Parallel Plane Walls Research Articles

Bifurcation of and ghost effect on the temperature field in the Bénard problem of a gas in the continuum limit

A gas in a time-independent state under a uniform weak gravity in a general domain is considered. The asymptotic behavior of the gas in the limit that the Knudsen number of the system tends to zero (or in the continuum limit) is investigated on the basis of the Boltzmann system for the case where the flow velocity vanishes in this limit, and the fluid-dynamic-type equations and their associated boundary conditions describing the behavior of the gas in the continuum limit are derived. The equations, different from the Navier–Stokes ones, contain thermal stress and infinitesimal velocity amplified by the inverse of the Knudsen number. The system is applied to analysis of the behavior of a gas between two parallel plane walls heated from below (Bénard problem), and a bifurcated strongly distorted temperature field is found in infinitesimal velocity and gravity. This is an example showing that the Navier–Stokes system fails to describe the correct behavior of a gas in the continuum limit.

Image system for Stokes-flow singularity between two parallel planar walls

Using a recently developed image representation for Stokes flow in a half-space bounded by a planar wall [Cichocki and Jones, Physica A 258, 273 (1998)], the image system is constructed for the flow field produced by a force multipole in the space bounded by two parallel walls. The image singularities are expressed in terms of products of double-reflection matrices, and the expansion is simplified using symmetries of the double-reflection operation. Our analysis yields recurrence relations for the strengths of the image multipoles. The relations are solved explicitly, and a complete image system is obtained for an arbitrary source-force multipole. Applications of our image representation for evaluating the hydrodynamic friction and mobility matrices of particles interacting with two parallel planar walls are indicated.

Stress fluctuations in sheared Stokesian suspensions.

We report an analysis, using the tools of nonlinear dynamics and chaos theory, of the fluctuations in the stress determined from simulations of shear flow of Stokesian suspensions. The simulations are for shear between plane parallel walls of a suspension of rigid identical spheres in a Newtonian fluid, over a range of particle concentration. By analyzing the time series of the stress, we find that the dynamics underlying these fluctuations is deterministic, low-dimensional, and chaotic. We use the dynamic and metric invariants of the underlying dynamics as a means of characterizing suspension behavior. The dimension of the chaotic attractor increases with particle concentration, indicating the increasing influence of multiple-body interactions on the rheology of the suspension with rise in particle concentration. We use our analysis to make accurate predictions of the short-term evolution of a stress component from its preceding time series, and predict the evolution of one component of the stress using the time series of another. We comment on the physical origin of the chaotic stress fluctuations, and on the implications of our results on the relation between the microstructure and the stress.

Diffusiophoresis of a colloidal sphere in nonelectrolyte gradients parallel to one or two plane walls

The quasisteady diffusiophoretic motion of a spherical particle in a fluid solution of a nonionic solute located between two infinite parallel plane walls is studied theoretically in the absence of fluid inertia and solute convection. The imposed solute concentration gradient is constant and parallel to the two plane walls, which may be either impermeable to the solute molecules or prescribed with the far-field concentration distribution. The particle–solute interaction layer at the particle surface is assumed to be thin relative to the particle radius and to the particle–wall gap widths, but the polarization effect of the diffuse solute in the thin interfacial layer caused by the strong adsorption of the solute is incorporated. The presence of the neighboring walls causes two basic effects on the particle velocity: first, the local solute concentration gradient on the particle surface is enhanced or reduced by the walls, thereby speeding up or slowing down the particle; secondly, the walls increase viscous retardation of the moving particle. To solve the continuity and momentum equations, the general solutions are constructed from the fundamental solutions in both the rectangular and the spherical coordinate systems. The boundary conditions are enforced first at the plane walls by the Fourier transforms and then on the particle surface by a collocation technique. Numerical results for the diffusiophoretic velocity of the particle relative to that under identical conditions in an unbounded fluid solution are presented for various values of the relaxation parameter of the particle as well as the relative separation distances between the particle and the two plates. For the special case of diffusiophoretic motions of a spherical particle parallel to a single plate and on the central plane of a slit, the collocation results agree well with the approximate analytical solutions obtained by using a method of reflections. The presence of the lateral walls can reduce or enhance the particle velocity, depending on the surface properties of the particle, the relative particle–wall separation distances, and the solutal boundary condition at the walls. In general, the boundary effect on diffusiophoresis is quite complicated and relatively weak in comparison with that on sedimentation.

Dynamical Simulation of the Flow of Suspensions: Wall-Bounded and Pressure-Driven Channel Flow

A boundary-element method is implemented to simulate the motion of two-dimensional particles with arbitrary shapes suspended in a viscous fluid under conditions of Stokes flow. Numerical discretization of an integral equation of the first kind for the particle surface traction results in a system of linear algebraic equations for the components of the traction over boundary elements distributed along the particle contours, as well as for the velocity of translation and the angular velocity of rotation of the particles about designated centers. The linear system is solved by the method of successive substitutions based on a physically motivated iterative method that involves the decomposition of the influence matrix into diagonal blocks consisting of physical particle clusters and then carrying out updates by matrix−vector multiplication using the inverses of the diagonal blocks. The iterations are found to converge as long as the minimum gap between the particles is larger than a threshold that depends on the particle shape and level of numerical discretization. To improve the numerical efficiency, the stiffness of the governing equations due to lubrication forces developing between intercepting particles is removed by preventing the particles from approaching one another to within less than a specified distance. Simulations of singly periodic suspensions of circular or elliptical particles in semi-infinite shear flow bounded by a plane wall and in pressure-driven flow in a channel bounded by two parallel plane walls are carried out for an extended period of time. The results confirm the occurrence of particle migration due to an effective hydrodynamic diffusivity and illustrate the dependence of the suspension effective viscosity and dynamics of the microstructure on the solid-phase areal fraction and particle aspect ratio.

