensuremath{\theta}}_{a}\ensuremath{\rightarrow}\ensuremath{\pi}$. In particular, we find that inflationary dark matter axions can have masses as high as $0.01--1\text{ }\text{ }\mathrm{meV}$, covering the whole thermal axion range, with values of ${H}_{I}$ up to ${10}^{9}\text{ }\text{ }\mathrm{GeV}$. Quantum fluctuations during inflation exclude dominant dark matter axions with masses above ${m}_{a}\ensuremath{\lesssim}1\text{ }\text{ }\mathrm{meV}$.

Electrical equivalent circuit of an ion-exchange membrane system

Usually, the current flowing through an electrochemical cell is divided into the faradaic current going to an electrochemical interface reaction, and the current charging electric double layer (EDL). This division leads to the Randles–Ershler equivalent circuit with an EDL capacitance in one branch, and the faradaic impedance in the other, specific for each particular system. However, the physics of the separation of the impedance into faradaic and capacitive components for different electrochemical systems is not sufficiently clear. The most of derivations resulting in the formal construction of the Randles–Ershler or similar equivalent circuits are based on the a priori separation of the electroneutral and the double-layer regions. In this paper, we derive an equation for the impedance of a three-layer system consisted of an ion-exchange membrane and two adjoining diffusion boundary layers (DBL) starting from the Poisson equation. The system is polarized by a constant electric current over which a small sinusoidal signal is applied. The equation shows that the impedance of the considered system can be formally interpreted via an equivalent circuit with a frequency dependent capacitance in one branch and a finite-length Warburg-type impedance in the other. To take into account this dependence, the impedance of the system may be presented as a series connection of five circuits. Three of them are consisted of a geometric capacitance connected in parallel with an ohmic resistance, respectively, for both diffusion layers and for the membrane bulk; the two others being the double-layer capacitance in parallel with the finite-length Warburg impedance for the left and the right interfaces, respectively. The comparison of the impedance spectra calculated within our analytical approach with those obtained by the full numerical solution of the Nernst–Planck–Poisson (NPP) equations shows a good agreement. Different possible situations, which might arise in real systems (different stationary current densities, different thicknesses of the left-hand and right-hand DBLs), are analysed when applying the approximate solution.

Non‐Newtonian transient thermoelastohydrodynamic lubrication analysis of an involute spur gear

AbstractThe lubrication design of gear is based on the assumption that the lubricant is an ideal Newtonian fluid. In fact, The Ree–Eyring fluid model is more in agreement with the lubricant rheological behaviour under medium or mild duty load. It is necessary for engineering lubrication design to study the non‐Newtonian transient thermal elastohydrodynamic lubrication (EHL) of gear. In the paper, a full numerical solution to the non‐Newtonian transient thermal EHL of gear is obtained by utilising the multi‐grid method, which has strong numerical stability, a quick convergence rate and a high degree of accuracy. Copyright © 2010 John Wiley & Sons, Ltd.

Theoretical optimization of optical fibre Raman–Brillouin hybrid sensors

This paper proposes a hybrid distributed fibre sensor which works on Raman and Brillouin scattering generated simultaneously from the same laser beam source. The problem is solved by an analytical and a numerical method to determine some important parameters associated with Raman–Brillouin hybrid distributed fibre sensors, by solving the governing differential equations. Optimization of the hybrid sensor's length is proposed and discussed. The governing equations consider pump depletion caused by all involved effects: Rayleigh, spontaneous and stimulated Raman and Brillouin. Solutions are discussed for three important cases: spontaneous Raman and Brillouin scattering, Brillouin stimulated and Raman spontaneous scattering, and finally both Raman and Brillouin stimulated scattering. It is demonstrated that the sensor's length can be maximized in order to obtain the maximum scattered intensity in two different cases, one of them in good agreement with the literature. It is discussed that the differences between the full numerical solution and the analytical solutions occur mainly due to the approximations made. Results may be used in the distributed sensor's design whose goal is to optimize the sensor's length to get the most intense scattering signal. The implementation of the sensor and its feasibility in all cases is also argued. All results can be applied to Raman amplification systems where Brillouin scattering also takes place.

Phase-controlled proximity effect in ferromagnetic Josephson junctions: Calculation of the density of states and the electronic specific heat

We study the thermodynamic properties of a dirty ferromagnetic S$\mid$F$\mid$S Josephson junction with s-wave superconducting leads in the low-temperature regime. We employ a full numerical solution with a set of realistic parameters and boundary conditions, considering both a uniform and non-uniform exchange field in the form of a Bloch domain wall ferromagnetic layer. The influence of spin-active interfaces is incorporated via a microscopic approach. We mainly focus on how the electronic specific heat and density of states (DOS) of such a system is affected by the \textit{proximity effect}, which may be tuned via the superconducting phase difference. Our main result is that it is possible to \textit{strongly modify the electronic specific heat} of the system by changing the phase difference between the two superconducting leads from 0 up to nearly $\pi$ at low temperatures. An enhancement of the specific heat will occur for small values $h\simeq\Delta$ of the exchange field, while for large values of $h$ the specific heat is suppressed by increasing the phase difference between the superconducting leads. These results are all explained in terms of the proximity-altered DOS in the ferromagnetic region, and we discuss possible methods for experimental detection of the predicted effect.

On the gluon spectrum in the glasma

We study the gluon distribution in nucleus–nucleus collisions in the framework of the Color Glass Condensate. Approximate analytical solutions are compared to numerical solutions of the nonlinear Yang–Mills equations. We find that the full numerical solution can be well approximated by taking the full initial condition of the fields in Coulomb gauge and using a linearized solution for the time evolution. We also compare k T -factorized approximations to the full solution.