Force Approximation Research Articles

Sphere-plate Casimir interaction in (D + 1) -dimensional spacetime

In this paper, we derive the formula for the Casimir interaction energy between a sphere and a plate in (D + 1)-dimensional Minkowski spacetime. It is assumed that the scalar field satisfies the Dirichlet or Neumann boundary conditions on the sphere and the plate. As in the D = 3 case, the formula is of TGTG type. One of our main contributions is deriving the translation matrices which express the change of bases between plane waves and spherical waves for general D. Using orthogonality of Gegenbauer polynomials, it turns out that the final TGTG formula for the Casimir interaction energy can be simplified to one that is similar to the D = 3 case. To illustrate the application of the formula, both large separation and small separation asymptotic behaviors of the Casimir interaction energy are computed. The large separation leading term is proportional to L−D+1 if the sphere is imposed with Dirichlet boundary condition, and to L−D−1 if the sphere is imposed with Neumann boundary condition, where L is distance from the center of the sphere to the plane. For the small separation asymptotic behavior, it is shown that the leading term is equal to the one obtained using proximity force approximation. The next-to-leading order term is also computed using perturbation method. It is shown that when the space dimension D is larger than 5, the next-to-leading order has sign opposite to the leading order term. Moreover, the ratio of the next-to-leading order term to the leading order term is linear in D, indicating a larger correction at higher dimensions.

Metal-impregnated carbon applied as adsorbent for removal of sulphur compounds using fixed-bed column technology

Metal-impregnated carbons (Cu–AC; Ag–AC and Pd–AC) were studied as adsorbents for the desulphurization of liquid fuels. A real gasoline was examined for sulphur compounds. Textural characteristics of adsorbents were determined by nitrogen adsorption/desorption isotherms at 77 K. The adsorption isotherms were obtained by frontal analysis in a single fixed bed at 30°C and 45°C. Breakthrough curves were simulated according to a mathematical model that assumed axially dispersed flow and mass transfer described by a linear driving force approximation and nonlinear adsorption equilibrium reached instantaneously on the external surface of the adsorbents particles. The model was solved numerically by orthogonal collocation in finite elements, using the commercial solver gPROMS. The proposed model matched experimental data reasonably well. Resistance to mass transfer was significant and thought to be due to intraparticle diffusion kinetics. The results confirmed the efficiency of the use of activated carbon (AC) in the adsorption of sulphur compounds, especially when its surface is modified with metals. Comparing adsorption capacities of sulphur compounds from real gasoline, AC–Pd material appeared more selective than other materials, even presenting a behaviour of rapid saturation explained by the presence of other components competing for adsorption sites, reducing their effectiveness in removing sulphur compounds. Both pristine AC and Pd–AC showed good regenerability. The regenerated Pd–AC sorbent can recover about 85% of the desulphurization capacity.

Flight Stability of Mosquitos: A Reduced Model

This paper investigates the stability of mosquito flight. We employ a reduced model that represents the mosquito as a rigid body with two rigid wings. Wing motions derived from analysis of high speed movies are used as inputs to a dynamical model of the mosquito body. Our simulations use a variational integrator with a quasi-steady approximation for aerodynamic forces. We study uniform flight, consisting of periodic wing motions in a body frame that drive a net translation of the body each period. These trajectories are periodic orbits in a suitable translating reference frame. We analyze the stability of body motions in this frame by computing a linearized return map of the periodic orbit along with its eigenvalues and singular values. Long time stability corresponds to eigenvalues all smaller than one in magnitude and depends upon model parameters. For our mosquito geometry, we find that hovering flight is unstable but locate parameter ranges in which forward flight is stable. We also investigate the ef...

Frequency- and time-domain methods related to flutter instability problem

Bridge flutter phenomenon presents an important criterion of instability, which should be considered in the bridge design phase. This paper presents different bridge flutter methods which can be used to solve the flutter problem. Most commonly used frequency-domain approach is based on the formulation of aero elastic forces with frequency dependent coefficients called flutter derivatives. The critical wind speed, as the main critical condition for the onset of flutter is obtained based on these aero elastic forces. Aero elastic forces can be also expressed in the time-domain, using so-called indicial functions. These functions are usually determined from the corresponding flutter derivatives. In situations when fluid-memory effects tend to become small the quasi-steady theory can be used as an approximation of aero elastic forces. A numerical example related to the typical bridge cross-section follows presented approaches. .

