Hard-sphere Equation Research Articles

Singular Limit for the Compressible Navier–Stokes Equations with the Hard Sphere Pressure Law on Expanding Domains

The article is devoted to the asymptotic limit of the compressible Navier–Stokes system with a pressure obeying a hard–sphere equation of state on a domain expanding to the whole physical space $$\textbf{R}^3$$ . Under the assumptions that acoustic waves generated in the case of ill-prepared data do not reach the boundary of the expanding domain in the given time interval and a certain relation between the Reynolds and Mach numbers and the radius of the expanding domain we prove that the target system is the incompressible Euler system on $$\textbf{R}^3$$ . We also provide an estimate of the rate of convergence expressed in terms of characteristic numbers and the radius of domains.

From Boltzmann Equation for Granular Gases to a Modified Navier–Stokes–Fourier System

In this paper, we give an overview of the results established in Alonso (http://arxiv.org/org/abs/2008.05173, 2020) which provides the first rigorous derivation of hydrodynamic equations from the Boltzmann equation for inelastic hard spheres in 3D. In particular, we obtain a new system of hydrodynamic equations describing granular flows and prove existence of classical solutions to the aforementioned system. One of the main issue is to identify the correct relation between the restitution coefficient (which quantifies the rate of energy loss at the microscopic level) and the Knudsen number which allows us to obtain non trivial hydrodynamic behavior. In such a regime, we construct strong solutions to the inelastic Boltzmann equation, near thermal equilibrium whose role is played by the so-called homogeneous cooling state. We prove then the uniform exponential stability with respect to the Knudsen number of such solutions, using a spectral analysis of the linearized problem combined with technical a priori nonlinear estimates. Finally, we prove that such solutions converge, in a specific weak sense, towards some hydrodynamic limit that depends on time and space variables only through macroscopic quantities that satisfy a suitable modification of the incompressible Navier–Stokes–Fourier system.

On the Effect of the Hard-sphere Term on the Statistical Associating Fluid Theory Equation of State

The hard-sphere system is the reference fluid for most versions of the statistical associating fluid theory (SAFT) equation of state (EoS). In the SAFT-type equations of state, the hard-sphere equation directly affects the reference term and through the radial distribution function, indirectly affects the chain and association terms. Although there are various EoSs for describing the thermodynamic behavior of the hard-sphere fluid, the Carnahan-Starling EoS has traditionally been used to express the hard-sphere contribution in SAFT-type models. In this work, we integrated eight hard-sphere EoS in the simplified SAFT EoS and parameterized the resulting EoS with pressure-density-temperature data. Then, using modified versions of the simplified SAFT (SSAFT), we calculated the thermodynamic properties including pressure, density, temperature, saturated vapor-pressure, isochoric heat capacity, isobaric heat capacity, and speed of sound, for ethane, butane, hexane, and hexatriacontane, and compared the results. In general, the results of calculations show that although the modified Carnahan-Starling, Kolafa, and Carnahan-Starling EoSs have had the best results, considering the simplicity and straightforwardness of the Carnahan-Starling EoS, this EoS is a reasonable choice for developing SAFT-type equations of state.

A combined density gradient theory with equation of state model for the study of surface tension of refrigerant fluids

A combined density gradient theory (DGT) with an equation of state (EoS) model has been employed for describing the surface tension of 26 pure refrigerant fluids. Two EoSs were considered to be employed in the model development. The first one is the Carnahan–Starling hard-sphere equation with the perturbation term of Dohm-Prausnitz (CS-DP EoS) and the second one is the modified version of that EoS, i.e., MCS-DP. Then two different DGT-based models (DGT + CS-DP and DGT + MCS-DP) were employed for calculating the surface tension of pure refrigerants. The surface tension data source was provided from NIST databank and several original papers, as well. Both DGT models used one unique value for the influence parameter for each fluid. The DGT + MCS-DP model with the average absolute deviations (AADs) of 4.23% was found to be almost as accurate as DGT + CS-DP model with AAD of 4.84% for the correlation of NIST data. But in case the calculation of experimental surface tensions, DGT + CS-DP and DGT + MCS-DP models led to AADs of 5.88% and 3.74%, respectively, which reveals very well the superiority of DGT + MCS-DP against DGT + CS-DP one. All calculations on the surface tension of studied refrigerants were carried out in 0.45–0.95 of reduced temperature. Then, the degree of accuracy of DGT + MCS-DP model has also been compared with some simple empirical equation based on the corresponding states principle.

