Constant-density thermodynamic perturbation theory for quantum fluids

A perturbation series is given which expresses the equilibrium thermodynamic quantities of a quantal fluid in terms of that of another (unperturbed) fluid having the same temperature and number density ρ. The general term of each series can be given; it is the integral of a product involving either the grand-canonical correlation functions of the unperturbed fluid or its fundamental probability distribution W^c_N(1,…,N) for sets of free particles and the modified Ursell-functions of the quantal fluid; each integral can be conveniently represented in the language of graphs.     

Acoustic scattering by a modified Werner method

A modified integral Werner method is used to calculate pressure scattered by an axisymmetric body immersed in a perfect and compressible fluid subject to a harmonic acoustic field. This integral representation is built as the sum of a potential of a simple layer and a potential of volume. It is equivalent to the exterior Helmholtz problem with Neumann boundary condition for all real wave numbers of the incident acoustic field. For elastic structure scattering problems, the modified Werner method is coupled with an elastodynamic integral formulation in order to account for the elastic contribution of the displacement field at the fluid/structure interface. The resulting system of integral equations is solved by the collocation method with a quadratic interpolation. The introduction of a weighting factor in the modified Werner method decreases the number of volume elements necessary for a good convergence of results. This approach becomes very competitive when it is compared with other integral methods that are valid for all wave numbers. A numerical comparison with an experiment on a tungsten carbide end-capped cylinder allows a glimpse of the interesting possibilities for using the coupling of the modified Werner method and the integral elastodynamic equation used in this research.     

A new integral equation for the radial distribution function of a hard sphere fluid

Based on a proposal by Shinomoto, a new integral equation is derived for the radial distribution function of a hard-sphere fluid using mainly geometric arguments. This integral equation is solved by a perturbation expansion in the density of the fluid, and the results obtained are compared with those from molecular dynamics simulations and from the Born-Green-Yvon (BGY) and Percus-Yevick (PY) theories. The present theory provides results for the radial distribution function which are intermediate in accuracy between those obtained from the BGY and from the PY theories.     

Analysis of the flow of non-Newtonian visco-elastic fluids in fractal reservoir with the fractional derivative

In both the oil reservoir engineering and seepage flow mechanics, heavy oil with relaxation property shows non-Newtonian rheological characteristics. The relationship between shear rate g& and shear stress t is nonlinear. Because of the relaxation phenomena of heavy oil flow in porous media, the equation of motion can be written as[1] 2,rrvpqkppqtrrtll秏骣+=-+琪抖桫 (1) where lv and lp are velocity relaxation and pressure retardation times. For most porous media, the above motion equation (1)…     

On the summation of the virial expansion for the local density of a gas in contact with a solid

We investigate the density profile of a fluid in contact with a wall. Our analysis is based on the summation of the virial expansion for the local density and the resulting integral equation for the density profile involves the two particle direct correlation function of a non-uniform fluid.     

Integral equation theory of a one dimensional hard-ellipse fluid between hard walls

Density functional theory is used to study the structure of a one dimensional fluid model of hard-ellipse molecules with their axes freely rotating in a plane, confined between hard walls. A simple Hypernetted chain (HNC) approximation is used for the density functional of the fluid and the integral equation for the density is obtained from the grand potential. The only required input is the direct correlation function of the one dimensional hard-ellipse fluid. For this model, the pressure, sum rule and the density at the walls are obtained. The Percus Yevick (PY), for lower density, and HNC, for higher density, integral equations are also solved to obtain the direct correlation function of hard-ellipse model introduced here. We obtain the average density at the wall as well as the radial density profile. We compare these with Monte Carlo simulations of the same model and find reasonable agreement.     

Wavelet method for solving integral equations of simple liquids

A new algorithm is developed to solve integral equations for simple liquids. The algorithm is based on the discrete wavelet transform of radial distribution functions. The Coifman 2 basis set is employed for the wavelet treatment. Using the algorithm, we have calculated structural and thermodynamic properties of a Lennard–Jones fluid in a wide range of energy and size parameters of the fluid.     

