Power-law Profile Research Articles

Scattering of flexural waves by a pit of quadratic profile inserted in an infinite thin plate

An acoustic black hole (ABH) is a pit of power law profile inserted in a plate with internal damping. Such a pit with varying thickness leads to local variations of wave propagation properties and has been found to be an efficient vibration damper. In this paper, the ABH is seen as a penetrable obstacle and is studied through its scattering properties. A wave based model is developed within the framework of the Kirchhoff theory for a thin plate of locally varying thickness. It is shown that analytical solutions can be found in the case of a pit of quadratic profile and for uniform damping properties (without added layer). A major outcome of the model is the derivation of the dispersion relations giving a detailed analysis of the behavior of the waves within the ABH. Particularly, cut-on frequencies below which no wave propagates within the ABH are derived and thus give interpretation of the well known typical ABH efficiency threshold frequency for damping vibrations. The analysis is led from numerical computations of the dispersion curves, the scattering diagrams and the scattering cross-section, which are compared to the case of a simple hole. The results show that the ABH behaves as a resonant scatterer, which is a key outcome of this study. The so-called trapped modes, which describe free damped oscillations of the ABH, are responsible for variations of the scattering cross-section with frequency. These variations are investigated thanks to a parametric study on the geometrical properties of the ABH.

Passive Vibration Control Based on Embedded Acoustic Black Holes

This paper uses finite element method to simulate the passive vibration control which is able to improve the overall performance and the operational bandwidth. The vibration control is based on dynamic structural tailoring achieved via acoustic black holes (ABH) with the local thickness varying according to power-law profile. The ABH is a passive technique which uses properties of wave propagation in structures with gradual decrease of thickness that leads to the decrease of phase and group velocities of flexural waves, which makes the ABH has the ability to reduce the structural vibrations after the wave pass through the ABH. However, because real manufacturing cannot develop ABH with zero residual thickness, this nonzero residual thickness will induce the corresponding reflection coefficients are far from zero. In this paper, two types of damping mechanism are attached to the surface of plate: (1) damping layers and (2) coupled electro–mechanical system in order to reduce the structure vibrations. The effects of different number of ABHs, different thickness of damping layers, and different configurations of electrical circuitry are also explored. In this study, the performances of ABH-based passive and semipassive vibration control are explored using numerical simulations of a two-dimensional plate with embedded ABHs. Results show that the ABH based design can enhance the performance of vibration control under steady-state response.

Linear relation between H i circular velocity and stellar velocity dispersion in early-type galaxies, and slope of the density profiles

We report a tight linear relation between the HI circular velocity measured at 6 $R_{\rm e}$ and the stellar velocity dispersion measured within 1 $R_{\rm e}$ for a sample of 16 early-type galaxies with stellar mass between $10^{10}$ and $10^{11}$ $\mathrm{M}_\odot$. The key difference from previous studies is that we only use spatially resolved $v_\mathrm{circ}$(HI) measurements obtained at large radius for a sizeable sample of objects. We can therefore link a kinematical tracer of the gravitational potential in the dark-matter dominated outer regions of galaxies with one in the inner regions, where baryons control the distribution of mass. We find that $v_\mathrm{circ}$(HI) = 1.33 $\sigma_\mathrm{e}$ with an observed scatter of just 12 percent. This indicates a strong coupling between luminous and dark matter from the inner- to the outer regions of early-type galaxies, analogous to the situation in spirals and dwarf irregulars. The $v_\mathrm{circ}$(HI)-$\sigma_\mathrm{e}$ relation is shallower than those based on $v_\mathrm{circ}$ measurements obtained from stellar kinematics and modelling at smaller radius, implying that \vcirc\ declines with radius -- as in bulge-dominated spirals. Indeed, the value of $v_\mathrm{circ}$(HI) is typically 25 percent lower than the maximum $v_\mathrm{circ}$ derived at $\sim0.2\ R_\mathrm{e}$ from dynamical models. Under the assumption of power-law total density profiles $\rho \propto r^{-\gamma}$, our data imply an average logarithmic slope $\langle\gamma\rangle=2.18\pm0.03$ across the sample, with a scatter of 0.11 around this value. The average slope and scatter agree with recent results obtained from stellar kinematics alone for a different sample of early-type galaxies.