Lateral Casimir Force Between Two Sinusoidally Corrugated Eccentric Cylinders Using Proximity Force Approximation

This paper is devoted to the presentation of the lateral Casimir force between two sinusoidally corrugated eccentric cylinders. Despite that applying scattering matrix method explains the problem exactly, procedure of applying this method is somehow complicated specially at nonzero temperature. Using the proximity force approximation (PFA) helps to achieve the lateral Casimir force in a truly explicit manner. We assume the cylinders to be slightly eccentric with similar radiuses and separations much smaller than corrugations' wave length for the validity of PFA. For such short distances the effect of finite conductivity would be non negligible. In addition to the effect of finite conductivity, we investigate thermal corrections of the lateral Casimir force to reduce the inaccuracy of the result obtained by PFA. Assuming the Casimir force density between two parallel plates, the normal Casimir force between two cylinders is obtained. With the aid of additive summation of the Casimir energy between cylinders without corrugation, we obtain the lateral Casimir force between corrugated cylinders.

Diffusion Constant of a Gold Particle Levitated by a Repulsive Casimir Force near a Silica Plate Immersed in Bromobenzene

Brownian motion of a gold particle levitated by a repulsive Casimir force at short distances from a silica plate in bromobenzene is studied. By using a proximity force approximation, the distribution function of the levitation height is calculated. Taking account for the effect of hindered diffusion near a wall, the diffusion constant of a gold particle is expressed as a function of the particle radius.

Chromatographic separation of prebiotic oligosaccharides. Case study: separation of galacto-oligosaccharides on a cation exchanger

Chromatographic separation of prebiotic oligosaccharides, galacto-oligosaccharides from a typical post-reaction mixture containing unreacted substrate lactose and by-products glucose and galactose was investigated. A commercial cation-exchange resin Dowex 50WX4 was chosen as a prospective preparative-scale adsorbent and separation performance of its four ionic forms, H+, Na+, K+ and Ca2+, was tested. Since adsorption equilibrium isotherms were linear within the entire investigated range of concentrations, they were determined by pulse experiments for all saccharides present in a typical post reaction mixture including tri- and tetragalacto-oligosaccharides. From the four counter ions listed above, hydrogen offered the best selectivity and column performance. The selected H+-form of the cation exchanger was further investigated by means of frontal analysis where high ionic strength and elevated viscosity appear and can play a substantial role. Breakthrough curves were measured for monosaccharides, lactose, glucose–lactose mixture and a multicomponent commercial mixture of galacto-oligosaccharides, Vivinal® GOS. The breakthrough curves were successfully described by the dispersive plug flow model with linear driving force approximation.

Energy Loss in the Impact of Elastic Spheres on a Rigid Half-Space in Presence of Adhesion

Adhesion can cause energy losses in asperities or particles coming into dynamic contact resulting in frictional dissipation, even if the deformation occurring is purely elastic. Such losses are of special significance in impact of nanoparticles and friction between surfaces under low contact pressure to hardness ratio. The objective of this work is to study the effect of adhesion during the normal impact of elastic spheres on a rigid half-space, with an emphasis on understanding the mechanism of energy loss. We use finite element method for modeling the impact phenomenon, with the adhesion due to van der Waals force and the short-range repulsion included as body forces distributed over the volume of the sphere. This approach, in contrast with commonly used surface force approximation, helps to model the interactions in a more precise way. We find that the energy loss in impact of elastic spheres is negligible unless there are adhesion-induced instabilities. Significant energy loss through elastic stress waves occurs due to jump-to-contact and jump-out-of-contact instabilities and can even result in capture of the elastic sphere on the half-space.

Casimir interaction between two magnetic metals in comparison with nonmagnetic test bodies

We present the complete results for the dynamic experiment on measuring the gradient of the Casimir force between magnetic (Ni-coated) surfaces of a plate and a sphere. Special attention is paid to the description of some details of the setup, its calibration, error analysis, and background effects. Computations are performed in the framework of the Lifshitz theory at nonzero temperature with an account of analytic corrections to the proximity force approximation and of surface roughness using both the Drude and the plasma model approaches. The theory of magnetic interaction between a sphere and a plate due to domain structure of their surfaces is developed for both out-of-plane and in-plane magnetizations in the absence and in the presence of spontaneous magnetization. It is shown that in all cases the magnetic contribution to the measured force gradients is much smaller than the total experimental error. The comparison between experiment and theory is done using the rigorous statistical method. It is shown that the theoretical approach taking into account dissipation of free electrons is excluded by the data at a 95% confidence level. The approach neglecting dissipation is confirmed by the data at more than 90% confidence level. We prove that the results of experiments with Ni-Ni, Ni-Au, and Au-Au surfaces taken together cannot be reconciled with the approach including free electrons dissipation by the introduction of any unaccounted background force, either attractive or repulsive.