On the Boundary Integral Equations for a Two-Dimensional Slowly Rotating Highly Viscous Fluid Flow

In this paper, the two-dimensional slowly rotating highly viscous fluid flow in small cavities is modelled by the triharmonic equation for the streamfunction. The Dirichlet problem for this triharmonic equation is recast as a set of three boundary integral equations which however, do not have a unique solution for three exceptional geometries of the boundary curve surrounding the planar solution domain. This defect can be removed either by using modified fundamental solutions or by adding two supplementary boundary integral conditions which the solution of the boundary integral equations must satisfy. The analysis is further generalized to polyharmonic equations.     

Analytical expressions for the correlation function of a hard sphere dimer fluid

A closed form expression is given for the correlation function of a hard sphere dimer fluid. A set of integral equations is obtained from Wertheim's multidensity Ornstein-Zernike integral equation theory with Percus-Yevick approximation. Applying the Laplace transformation method to the integral equations and then solving the resulting equations algebraically, the Laplace transforms of the individual correlation functions are obtained. By the inverse Laplace transformation, the radial distribution function (RDF) is obtained in closed form out to 3D (D is the segment diameter). The analytical expression for the RDF of the hard dimer should be useful in developing the perturbation theory of dimer fluids.     

The correlations in a star molecule fluid. Integral equation theory and Monte Carlo study

The structure of a starlike molecule (SLM) fluid with four arms of different length is studied by applying the associative Percus–Yevick integral equation (IE) theory and canonical Monte Carlo (MC) simulations. In the IE study the SLM fluid is modelled by a fluid of hard spheres with four associative sites on each sphere while the MC has been performed for a freely-joined tangent hard sphere fluid. The total radial distribution functions have been calculated in both approaches for different volume fraction regimes and different arm lengths. It is shown that the associative IE theory predicts the structure of SLM fluid best for relatively long arms and at high densities. Additionally, the dependence of the SLM centre–centre correlations on the functionality and fluid particle density has been analysed using the MC results.     

Exact solution of the Percus-Yevick equation for a hard-core fluid in odd dimensions

It is shown that the Percus-Yevick integral equation for the pair distribution function of a fluid interacting with a hard-core potential can be solved not only in one and three dimensions, where the solution is well known, but more generally in all odd dimensions. The nonlinear integral equation is reduced to an algebraic equation of order d?3 for odd dimensions d greater than three. As an example the direct correlation function in five dimensions is derived explicitly.     

The invariant integral of physical mesomechanics as the foundation of mathematical physics: Some applications to cosmology,electrodynamics, mechanics and geophysics

Phe general invariant integral based on the energy conservation law is introduced into physical mesomechanics, with taking into account the cosmic, gravitational, mass, elastic, thermal and electromagnetic energy of matter. Phe physical mesomechanics thus becomes a mega-mechanics embracing most of the scales of nature. Some basic laws following from the general invariant integral are indicated, including Coulomb’s law of electricity generalized for moving electric charges, Newton’s law of gravitation generalized for coupled gravitational/cosmic field, the Archimedes’ law of buoyancy generalized for bodies partially submerged in water, and others. Using the invariant integral the temperature track behind moving cracks and dislocations is found out, and the coupling of elastic and thermal energies is set up in fracturing and plastic flow, namely for opening mode cracks and edge dislocations. For porous materials saturated with a fluid or gas, the notion of binary continuum is used to introduce the corresponding invariant integrals. As applied to the horizontal drilling and hydrofracturing of boreholes in the Earth’ crust, the field of pressure and flow rate as well as the fluid output from both a horizontal borehole and a diskshape fracture issuing the borehole, are derived in the fluid extraction regime. A theory of fracking in shale gas/oil reservoirs is suggested for three basic regimes of the drill mud permeation into the multiply fractured rock region, with calculating the shape and volume of this region in terms of the geometry parameters and pressures of rock, drill mud and shale gas. Phe method of functional equations in the theory of a complex variable and the boundary layer method are also used to solve these problems.     

Computer simulation and replica Ornstein—Zernike integral equation studies of a hard-sphere dipolar fluid adsorbed into disordered porous media

By means of extensive grand canonical Monte Carlo simulations and replica Ornstein-Zernike integral equation calculations, we explore the thermodynamics and dielectric behaviour of a dipolar fluid confined in disordered matrices. Different matrix topologies are modelled using, on the one hand, quenched hard-sphere configurations and, on the other, randomly positioned spheres. This illustrates the influence of the pore size and shape on the properties of the adsorbed fluid. For the same purpose, various sizes of the matrix particles have been considered. The integral equation calculations in the hypernetted chain approximation agree quantitatively with the simulation results. As in other studies on quenched disorder, one observes that the effect of confinement when considering hard-sphere matrices is rather limited, exhibiting however a tendency to facilitate the transition to ferroelectric states, as well as favouring the gas-liquid transition. Additionally, one observes that the sensitivity of the fluid properties to changes in the size of the matrix particles is considerably larger when these are randomly positioned.     