Comparison of cosmological models using standard rulers and candles

In this paper, we used standard rulers and standard candles (separately and jointly) to explore five popular dark energy models under the assumption of the spatial flatness of the Universe. As standard rulers, we used a data set comprised of 118 galactic scale strong lensing systems (individual standard rulers if properly calibrated for the mass density profile) combined with BAO diagnostics (statistical standard ruler). Type Ia supernovae served as standard candles. Unlike most previous statistical studies involving strong lensing systems, we relaxed the assumption of a singular isothermal sphere (SIS) in favor of its generalization: the power-law mass density profile. Therefore, along with cosmological model parameters, we fitted the power law index and its first derivative with respect to the redshift (thus allowing for mass density profile evolution). It turned out that the best fitted γ parameters are in agreement with each other, irrespective of the cosmological model considered. This demonstrates that galactic strong lensing systems may provide a complementary probe to test the properties of dark energy. The fits for cosmological model parameters which we obtained are in agreement with alternative studies performed by other researchers. Because standard rulers and standard candles have different parameter degeneracies, a combination of standard rulers and standard candles gives much more restrictive results for cosmological parameters. Finally, we attempted an analysis based on model selection using information theoretic criteria (AIC and BIC). Our results support the claim that the cosmological constant model is still best and there is no (at least statistical) reason to prefer any other more complex model.

CLASH: JOINT ANALYSIS OF STRONG-LENSING, WEAK-LENSING SHEAR, AND MAGNIFICATION DATA FOR 20 GALAXY CLUSTERS*

ABSTRACT We present a comprehensive analysis of strong-lensing, weak-lensing shear and magnification data for a sample of 16 X-ray-regular and 4 high-magnification galaxy clusters at 0.19 ≲ z ≲ 0.69 selected from Cluster Lensing And Supernova survey with Hubble (CLASH). Our analysis combines constraints from 16-band Hubble Space Telescope observations and wide-field multi-color imaging taken primarily with Suprime-Cam on the Subaru Telescope, spanning a wide range of cluster radii (10″–16′). We reconstruct surface mass density profiles of individual clusters from a joint analysis of the full lensing constraints, and determine masses and concentrations for all of the clusters. We find the internal consistency of the ensemble mass calibration to be ≤5% ± 6% in the one-halo regime (200–2000 kpc h −1) compared to the CLASH weak-lensing-only measurements of Umetsu et al. For the X-ray-selected subsample of 16 clusters, we examine the concentration–mass (c–M) relation and its intrinsic scatter using a Bayesian regression approach. Our model yields a mean concentration of c ∣ z = 0.34 = 3.95 ± 0.35 at M 200c ≃ 14 × 1014 M ⊙ and an intrinsic scatter of σ ( ln c 200 c ) = 0.13 ± 0.06 , which is in excellent agreement with Λ cold dark matter predictions when the CLASH selection function based on X-ray morphological regularity and the projection effects are taken into account. We also derive an ensemble-averaged surface mass density profile for the X-ray-selected subsample by stacking their individual profiles. The stacked lensing signal is detected at 33σ significance over the entire radial range ≤4000 kpc h −1, accounting for the effects of intrinsic profile variations and uncorrelated large-scale structure along the line of sight. The stacked mass profile is well described by a family of density profiles predicted for cuspy dark-matter-dominated halos in gravitational equilibrium, namely, the Navarro–Frenk–White (NFW), Einasto, and DARKexp models, whereas the single power-law, cored isothermal and Burkert density profiles are disfavored by the data. We show that cuspy halo models that include the large-scale two-halo term provide improved agreement with the data. For the NFW halo model, we measure a mean concentration of c 200 c = 3.79 − 0.28 + 0.30 at M 200 c = 14.1 − 1.0 + 1.0 × 10 14 M ⊙ , demonstrating consistency between the complementary analysis methods.