Predicting the Surface Tension of Refrigerants from Density Gradient Theory and Perturbed Hard-sphere Equation of State

The surface tensions of pure refrigerants were predicted using a density gradient theory (DGT) coupled with a perturbed hard-sphere equation of state (EoS) approach. The EoS is taken from the Carnahan –Starling hard-sphere equation with the perturbation term of Dohm-Prausnitz (CS-DP EoS), in which the relevant parameters to the molecular size and energies are universal functions of temperature in describing the equilibrium bulk properties. Then DGT+CS-DP EoS model has been employed for predicting the surface tension of 26 pure refrigerants taken from a Chemistry Webbook provided by NIST. Our calculations on the surface tension data from the DGT+CS-DP model led to an average absolute deviation of 4.84%. Then, the degree of accuracy of DGT+CS-DP model has also been compared with other DGT-based model.

Soft congestion approximation to the one-dimensional constrained Euler equations

This article is concerned with the analysis of the one-dimensional compressible Euler equations with a singular pressure law, the so-called hard sphere equation of state. We provide a detailed description of the effect of the singular pressure on the breakdown of the smooth solutions. Moreover, we rigorously justify the singular limit for smooth solutions towards the free-congested Euler equations, where the compressible (free) dynamics is coupled with the incompressible one in the constrained (i.e. congested) domain.

Beyond Salsburg–Wood: Glass equation of state for polydisperse hard spheres

We reconstruct glass equations of state for polydisperse hard spheres with the help of computer simulations. To perform the reconstructions, we assume that hard-sphere glass equations of state have the form Zg(φ, φJ) = Zg(φJ/φ), where Zg, φ, and φJ are the reduced glass pressure (PV/NkBT), sphere volume fraction (packing density), and jamming density of the current basin of attraction, respectively. Specifically, we use the form X = ∑iciYi, where X = (φJ/φ) − 1 and Y = 1/(Zg − 1). Our reconstructions converge to the well-known Salsburg–Wood and free volume equations of state in the limit φ → φJ, but they are also applicable for values of φ ≪ φJ. We support the ansatz Zg(φ, φJ) = Zg(φJ/φ) with extensive computer simulations. We use log-normal distributions of particle radii (r) and polydispersities δ=⟨Δr2⟩/⟨r⟩=0.1−0.3 in steps of 0.05. By supplying the fluid equation of state (EOS) into the new glass EOS, we evaluate equilibrium jamming densities φEJ for a range of φ. By using the ideal glass transition densities φg as an input φ, we estimate the corresponding glass close packing limits φGCP = φEJ(φg). We use the Boublík–Mansoori–Carnahan–Starling–Leland fluid EOS, and we estimate φg from the Vogel–Fulcher–Tammann fits—but our method can work with any choice of the fluid EOS and φg estimates. We show that our glass EOS leads to much better predictions for φEJ(φ) than the standard Salsburg–Wood glass EOS.

Viscosities of 2-hydroxy ethylammonium formate ionic liquid from 0.1 MPa to 40 MPa: Measurement and modelling

This work extended the measurements of dynamic viscosities of 2-hydroxy ethylammonium formate (2-HEAF) ionic liquid to high-pressure. The dynamic viscosities were measured in (298.15–328.15) K range and pressures from (0.1 to 40) MPa with the uncertainties within ±(0.85–5.3) mPa s. The experimental data were correlated/predicted using a two-parameter empirical equation as well as semi-theoretical model based on the rough-hard-sphere theory (RHS) with the average absolute relative deviation of 1.14% and 2.18%, respectively. The RHS model used two molecular parameters as well as the liquid densities, for which the values were estimated from perturbed hard-sphere equation of state. The experimental results have also been compared with those coming from the previously reported molecular dynamics results. Additionally, molecular dynamics results allowed a nanoscopic level analysis of the main features controlling viscosity in terms of hydrogen bonding properties and changes with temperature and pressure.