Hamiltonian structure for Alfvén wave turbulence equations

A hamiltonian formulation using a noncanonical Poisson bracket is presented for a nonlinear fluid system that includes reduced magnetohydrodynamics and the Hasegawa-Mima equation as limiting cases. Nonlinear integral invariants for the system are found to be in the kernel of the noncanonical Poisson bracket. This Poisson bracket is given a Lie algebraic interpretation.     

Justification of Shallow-Water Theory

The basic conservation laws of shallow-water theory are derived from multidimensional mass and momentum integral conservation laws describing the plane-parallel flow of an ideal incompressible fluid above the horizontal bottom. This conclusion is based on the concept of hydrostatic approximation, which generalizes the concept of long-wavelength approximation and is used for justifying the applicability of the shallow-water theory in the simulation of wave flows of fluid with hydraulic bores.     

Phase stability of the penetrable-square-well model: an integral equation approach

We present an extensive investigation, based on the hypernetted-chain integral equation, of the topology of mechanically stable and unstable regions of the thermodynamic plane for the penetrable-square-well fluid. The results obtained are in qualitative agreement with Monte Carlo numerical simulation [R. Fantoni, A. Malijevský, A. Santos, and A. Giacometti, Europhys. Lett. 93, 26002 (2011)] and confirm the existence of a fluid–fluid transition between thermodynamically metastable phases. Moreover, we were able to investigate regimes where simulation is impractical, and showed that as the width of the attractive well vanishes, the metastable fluid–fluid transition disappears.     

The pressure of a hard sphere fluid on a curved surface

Utilizing the integral equation approach to the hard sphere fluid system developed in the preceding paper, the hard sphere-hard wall interaction is studied. For the case of a flat wall, perturbation solutions of the integral equation valid to second and third order in the packing fraction,y, are derived. For a surface of arbitrary curvature, an equation of state valid to second order in the packing fraction is also derived. When applied to very small cavities, it is found that the pressure at high densities is significantly higher than it would be for a flat wall.     

An Integral Equation Theory for the Widom–Rowlinson Mixture

The Widom–Rowlinson mixture is a two-component fluid in which like species do not interact and unlike species interact via a hard-core repulsion. As the density is increased, this fluid phase separates. Standard integral equation approaches, such as the Percus–Yevick or hypernetted chain, or thermodynamically self-consistent hybirds of these two, make very inaccurate predictions for the location of this critical point in the three-dimensional model. In this article we suggest a family of new approximations for this model that rely on incorporating terms in the density expansion of the direct correlation function into the closure approximation. We show that the simplest of these closures is significantly more accurate than previous theories for the structure and thermodynamics of the fluid.     

The O-H distance in ice

The thermodynamic properties of normal and para-hydrogen are computed from multiple time-step path integral hybrid Monte Carlo (PIHMC) simulations. Four different isotropic pair potentials are evaluated by comparing simulation results with experimental data. The Silvera–Goldman potential is found to be the most accurate of the potentials tested for computing the density and internal energy of fluid hydrogen. Using the Silvera–Goldman potential, simulation and experimental data are compared on isobars ranging from 0.1 to 100 MPa and for temperatures from 18 to 300 K. The Gibbs free energy is calculated from the PIHMC simulations by an adaptation of Widom's particle insertion technique to a path integral fluid. A new method is developed for computing phase equilibria for quantum fluids directly by combining PIHMC with the Gibbs ensemble technique. This Gibbs–PIHMC method is used to calculate the vapour–liquid phase diagram of hydrogen from simulations. Agreement with experimental data is good.     

A hard sphere fluid model for superionic conductors

We develop a theory for the mobile constituent of a superionic conductor using the Ornstein-Zernike integral equation for the pair correlation function of an inhomogeneous fluid. We solve this equation in the Percus-Yevick approximation using a simple decoupling procedure and hard core potentials. Comparison is made with molecular dynamics calculations on α-AgI.