On wetting characteristics of droplet on a spherical particle in film boiling regime

This study reports droplet-particle interaction of size ratio less than unity in the film boiling regime on a highly thermally conductive spherical particle surface. Specifically, the effects of impact Weber number (We) of subcooled state droplets comprising water (We=3.9–103.6) and isopropyl alcohol (IPA) (We=8.6–194.6) were studied using high speed imaging technique in the particle temperature range of 250–350°C. In general, non-wetting interaction behaviour was observed with two distinct outcomes – rebound and complete disintegration demarcated by a critical Weber number range instead of a single threshold value. Extent of surface wetting was characterised by the maximum droplet spread diameter parameter which was found to scale with impact Weber number in a power law form which agrees with the theoretical scaling argument. Additionally, an energy balance model was developed to compute this parameter which provided good agreement with the experimental measurements in the lower Weber number regime, however, higher deviations were noted near the transition regime. Also quantified from experiments was the droplet-particle contact time which exhibited a power law dependency on Weber number in the rebound regime, however, was noted to be almost independent of Weber number in the disintegration regime. Particle surface wettability was characterised by the experimentally measured dynamic contact angles which were found to vary in the range of 120–160o in low Weber number regime manifesting the hydrophobic nature of particle surface in film boiling regime. Also, all the parameters such as contact line velocity, particle temperature and droplet size apparently had relatively insignificant influence on the variation of dynamic contact angle. Temporal variation of non-dimensional spreading parameter exhibited a self-similar behaviour wherein all data collapsed on a single power law profile. It was further shown that the behaviour could also be described by a recovery type exponential profile through suitable non-dimensionalization and both profiles can be utilized to produce a spreading kinetics.

Slots of Power-Law Profile as Acoustic Black Holes for Flexural Waves in Metallic and Composite Plates

A new method of damping flexural vibrations in plate-like structures based on the ‘acoustic black hole effect’ has been recently developed and investigated. As ‘acoustic black holes’, one-dimensional elastic wedges of power-law profile covered by narrow strips of absorbing layers near sharp edges have been used initially. The addition of such power-law profiled wedges to edges of rectangular plates or strips results in substantial increase in damping of resonant flexural vibrations in such plates or strips due to the more efficient absorption of flexural waves at the tips of power-law wedges. One of the problems faced by this method of damping is having the wedge tips exposed on the outer edges of the plate or strip. One of the solutions to this problem is to move the wedges inside a plate, so that they form edges of power-law slots within the plate. The present paper reports the results of the experimental investigations into the effects of such slots on damping flexural vibrations. The obtained experimental results show that introducing power-law profiled slots within plates represents an effective method of damping flexural vibrations, which is comparable with the method using power-law wedges at plate edges.

SNe Ia AND THEIR ENVIRONMENT: THEORY AND APPLICATIONS TO SN 2014J

ABSTRACT We present theoretical semi-analytic models for the interaction of stellar winds with the interstellar medium (ISM) or prior mass loss implemented in our code SPICE, assuming spherical symmetry and power-law ambient density profiles and using the Π-theorem. This allows us to test a wide variety of configurations, their functional dependencies, and to find classes of solutions for given observations. Here, we study Type Ia Supernova (SN Ia) surroundings of single and double degenerate systems, and their observational signatures. Winds may originate from the progenitor prior to the white dwarf (WD) stage, the WD, a donor star, or an accretion disk (AD). For M Ch explosions, the AD wind dominates and produces a low-density void several light years across, surrounded by a dense shell. The bubble explains the lack of observed interaction in late time SN light curves for, at least, several years. The shell produces narrow ISM lines Doppler shifted by 10–100 km s−1, and equivalent widths of ≈100 mÅ and ≈1 mÅ in cases of ambient environments with constant density and produced by prior mass loss, respectively. For SN2014J, both mergers and M Ch mass explosions have been suggested based on radio and narrow lines. As a consistent and most likely solution, we find an AD wind running into an environment produced by the red giant wind of the progenitor during the pre-WD stage, and a short delay, 0.013–1.4 Myr, between the WD formation and the explosion. Our framework may be applied more generally to stellar winds and star formation feedback in large scale galactic evolution simulations.

Highly buoyant bent-over plumes in a boundary layer

Highly buoyant plumes, such as wildfire plumes, in low to moderate wind speeds have initial trajectories that are steeper than many industrial waste plumes. They will rise further into the atmosphere before bending significantly. In such cases the plume's trajectory will be influenced by the vertical variation in horizontal velocity of the atmospheric boundary layer. This paper examined the behavior of a plume in an unstratified environment with a power-law ambient velocity profile. Examination of previously published experimental measurements of plume trajectory show that inclusion of the boundary layer velocity profile in the plume model often provides better predictions of the plume trajectory compared to algebraic expressions developed for uniform flow plumes. However, there are many cases in which uniform velocity profile algebraic expressions are as good as boundary layer models. It is shown that it is only important to model the role of the atmospheric boundary layer velocity profile in cases where either the momentum length (square root of source momentum flux divided by the reference wind speed) or buoyancy length (buoyancy flux divided by the reference wind speed cubed) is significantly greater than the plume release height within the boundary layer. This criteria is rarely met with industrial waste plumes, but it is important in modeling wildfire plumes.

