System Of Solid Spheres Research Articles

Modification of the singlet equation for a molecular system of solid spheres near a solid surface in the Percus-Yevick approximation

An analysis of theoretical methods for studying the molecular structure of liquids bordering a solid surface was carried out. It was established that the currently existing nonlinear integral equations for partial distribution functions do not have an analytical solution for spatially inhomogeneous liquids. A linear integral equation having an analytical solution for a system of solid spheres near a solid surface was proposed.

Free Energy and the Equation of State of a System of Solid Spheres in Narrow Cylindrical Pores

In this work, the free energy and the equation of state of a system of solid spheres in narrow cylindrical pores are determined using the combined method of accelerated convergence of series. This method includes the Euler method and a method based on taking the behavior of the system into account at high densities near the dense packing and on the idea of the effective number of the nearest neighbors. The results are compared with the data of a computer experiment for three different values of pore size and in all cases good agreement between theory and experiment was obtained. These are much better than the results of virial expansion, and when the transverse pore sizes increase, better than the results found on the basis of a series in powers of pressure as well. The method makes it possible to estimate the limits of applicability of the method of convergence acceleration, based on the transition from a series in density to a series in terms of pressure. To do this, the density with close packing and the maximum allowable density found by the method of Euler are compared. If these densities are close, then only the Euler method can be used. In the case of a significant difference, a combined method of accelerated convergence is required.

Mesoscale study of particle sedimentation with inertia effect using dissipative particle dynamics

Mesoscale dispersed two-phase flows often involve complicated dynamic behaviors. Grid-based methods within the framework of continuum mechanics are usually difficult to capture certain degree of molecular-level effect, while the molecular dynamics is only practical at extremely small temporal and spatial scales. In this paper, dissipative particle dynamics (DPD) is extended to the investigation of a fluid–solid sphere system with inertia effect within two parallel plates through the modification of DPD weighting function and hence the dynamic parameters. The sphere and walls are composed of frozen DPD particles that are first treated to reach equilibrium state in the simulation. The force on the solid sphere is obtained from all the particles included in the sphere. The drag coefficient of the frozen sphere is evaluated and compared with the classical correlations. The initial value problem for the sedimentation of the sphere is then solved at certain Reynolds numbers, which is consistent with our direct numerical simulation results.

Features of the chemical equilibrium of dimerization in a system of solid spheres

Statistical and thermodynamic analyses of the equilibrium of dimerization in solid-phase systems is performed for a model of the Van der Waals type and the Perkus-Yevik approximation. For the model of Van der Waals type, the simple equation γ = exp[p0(2−λ3)] is obtained for an average activity coefficient (p0 is reduced pressure and λ is bond length in dimer) that describes both positive and negative deviations from the ideal, depending on the change in volume after the elementary act of chemical reaction. It is found that the Perkus-Yevik approximation predicts similar results with more pronounced deviations from the ideal, and the activity coefficient depends on the degree of dissociation as well.

Milk Gelation Studied with Small Angle Neutron Scattering Techniques and Monte Carlo Simulations

The sol-gel transition of fat-free milk by acidification was studied with neutron scattering experiments and Monte Carlo simulations. Spin-echo small angle neutron scattering (SESANS) and ultrasmall angle neutron scattering (USANS) experiments were performed to measure the static structure of milk and yogurt, as well as the aggregation kinetics. Colloidal gelation was simulated from a reaction limited domain (RLCA) to the diffusion limited regime (DLCA) as cluster-cluster aggregation of adhesive, hard spheres on a three-dimensional lattice. Comparisons were drawn between experimental and numerical correlation functions. Milk was modeled as a suspension of casein micelles in water, and its structure was described with a dilute system of solid spheres with a log-normal distribution of sizes. The structure and formation of yogurt were described with a self-affine model, used for systems containing heterogeneities with a wide range of sizes. Simulation speed was increased by 1 order-of-magnitude using a new algorithm to eliminate dead time. Observations by SESANS and USANS of milk particle sizes and yogurt length scales were consistent and agreed well with literature. Kinetic USANS data yielded reliable information about the growth of typical length scale during aggregation. The simulation model predicted the measurement data qualitatively best staying close to the RLCA regime until large structures had formed. Correlation lengths were in good quantitative agreement, but longest simulated length scales were a of factor 2(1)/(2) below experimental findings. We conclude that small, mobile aggregates are formed during the first 3 h, mostly influencing the dimensionality of the system and that large, inert structures are formed from 2 up to 8 h, which determine the typical length scale.