Material dependence of Casimir interaction between a sphere and a plate: First analytic correction beyond proximity force approximation

We derive analytically the asymptotic behavior of the Casimir interaction between a sphere and a plate when the distance between them, $d$, is much smaller than the radius of the sphere, $R$. The leading-order and next-to-leading-order terms are derived from the exact formula for the Casimir interaction energy. They are found to depend nontrivially on the dielectric functions of the objects. As expected, the leading-order term coincides with that derived using the proximity force approximation. Numerical results are presented when the dielectric functions are given by the plasma model or the Drude model, with the plasma frequency (for plasma and Drude models) and relaxation frequency (for Drude model) given by the conventional values used for gold metal. It is found that if plasma model is used instead of the Drude model, the error in the sum of the first two leading terms is at most 2%, while the error in ${\ensuremath{\theta}}_{1}$, the ratio of the next-to-leading-order term divided by $d/R$ to the leading-order term, can go up to 4.5%.

A Study on the Kinetics of Propane-Activated Carbon: Theory and Experiments

Experimental kinetics results of propane in Maxsorb III activated carbon is obtained at temperatures of 10°C and 30°C, and pressures up to 800kPa using a magnetic suspension balance. A multi-gradient linear driving force (LDF) approximation is used for adsorbate uptake as a function of time. The LDF mass-transfer-rate coefficients were thus determined. Using this approach, the experimentally derived LDF coefficients based on independently measured kinetic parameters for propane in the activated-carbon bed agree very well with experimental results. The computational efficiency is gained by adopting this extended LDF model.

Generalized Kutta–Joukowski theorem for multi-vortex and multi-airfoil flow (a lumped vortex model)

For purpose of easy identification of the role of free vortices on the lift and drag and for purpose of fast or engineering evaluation of forces for each individual body, we will extend in this paper the Kutta–Joukowski (KJ) theorem to the case of inviscid flow with multiple free vortices and multiple airfoils. The major simplification used in this paper is that each airfoil is represented by a lumped vortex, which may hold true when the distances between vortices and bodies are large enough. It is found that the Kutta–Joukowski theorem still holds provided that the local freestream velocity and the circulation of the bound vortex are modified by the induced velocity due to the outside vortices and airfoils. We will demonstrate how to use the present result to identify the role of vortices on the forces according to their position, strength and rotation direction. Moreover, we will apply the present results to a two-cylinder example of Crowdy and the Wagner example to demonstrate how to perform fast force approximation for multi-body and multi-vortex problems. The lumped vortex assumption has the advantage of giving such kinds of approximate results which are very easy to use. The lack of accuracy for such a fast evaluation will be compensated by a rigorous extension, with the lumped vortex assumption removed and with vortex production included, in a forthcoming paper.

Investigation of depth effects on the wave exciting low frequency drift forces by different approximation methods

The paper presents an experimental and numerical investigation on the slowly varying wave exciting drift forces acting on a body of simple geometry subjected to bichromatic waves and long crested irregular seas. The body is axis-symmetric about the vertical axis, like a vertical cylinder with a rounded bottom and it is restrained from moving. Three different water depths representing shallow, intermediate and deep waters are considered.The objective is, firstly, to investigate the relation between the water depth, the incident wavelength and the resulting slowly varying drift force. The other objective is to assess the accuracy of different approximations to the solution of the second order problem. With this aim in mind, the quadratic transfer functions are calculated with a boundary element method using several different levels of approximation for the second order forces, as described in the following sentences. The most complete approximation solves the boundary value problem completely up to the second order. The first-order approximation is similar to the former; however, the second order boundary value problem is simplified by neglecting the free surface forcing. The simpler method is of zeroth-order with respect to the difference frequency and it is commonly known as Newman's approximation. A fourth approximation is evaluated, which combines Newman's approximation with a contribution from the second order incident wave potential.A fifth approximation is applied to the vertical second order forces only and it consists on, first, calculate the steady vertical second order force in monochromatic waves corrected by an additional setdown, and second, apply Newman's approximation together with the corrected second order steady vertical forces. The additional setdown is derived from the second order incident bichromatic wave potential as the difference frequency tends to zero.Second order WAMIT code is used for the frequency domain hydrodynamic calculations. The numerical results are compared with experimental data in bichromatic waves and in irregular waves.This study shows that the contribution from the second order velocity potential must be considered for shallow waters calculations to achieve accurate results. For small difference frequencies, second order scattering potential effects are small; therefore, in this case, a good practical approximation consists on considering the second order potential is contributed by the incident waves only.

Demonstration of Angle-Dependent Casimir Force between Corrugations

The normal Casimir force between a sinusoidally corrugated gold coated plate and a sphere was measured at various angles between the corrugations using an atomic force microscope. A strong dependence on the orientation angle of the corrugation is found. The measured forces were found to deviate from the proximity force approximation and are in agreement with the theory based on the gradient expansion including correlation effects of geometry and material properties. We analyze the role of temperature. The obtained results open new opportunities for control of the Casimir effect in micromechanical systems.