The Mathematical Treatment for Air Pollutant Diffusion Using Laplace Adomain Decomposition Method with Ground Surface

The method of Laplace Adomain Decomposition has been used to obtain a semi-analytical solution of the threedimensional steady state advection diffusion equation for dispersion of air pollutant from a point source. The present treatment takes into account a realistic boundary condition which considers the ground surface as an absorberreflector surface for the pollutant, simultaneously. This physical consideration is achieved by assuming that the vertical eddy diffusivity coefficient should be non-zero at the ground surface for vertical diffusion to be possible. The wind prevailing speed is parameterized in terms of vertical height using the power law profile. An upper boundary condition assuming capping inversion is considered which means that pollutant is subjected to a boundary Condition of zero flux. The present model calculations are compared with the available data of the atmospheric dispersion experiments that were carried out in the Copenhagen area (Denmark) and the semi-empirical model for Gaussian plume model with the same input data. In both comparison tasks, the results are reasonably good which indicates that the present treatment performs well as a simple analytical dispersion model.

Exact Axisymmetric Solutions of the 2-D Lane-Emden Equations with Rotation

We have derived exact axisymmetric solutions of the two-dimensional Lane-Emden equations with rotation. These solutions are intrinsically favored by the differential equations regardless of any adopted boundary conditions and the physical solutions of the Cauchy problem are bound to oscillate about and remain close to these intrinsic solutions. The isothermal solutions are described by power-law density profiles in the radial direction, whereas the polytropic solutions are described by radial density profiles that are powers of the zeroth-order Bessel function of the first kind. Both families of solutions decay exponentially in the vertical direction and both result in increasing or nearly flat radial rotation curves. The results are applicable to gaseous spiral-galaxy disks that exhibit flat rotation curves and to the early stages of protoplanetary disk formation before the central star is formed.

The Case against Dark Matter and Modified Gravity: Flat Rotation Curves Are a Rigorous Requirement in Rotating Self-Gravitating Newtonian Gaseous Discs

By solving analytically the various types of Lane-Emden equations with rotation, we have discovered two new coupled fundamental properties of rotating, self-gravitating, gaseous disks in equilibrium: Isothermal disks must, on average, exhibit strict power-law density profiles in radius $x$ on their equatorial planes of the form $A x^{k-1}$, where $A$ and $k-1$ are the integration constants, and "flat" rotation curves precisely such as those observed in spiral galaxy disks. Polytropic disks must, on average, exhibit strict density profiles of the form $\left[\ln(A x^k)\right]^n$, where $n$ is the polytropic index, and "flat" rotation curves described by square roots of upper incomplete gamma functions. By "on average," we mean that, irrespective of the chosen boundary conditions, the actual profiles must oscillate around and remain close to the strict mean profiles of the analytic singular equilibrium solutions. We call such singular solutions the "intrinsic" solutions of the differential equations because they are demanded by the second-order equations themselves with no regard to the Cauchy problem. The results are directly applicable to gaseous galaxy disks that have long been known to be isothermal and to protoplanetary disks during the extended isothermal and adiabatic phases of their evolution. In galactic gas dynamics, they have the potential to resolve the dark matter--modified gravity controversy in a sweeping manner, as they render both of these hypotheses unnecessary. In protoplanetary disk research, they provide observers with powerful new probing tool, as they predict a clear and simple connection between the radial density profiles and the rotation curves of self-gravitating disks in their very early (pre-Class 0 and perhaps the youngest Class Young Stellar Objects) phases of evolution.