Energy extremals and nonlinear stability in a variational theory of a coupled barotropic flow - Rotating solid sphere system

A new variational principle—extremizing the fixed frame kinetic energy under constant relative enstrophy—for a coupled barotropic flow—rotating solid sphere system is introduced with the following consequences. In particular, angular momentum is transferred between the fluid and the solid sphere through a modeled torque mechanism. The fluid’s angular momentum is therefore not fixed but only bounded by the relative enstrophy, as is required of any model that supports super-rotation. The main results are: At any rate of spin Ω and relative enstrophy, the unique global energy maximizer for fixed relative enstrophy corresponds to solid-body super-rotation; the counter-rotating solid-body flow state is a constrained energy minimum provided the relative enstrophy is small enough, otherwise, it is a saddle point. For all energy below a threshold value which depends on the relative enstrophy and solid spin Ω, the constrained energy extremals consist of only minimizers and saddles in the form of counter-rotating states. Only when the energy exceeds this threshold value can pro-rotating states arise as global maximizers. Unlike the standard barotropic vorticity model which conserves angular momentum of the fluid, the counter-rotating state is rigorously shown to be nonlinearly stable only when it is a local constrained minima. The global constrained maximizer corresponding to super-rotation is always nonlinearly stable.

Phase transitions of the energy-relative enstrophy theory for a coupled barotropic fluid–rotating sphere system

The statistical equilibrium of a coupled barotropic fluid–rotating solid sphere system is simulated using a energy-relative enstrophy spherical model in a wide range of parameter space by Monte Carlo (MC) methods [J.M. Hammersley, D.C. Handscomb, Monte Carlo Methods, Methuen & Co, London, Wiley, New York City, 1964; C.C. Lim, J. Nebus, Vorticity, Statistical Mechanics and Simulations, Springer, Berlin, 2006]. The energy-relative enstrophy model does not have the low temperature defect of the classical energy–enstrophy theory [R.H. Kraichnan, Statistical dynamics of two-dimensional flows, J. Fluid Mech. 67 (1975) 155–175] because of its microcanonical constraint on relative enstrophy. This model also differs from previous work in not fixing the angular momentum. A family of spin–lattice models are derived as convergent finite dimensional approximations to the total kinetic energy. MC simulations are used to calculate the mean nearest neighbor parity as order parameter or indicator of phase transitions in the system. We find that at extremely high energy levels or small negative temperatures, the preferred state is a super-rotational equilibrium state aligned with the planetary spin. This phenomenon has been observed in the Venusian middle atmosphere for many decades but remains difficult to explain [ 〈 http://www-atm.physics.ox.ac.uk/project/virtis/venus-super.html 〉 ]. For large values of the planetary spin compared with the relative enstrophy there appears to be two phase transitions when the temperature varies from a numerically large negative value to a large positive value. The latter arises as constrained energy minimizers.

Analytic Solution for Diffusional Filtration across Granular Beds in Low Knudsen Number Regime

AbstractThe collection efficiency of aerosols in the low Knudsen number region was studied using a system of multiple spheres. Kuwabara's free vorticity model was expanded to include the effects of gas slip at the collector surface, with the collection efficiency due to diffusion obtained analytically and compared with existing experimental results. The results showed that the diffusional collection efficiency increases as the Knudsen number increases due to gas slippage at the collector surface. The obtained analytical solution converged to the existing collection efficiency of a solid sphere system with a Knudsen number of zero, and that of a bubble with an infinite Knudsen number. The comparison of the experimental results with analytic solution in this study shows that the trends agree well. Therefore, this study is a subsequent expansion of the collection efficiency in the finite Knudsen number region, and can be used for a broad range of collector sizes, pressures and temperatures.