Polynomial force approximations and multifrequency atomic force microscopy

We present polynomial force reconstruction from experimental intermodulation atomic force microscopy (ImAFM) data. We study the tip–surface force during a slow surface approach and compare the results with amplitude-dependence force spectroscopy (ADFS). Based on polynomial force reconstruction we generate high-resolution surface-property maps of polymer blend samples. The polynomial method is described as a special example of a more general approximative force reconstruction, where the aim is to determine model parameters that best approximate the measured force spectrum. This approximative approach is not limited to spectral data, and we demonstrate how it can be adapted to a force quadrature picture.

The wave excitation forces on a truncated vertical cylinder in water of infinite depth

When carrying out any numerical modelling it is useful to have an analytical approximation available to provide a check on the accuracy of the numerical results and to give insight into the underlying physics of the system. The numerical modelling of wave energy converters is an efficient and inexpensive method of undertaking initial optimisation and experimentation. Therefore, the main objective of this paper is to determine an analytical approximation for the wave excitation forces on a floating truncated vertical cylinder in water of infinite depth. The approximation is developed by solving appropriate boundary value problems using the method of separation of variables. A graphical representation of the analytical approximation for the truncated vertical cylinder and the cylinder of infinite depth are presented and are compared to the results from a computational fluid dynamics analysis, using a commercial boundary element package. The presented analytical approximation and the computational fluid dynamics analysis results were found to be in good agreement. Furthermore, the presented analytical approximation was found to be in good agreement with independent experimental data.

Thermal Casimir force between nanostructured surfaces

We present detailed calculations for the Casimir force between a plane and a nanostructured surface at finite temperature in the framework of the scattering theory. We then study numerically the effect of finite temperature as a function of the grating parameters and the separation distance. We also infer non-trivial geometrical effects on the Casimir interaction via a comparison with the proximity force approximation. Finally, we compare our calculations with data from experiments performed with nanostructured surfaces.

A toroidal approximation of capillary forces in polydisperse granular assemblies

In this paper we propose a numerical procedure for the prediction of capillary forces in polydisperse granular assemblies at a degree of moisture content that corresponds to the so-called pendular regime. The capillary force model is adopted within the Laplace–Young framework with a toroidal approximation of the liquid bridge geometry. Governing equations are first derived in a form that highlights the role of intrinsic parameters such as inter-particle separation distance, ratio of particle radii and liquid volume. A proper scaling of these equations is adopted so that the solution applies to any particle pair configuration. Numerical integration algorithms are provided in a way that facilitates implementations in macroscopic procedures for computer simulations. A qualitative evaluation is undertaken to highlight model predictions of the effects on capillary force of various intrinsic parameters that characterise the particle pair and liquid bridge. The model is validated against the experimental results provided by Willet et al. (Langmuir 16:9396–9405, 2000) for a wide range of liquid volumes and particle-pair polydispersity.

Dynamic properties in CO2 adsorption on amine-modified MCM-41

Adsorption is an effective method for separating CO2 from flue gases. The fixed-bed adsorption and desorption of CO2 on MCM-41 and amine-modified MCM-41 (APTS-MCM-41) were studied. The dual-site Langmuir model, which considered both chemical adsorption and physical adsorption, was used to fit the experimental data. The resulting thermodynamic parameters, such as the saturated adsorption capacity, equilibrium constant, enthalpy and entropy, suggested the feasibility of the CO2 adsorption on APTS-MCM-41, as well as the reality of the adsorption mechanism. A dynamic model based on the linear driving force approximation for the mass transfer was used to describe the adsorption kinetics of CO2. The influences of the operating conditions on the breakthrough curve were investigated using both experimental and theoretical methods. Meanwhile, the impacts of changing values of the axial dispersion coefficient and the mass transfer coefficient were investigated. It showed that the amine-modified mesoporous materials would be a promising sorbent in the future CO2 capture.

A simple approximation for forces exerted on an AFM tip in liquid

The critical quantity in understanding imaging using an atomic force microscope is the force the sample exerts on the tip. We put forward a simple one-to-one force to water density relationship, explain exactly how it occurs, and in which circumstances it holds. We argue that two wide classes of atomic force microscope (AFM) tip should lead to at least qualitative agreement with our model and represent a significant fraction of AFM tips as currently prepared. This connection between the short-range force and the unperturbed equilibrium water density removes the need to perform simulations for each tip location, conservatively speeding up simulations by around three orders of magnitude compared to current methods that explicitly calculate the force on a tip model at each point in space.