An Analytical Dispersion Model for Sources in the Atmospheric Surface Layer with Dry Deposition to the Ground Surface

ABSTRACTThis study presents a dispersion model based on an analytical solution of two-dimensional advection-diffusion equation to simulate the crosswind integrated concentration in the bounded atmospheric boundary layer. The dispersion model considers the power law profiles of wind speed and vertical eddy diffusion coefficient and deposition velocity at the ground surface. A closed form analytical solution of the advection-diffusion equation for these parameterizations of wind and eddy diffusivity was derived by considering the deposition through lower boundary condition at the ground surface. A sensitivity analysis of the dispersion model for ground level concentrations with varying deposition velocities at the ground surface was performed. Hanford diffusion experiment for a depositing tracer was used to evaluate the dispersion model in stable condition. Model was evaluated by accounting both deposition and non-deposition at the ground surface. The simulated concentrations from the proposed dispersion model with dry deposition are in good agreement with those observed and the model predicts ~72% cases within a factor of two. It was shown that the consideration of no-deposition condition in the model for a depositing tracer gives rise to a larger over-prediction to the observations in comparison to with deposition.

Dark matter distribution and annihilation at the Galactic center

We describe a promising method for measuring the total dark matter mass near a supermassive black hole at the Galactic center based on observations of nonrelativistic precession of the orbits of fast S0 stars. An analytical expression for the precession angle has been obtained under the assumption of a power-law profile of the dark matter density. The awaited weighing of the dark matter at the Galactic center provides the strong constraints on the annihilation signal from the neuralino dark matter particle candidate. The mass of the dark matter necessary for the explanation of the observed excess of gamma-radiation owing to the annihilation of the dark matter particles has been calculated with allowance for the Sommerfeld effect.

Characteristics of the Drag Coefficient over a Tropical Environment in Convective Conditions*

Abstract The turbulent data over a tropical region are utilized to analyze the observational behavior of the drag coefficient with respect to wind speed U and the stability parameter in convective conditions. The drag coefficient is observed to follow the power-law profile with respect to U, with large values in low winds and relatively lower values with moderate-wind conditions. Depending on the stability regimes, regression curves for with U are proposed. The variation of with is bounded by a curve. This curve first shows increasing behavior with until it reaches a peak at and then decreases with increasing instability. A mathematical analysis based on Monin–Obukhov similarity (MOS) reveals that increases monotonically with increasing instability. This suggests that MOS theory is able to capture the increasing nature of in weakly to moderately unstable conditions. However, it is unable to explain the observed decreasing behavior of with in moderately to strongly unstable conditions in the tropics within the framework of commonly used similarity functions.

Scattering effects induced by imperfections on an acoustic black hole placed at a structural waveguide termination

The so-called “acoustic black hole” (ABH) effect is a passive vibration control technique based on the flexural waves properties in thin structure of varying thickness. A usual implementation consists in using a plate with tapered extremity with a power-law profile, covered with a thin damping layer. The inhomogeneity of the structure leads to a decrease of flexural wave speed and an increase of their amplitude, therefore resulting in an efficient energy dissipation if damping layer is placed where the thickness is minimal. The manufacture of an efficient extremity is difficult because of the small thickness, and often generates imperfections and tearing. Moreover, previous works suggest that multiple flexural modes are propagating across the width of the ABH tip. A model of an ABH multimodal waveguide taking into account an imperfect termination is developed. It shows that an elementary imperfection can affect the reflection coefficient of the extremity and reduce it. Scattering and propagation properties of the extremity are also studied. An incident mode excites several modes that are localised in the tapered region and local resonances explain the drops in the reflection coefficient. Experimental evidence of the influence of the imperfection on the reflection coefficient is provided. A key result of the paper is that manufacturing imperfections are not detrimental to the ABH effect.

Damage imaging in a laminated composite plate using an air-coupled time reversal mirror

We demonstrate the possibility of selectively imaging the features of a barely visible impact damage in a laminated composite plate by using an air-coupled time reversal mirror. The mirror consists of a number of piezoelectric transducers affixed to wedges of power law profiles, which act as unconventional matching layers. The transducers are enclosed in a hollow reverberant cavity with an opening to allow progressive emission of the ultrasonic wave field towards the composite plate. The principle of time reversal is used to focus elastic waves at each point of a scanning grid spanning the surface of the plate, thus allowing localized inspection at each of these points. The proposed device and signal processing removes the need to be in direct contact with the plate and reveals the same features as vibrothermography and more features than a C-scan. More importantly, this device can decouple the features of the defect according to their orientation, by selectively focusing vector components of motion into the object, through air. For instance, a delamination can be imaged in one experiment using out-of-plane focusing, whereas a crack can be imaged in a separate experiment using in-plane focusing. This capability, inherited from the principle of time reversal, cannot be found in conventional air-coupled transducers.