Permittivity of a randomly inhomogeneous medium

A theory of permittivity of suspension-type systems is developed that allows one to calculate such optical parameters of inhomogeneous systems as the length of scattering and the transport length. It is shown that, in the Born approximation, which takes into account two-particle correlations in the arrangement of scattering particles, the theoretical and experimental data are in agreement only to within tens of percent. The contribution of three-particle correlations to the permittivity of a system of solid spheres is determined. It is shown that, in describing the optical properties of suspensions with a large difference between the refractive indices of the medium and the particles, it does not suffice to replace the Rayleigh-Gans form factor by the Mie form factor, even under a restriction to two-particle correlations.

Liquid-solid phase transition in a heterogeneous system of solid spheres

A molecular dynamics method is used to study the influence of a heavy dispersed particle on a liquid-solid phase transition in a molecular system of solid spheres. It is shown that the presence of a dispersed particle shifts the transition toward higher densities and pressures. In addition, in the liquid state the heterogeneous system has a lower pressure, whereas in the solid state the pressure is higher than that in the corresponding states of a homogeneous medium. It is established that the pressures of the heterogeneous mixture in the regions before and after the phase transition can be converted to the pressure of the single-component system by introducing various scaling factors (effective densities of the medium).

Convective Mass Transfer onto Multiple Gas Bubbles

An analysis has been performed on the fluid flow and convective mass transfer in a system consisting of many gas bubbles. Both interference effects of neighboring bubbles and the influence of the circulation induced inside the bubbles on the flow field and the mass transfer rate were taken into account in the analysis. Analytical solutions were obtained for the mass transfer rate. The results indicate that both the packing density and the viscosity ratio of fluid spheres and surrounding fluid have significant effects on the flow field and the mass transfer rate. It was found that the mass transfer rate in a system of bubbles becomes much higher than that for a system of solid spheres. Based on the analytical results, discussions are provided concerning the mass transfer onto gas bubbles from liquid flow.

Filtration of Fine Particles by Multiple Liquid Droplet and Gas Bubble Systems

The collection of small particles by a system consisting of multiple fluid spheres such as water droplets or gas bubbles was studied analytically. Kuwabara's free vorticity model based on the solid particle system was extended to include the effects of induced internal circulation inside a liquid droplet or gas bubble system on the flow and the mass transfer rate. Using the resolved flow field, analytic solutions were obtained for the particle collection efficiencies due to diffusion and interception. The results were applied to the problem of particle capture by gas bubbles in liquid or by droplets suspended in gas. The results indicated that the particle collection efficiency by the multiple fluid sphere system is higher than that by a solid sphere system. This is due to the internal circulation which develops inside the fluid spheres.

Pseudopotential study of the thermodynamic characteristics of supercooled Mg and Zn melts

We use a pseudopotential and thermodynamic perturbation theory to study the thermodynamic characteristics of metallic Mg and Zn melts. A system of solid spheres is selected, to be the ground state. Our calculations of the melting temperature, entropy, and volume are in good agreement with experiment, and the calculated results are insensitive to the choice of pseudopotential. As the temperature is lowered the diameter of the solid spheres and the packing density increase. In addition, there is a marked decrease in the melt's self diffusion coefficient, reflecting the possibility of forming an amorphous structure.

Simple method of optimizing structural parameters of system of solid spheres for the calculation of liquid radial function

A thermodynamically consistent representation, in terms of the equation of state, is obtained for the coefficients of the Wertheim-Till direct correlation functions for a system of solid spheres modeling the microstructure of a simple liquid. It is shown that use of the equation of state in Karnahan — Sterling form significantly improves the results for the radial function, giving good agreement with the results of the semiempirical Verlet-Weis approximation for the data of a numerical experiment on a system of solid spheres. Optimization of the sphere diameter using the entropic method proposed earlier is shown to ensure agreement of the nodes of the radial function of the liquid with experimental data. The agreement of the method proposed with other known methods for the calculation and optimization of the radial function of solid spheres is discussed, and its advantages are noted.