Correction to: A simplified equation of state for polymer melts from perturbed Yukawa hard-sphere chain

After a careful reconsideration of Ref. [1] article, several typos appeared in writing the general formalism of PYC EoS, Eqs. 2 and 4 of Ref. [1]. Since those typos were appearing in the key equations of this paper, they would cause the significant errors in the calculation of properties of studied molten polymers using “Yukawa hard-sphere chain equation of state”. Regarding this, the present Erratum aims to fix this issue.

Dispersion activity coefficient models. Part 1: Cubic equations of state

An explicit expression for dispersion in activity coefficient models can be derived from cubic equations of state (cEoS). Here we show that all the two-parameter cEoS deliver a van Laar type of equation. The difference between these equations can be characterized by a single parameter K, which can be computed directly from the cEoS characteristic parameters. The theoretical values for K are always higher than experimental activity coefficient data of alkane mixtures indicate. We show that mixtures of linear and branched alkanes require K=4.13 and K=3.04, respectively, while the lowest theoretical value, K=9, is given by the van der Waals equation. This mismatch in results is caused by the assumptions, which are made in the derivation of the van der Waals equation of state and which remain present in later developed cEoS. One of these is that all molecules are spherical, which leads to the inconsistency that the ratio of the covolume and the van der Waals volume is always 4, while this ratio for linear alkanes decreases rapidly to nearly 2 with increasing chain length. Another assumption is that all molecules experience the same number of external interactions, which neglects the fact that polyatomic molecules have less intermolecular interactions per spherical segment due to presence of covalent bonds and the occurrence of intramolecular interaction. Therefore, the van Laar type of activity coefficient equations are limited in their use as predictive model for dispersion. Perturbed hard-sphere chain equation of state will be discussed in part 2.

Local Well-Posedness for Boltzmann’s Equation and the Boltzmann Hierarchy via Wigner Transform

We use the dispersive properties of the linear Schrodinger equation to prove local well-posedness results for the Boltzmann equation and the related Boltzmann hierarchy, set in the spatial domain $${\mathbb{R}^d}$$ for $${d\geq 2}$$ . The proofs are based on the use of the (inverse) Wigner transform along with the spacetime Fourier transform. The norms for the initial data f0 are weighted versions of the Sobolev spaces $${L^2_v H^\alpha_x}$$ with $${\alpha \in \left( \frac{d-1}{2},\infty\right)}$$ . Our main results are local well-posedness for the Boltzmann equation for cutoff Maxwell molecules and hard spheres, as well as local well-posedness for the Boltzmann hierarchy for cutoff Maxwell molecules (but not hard spheres); the latter result holds without any factorization assumption for the initial data.

Impact of roughness on the instability of a free-cooling granular gas.

A linear stability analysis of the hydrodynamic equations with respect to the homogeneous cooling state is carried out to identify the conditions for stability of a granular gas of rough hard spheres. The description is based on the results for the transport coefficients derived from the Boltzmann equation for inelastic rough hard spheres [Phys. Rev. E 90, 022205 (2014)PLEEE81539-375510.1103/PhysRevE.90.022205], which take into account the complete nonlinear dependence of the transport coefficients and the cooling rate on the coefficients of normal and tangential restitution. As expected, linear stability analysis shows that a doubly degenerate transversal (shear) mode and a longitudinal ("heat") mode are unstable with respect to long enough wavelength excitations. The instability is driven by the shear mode above a certain inelasticity threshold; at larger inelasticity, however, the instability is driven by the heat mode for an inelasticity-dependent range of medium roughness. Comparison with the case of a granular gas of inelastic smooth spheres confirms previous simulation results about the dual role played by surface friction: while small and large levels of roughness make the system less unstable than the frictionless system, the opposite happens at medium roughness. On the other hand, such an intermediate window of roughness values shrinks as inelasticity increases and eventually disappears at a certain value, beyond which the rough-sphere gas is always less unstable than the smooth-sphere gas. A comparison with some preliminary simulation results shows a very good agreement for conditions of practical interest.

Condensation and dissociation rates for gas phase metal clusters from molecular dynamics trajectory calculations.