Measuring angular diameter distances of strong gravitational lenses

The distance-redshift relation plays a fundamental role in constraining cosmological models. In this paper, we show that measurements of positions and time delays of strongly lensed images of a background galaxy, as well as those of the velocity dispersion and mass profile of a lens galaxy, can be combined to extract the angular diameter distance of the lens galaxy. Physically, as the velocity dispersion and the time delay give a gravitational potential (GM/r) and a mass (GM) of the lens, respectively, dividing them gives a physical size (r) of the lens. Comparing the physical size with the image positions of a lensed galaxy gives the angular diameter distance to the lens. A mismatch between the exact locations at which these measurements are made can be corrected by measuring a local slope of the mass profile. We expand on the original idea put forward by Paraficz and Hjorth, who analyzed singular isothermal lenses, by allowing for an arbitrary slope of a power-law spherical mass density profile, an external convergence, and an anisotropic velocity dispersion. We find that the effect of external convergence cancels out when dividing the time delays and velocity dispersion measurements. We derive a formula for the uncertainty in the angular diameter distance in terms of the uncertainties in the observables. As an application, we use two existing strong lens systems, B1608+656 (zL=0.6304) and RXJ1131−1231 (zL=0.295), to show that the uncertainty in the inferred angular diameter distances is dominated by that in the velocity dispersion, σ2, and its anisotropy. We find that the current data on these systems should yield about 16% uncertainty in DA per object. This improves to 13% when we measure σ2 at the so-called sweet-spot radius. Achieving 7% is possible if we can determine σ2 with 5% precision.

Triangular libration points in the R3BP under combined effects of oblateness, radiation and power-law profile

We study the effects of oblateness up to J4 of the primaries and power-law density profile (PDP) on the linear stability of libration location of an infinitesimal mass within the framework of restricted three body problem (R3BP), by using a more realistic model in which a disc with PDP is rotating around the common center of the system mass with perturbed mean motion. The existence and stability of triangular equilibrium points have been explored. It has been shown that triangular equilibrium points are stable for 0<μ<μc and unstable for μc⩽μ≤1/2, where μc denotes the critical mass parameter. We find that, the oblateness up to J2 of the primaries and the radiation reduces the stability range while the oblateness up to J4 of the primaries increases the size of stability both in the context where PDP is considered and ignored. The PDP has an effect of about ≈0.01 reduction on the application of μc to Earth–Moon and Jupiter–Moons systems. We find that the comprehensive effects of the perturbations have a stabilizing proclivity. However, the oblateness up to J2 of the primaries and the radiation of the primaries have tendency for instability, while coefficients up to J4 of the primaries have stability predisposition. In the limiting case c=0, and also by setting appropriate parameter(s) to zero, our results are in excellent agreement with the ones obtained previously. Libration points play a very important role in space mission and as a consequence, our results have a practical application in space dynamics and related areas. The model may be applied to study the navigation and stationkeeping operations of spacecraft (infinitesimal mass) around the Jupiter (more massive) -Callisto (less massive) system, where PDP accounts for the circumsolar ring of asteroidal dust, which has a cloud of dust permanently in its wake.

Capillary bridges and capillary forces between two axisymmetric power–law particles

Capillary interactions are fundamentally important in many scientific and industrial fields. However, most existing models of the capillary bridges and capillary forces between two solids with a mediated liquid, are based on extremely simple geometrical configurations, such as sphere–plate, sphere–sphere, and plate–plate. The capillary bridge and capillary force between two axisymmetric power–law profile particles with a mediated constant-volume liquid are investigated in this study. A dimensionless method is adopted to calculate the capillary bridge shape between two power–law profile particles based on the Young–Laplace equation. The critical rupture criterion of the liquid bridge is shown in four forms that produce consistent results. It was found that the dimensionless rupture distance changes little when the shape index is larger than 2. The results show that the power–law index has a significant influence on the capillary force between two power–law particles. This is directly attributed to the different shape profiles of power–law particles with different indices. Effects of various other parameters such as ratio of the particle equivalent radii, liquid contact angle, liquid volume, and interparticle distance on the capillary force between two power–law particles are also examined.