In gas phase synthesis systems, clusters form and grow via condensation, in which a monomer binds to an existing cluster. While a hard-sphere equation is frequently used to predict the condensation rate coefficient, this equation neglects the influences of potential interactions and cluster internal energy on the condensation process. Here, we present a collision rate theory-molecular dynamics simulation approach to calculate condensation probabilities and condensation rate coefficients. We use this approach to examine atomic condensation onto 6-56-atom Au and Mg clusters. The probability of condensation depends upon the initial relative velocity (v) between atom and cluster and the initial impact parameter (b). In all cases, there is a well-defined region of b-v space where condensation is highly probable, and outside of which the condensation probability drops to zero. For Au clusters with more than 10 atoms, we find that at gas temperatures in the 300-1200 K range, the condensation rate coefficient exceeds the hard-sphere rate coefficient by a factor of 1.5-2.0. Conversely, for Au clusters with 10 or fewer atoms and for 14- and 28-atom Mg clusters, as cluster equilibration temperature increases, the condensation rate coefficient drops to values below the hard-sphere rate coefficient. Calculations also yield the self-dissociation rate coefficient, which is found to vary considerably with gas temperature. Finally, calculations results reveal that grazing (high b) atom-cluster collisions at elevated velocity (>1000 m s-1) can result in the colliding atom rebounding (bounce) from the cluster surface or binding while another atom dissociates (replacement). The presented method can be applied in developing rate equations to predict material formation and growth rates in vapor phase systems.

Solubility Determination and Correlation of Gatifloxacin, Enrofloxacin, and Ciprofloxacin in Supercritical CO2

The solubilities of gatifloxacin, enrofloxacin, ciprofloxacin in SC–CO2 were determined by the dynamic method under pressures of 12–36 MPa and temperature of 313, 323, and 333 K. The consistency of the resulting solubility data was verified by the Mendez-Santiago and Teja model, Chrastil model, Bartle model, and K-J model and these models give comparable AARDs (between 6.70% and 13.51%). The compressed gas model and modified expanded liquid model were also used to correlate the resulting solubilities of the three fluoroquinolone drugs. By introducing the reference solubility and calculating the fugacity coefficient of the solute using the Carnahan–Starling-VDW hard sphere equation of state (CS-VDW EoS), the compressed gas model gives the AARDs between 0.86% and 23.65%. The intrinsic proximity of the model parameters in this model is also confirmed for the structurally similar fluoroquinolone drugs. This intrinsic proximity can be used in mutual solubility prediction between the fluoroquinolone drugs once the reference solubilities were known. The modified expanded liquid model gives the AARDs between 10.88% and 16.65% in solubility correlation. However, the predictive capability of the modified expanded liquid model based on the group contribution method needs to be improved further.

On the rough hard-sphere-based model for transport properties of nanofluids

A semi-theoretical model based on rough hard-sphere (RHS) theory has been employed to correlate and predict the transport properties of 10 nanofluids. The proposed model comprises smooth hard-sphere taken from modified Enskog's theory and a coupling parameter of translational and rotational motions. The molecular scaling constants to be used in new RHS model have been taken from previously developed perturbed hard-sphere equation of state (Hosseini, S.M., Alavianmehr, M.M. and Moghadasi, J., Fluid Phase Equilibria, 423, (2016) 181–189). The new idea of the present study is to introduce a nanoparticle concentration-dependent function into the RHS expression of transport property of the base fluid. The performance of model has been checked by correlating the atmospheric (0.1 MPa) dynamic viscosities and thermal conductivities over the temperature range within 283–343 K. From 539 experimental data points examined, the average absolute relative deviation (AARD) of the correlated and predicted dynamic viscosities was found to be 2.80%. In the case of thermal conductivities, the RHS-based model correlated and predicted 201 experimental data points with AARD equal to 2.47%.

Shape factors in hard convex body equations of state

The analysis using the Carnahan Muller shape factor approach for the original chain-SAFT equation of state, which is based on the Carnahan-Starling hard-sphere reference equation of state, has been performed. The performance of the second order approximation for shape factor of every proposed equation of state of hard convex fluid has been found to be of comparable accuracy when compared to their respective original equation of state.

Transport properties of pure and mixture of ionic liquids from new rough hard-sphere-based model

This paper aims to develop a new semi-theoretical model based on the rough hard-sphere (RHS) theory to correlate and predict the transport properties of 17 ionic liquids (ILs). The proposed model comprises smooth hard-sphere taken from modified Enskog's theory and a coupling parameter of translational and rotational motions. The molecular scaling constants to be used in the new RHS model were taken from previously developed perturbed hard-sphere equation of state for ILs. The performance of model has been checked over the temperature range within 263–388 K and pressures up to 249.6 MPa. From 1195 experimental data points examined, the AARD of correlated and predicted dynamic viscosities was found to be 3.67%. In the case of thermal conductivities, the proposed model correlated and predicted 130 experimental data points with AARD equal to 2.15%. The comparison of the new RHS-based model with the previous RHS model and group-contribution method demonstrated the superiority of the present work over the literature ones. The proposed RHS-based model has also been extended to14 binary mixtures formed by IL + IL and IL + molecular solvent. From 736 data points examined for the mentioned mixtures, the AARD of predicted viscosities was found to be 5.08%.

Application of perturbed hard-sphere equation of state to the study of volumetric properties of nano-fluids

This paper aimed to examine the application of a perturbed hard-sphere (PHS) scheme for modeling the volumetric properties of nanofluids. In this regard, PHS scheme has been employed to develop an analytical equation of state (EOS) to correlate and predict the volumetric properties of some nanofluids containing SnO2,ZnO, Co3O4, TiO2-anatase (-A), TiO2-rutile (-R), CuO and Al2O3 as nanoparticles dispersed in ethylene glycol, poly ethylene glycol, water, poly ethylene glycol + water and ethylene glycol + water as base fluids. Two temperature-dependent parameters appeared in the EOS were expressed in terms of molecular scaling constants σ, the effective hard-sphere diameter, and ε, the non-bonded interaction energy. The aforementioned scaling constants were correlated with melting temperatures and true densities which demonstrated the rationality of these constants. The performance of the proposed model was assessed by predicting 1348 density data, for which their measured values were available in the literature over the pressure range from 0.1 to 45 MPa and temperature range from 273 to 363 K. The overall average absolute deviation (AAD) of the correlated (at 0.1 MPa) and predicted (at elevated pressures) densities of 9 studied nanofluids from the experimental data was found to be 0.44%. Generally, this work showed that PHS is an appropriate scheme for the correlation and prediction of the properties of this class of fluids by the help of crossed interaction parameters between nanoparticles and base fluid molecules. The isothermal compressibility coefficients and excess volumes of studied nanofluids have also been investigated by the proposed model.

Exponential trend to equilibrium for the inelastic Boltzmann equation driven by a particle bath

We consider the spatially homogeneous Boltzmann equation for inelastic hard spheres (with constant restitution coefficient ) under the thermalization induced by a host medium with a fixed Maxwellian distribution. We prove that the solution to the associated initial-value problem converges exponentially fast towards the unique equilibrium solution. The proof combines a careful spectral analysis of the linearised semigroup as well as entropy estimates. The trend towards equilibrium holds in the weakly inelastic regime in which α is close to 1, and the rate of convergence is explicit and depends solely on the spectral gap of the elastic linear collision operator.

Boltzmann equation for granular media with thermal force in a weakly inhomogeneous setting

In this paper, we consider the spatially inhomogeneous diffusively driven inelastic Boltzmann equation in different cases: the restitution coefficient can be constant or can depend on the impact velocity (which is a more physically relevant case), including in particular the case of viscoelastic hard spheres. In the weak thermalization regime, i.e. when the diffusion parameter is sufficiently small, we prove existence of global solutions considering the close-to-equilibrium regime as well as the weakly inhomogeneous regime in the case of a constant restitution coefficient. It is the very first existence theorem of global solution in an inelastic “collision regime” (that is excluding [1] where an existence theorem is proven in a near to the vacuum regime). We also study the long-time behavior of these solutions and prove a convergence to equilibrium with an exponential rate. The basis of the proof is the study of the linearized equation. We obtain a new result on it, we prove existence of a spectral gap in weighted (stretched exponential and polynomial) Sobolev spaces and a result of exponential stability for the semigroup generated by the linearized operator. To do that, we develop a perturbative argument around the spatially inhomogeneous equation for elastic hard spheres and we take advantage of the recent paper [14] where this equation has been considered. We then link the linearized theory with the nonlinear one in order to handle the full non-linear problem thanks to new bilinear estimates on the collision operator that we establish. As far as the case of a constant coefficient is concerned, the present paper largely improves similar results obtained in [20] in a spatially homogeneous framework. Concerning the case of a non-constant coefficient, this kind of results is new and we use results on steady states of the linearized equation from [5] .