Thermocapillary motion of a fluid droplet parallel to two plane walls

The steady thermocapillary migration of a fluid droplet located between two infinite parallel plane walls is studied theoretically in the absence of fluid inertia and thermal convection. The imposed temperature gradient is constant and parallel to the two plates, and the droplet is assumed to retain a spherical shape. The plane walls may be either insulated or prescribed with the far-field temperature distribution. The presence of the neighboring walls causes two basic effects on the droplet velocity: first, the local temperature gradient on the droplet surface is enhanced or reduced by the walls, thereby speeding up or slowing down the droplet; secondly, the walls increase viscous retardation of the moving droplet. To solve the thermal and hydrodynamic governing equations, the general solutions are constructed from the fundamental solutions in both the rectangular and spherical coordinate systems. The boundary conditions are enforced first at the plane walls by the Fourier transforms and then on the droplet surface by a collocation technique. Numerical results for the thermocapillary migration velocity of the droplet relative to that under identical conditions in an unbounded medium are presented for various values of the relative viscosity and thermal conductivity of the droplet as well as the relative separation distances between the droplet and the two plates. For the special cases of thermocapillary motions of a spherical droplet parallel to a single plate and on the central plane of a slit, the collocation results agree well with the approximate analytical solutions obtained by using a method of reflections. The presence of the lateral walls can reduce or enhance the droplet velocity, depending upon the relative transport properties of the droplet, the relative droplet-wall separation distances, and the thermal boundary condition at the walls. In general, the boundary effect on thermocapillary migration is quite complicated and relatively weak in comparison with that on sedimentation.

Investigation of the Relative Motion of a Viscous Fluid and a Porous Medium Using the Brinkman Equations

Exact solutions are obtained for the following three problems in which the Brinkman filtration equations are used: laminar fluid flow between parallel plane walls, one of which is rigid while the other is a plane layer of saturated porous medium, motion of a plane porous layer between parallel layers of viscous fluid, and laminar fluid flow in a cylindrical channel bounded by an annular porous layer.

An Integrated Approach to the Determination of Permeability Tensors for Naturally Fractured Reservoirs

Abstract The use of permeability tensors is required when modelling fluid flow in anisotropic and heterogeneous reservoirs presenting multiple zones of directional permeability, or those categorized as naturally fractured reservoirs. A general procedure for characterizing complex reservoirs utilizing their permeability tensor is being developed by integrating data and methods from different disciplines. Permeability tensors for geologically defined fracture patterns are derived, and finally these small-scale descriptors are incorporated into a reservoir simulation program capable of handling full tensor permeabilities. The application and convenience of the method presented in this paper is illustrated with a field example from a naturally fractured reservoir. Introduction For many years, substantial research has been conducted in the areas of geosciences and engineering in order to characterize naturally fractured reservoirs. Numerous approaches have been presented to properly overcome this difficult task. Geoscientists have focused their research towards understanding the process of fracturing (rock mechanics), and the subsequent description of fracture characteristics such as density and orientation. Engineers, on the other hand, have focused their attention to the description of the fluid flow in the fracture systems, and in the development of accurate models (reservoir simulators) to reproduce the history, and predict the hydrocarbon production, for these complex systems. One of the first matrix-fracture models was presented by Warren and Root(1), who presented an idealized sugar cube model with two classes of porosity, a primary porosity that is intergranular, and a secondary porosity that is induced by fractures. Even though the sugar cube model has been widely accepted as the forerunner of the modern interpretation of dual-porosity systems, its limitations in describing the behaviour of some complex fractured reservoir systems have been observed. It is now well known that almost all fracture systems are much more complicated than the suggested model of three orthogonal sets of uniform fractures. One of the most important factors that has been identified as a necessary addition to improve the overall description of such complex reservoirs is the definition of a nine-component permeability tensor for the fracture system. This tensor is used to model fluid flow in complex reservoirs with multiple zones of directional permeability, where the orientation and magnitude of the principal permeabilities may vary between different zones in the reservoir. Snow(2, 3) studied the convenience of using mathematical equivalents of parallel plate openings to simulate fractures dispersed in orientation, distributed in aperture, and of arbitrary spacing. Models for fractured media which contained any number of planar conductors of any orientation and any fine aperture were presented. A key assumption was that all of the conduits have smooth parallel plane walls of indefinite extent (infinite fractures), and an rbitrary aperture. As a result, a permeability tensor could be obtained by superposition of contributions due to the fractures, and due to the permeable matrix. Long et al.(4, 5) addressed the more realistic scenario of finite or discrete fracture systems, where properties such as shape, orientation and location of the fractures in an impermeable matrix were considered to be random variables.

Slow motion of a droplet between two parallel plane walls

A combined analytical–numerical study for the creeping flow caused by a fluid sphere translating in a second, immiscible fluid parallel to two flat plates at an arbitrary position between them is presented. To solve the Stokes equations for the fluid velocity fields inside and outside the spherical droplet, a general solution is constructed from fundamental solutions in both rectangular and spherical coordinate systems. Boundary conditions are enforced first at the plane walls by the Fourier transforms and then on the droplet surface by a collocation technique. Numerical results for the hydrodynamic drag force acting on the droplet are obtained with good convergence for various relative viscosities of the droplet and separation distances between the droplet and the walls. For the motion of a solid sphere (droplet with infinite viscosity) parallel to a single plane wall or to two walls, our drag results are in perfect agreement with the available solutions in the literature for all particle-to-wall spacings. The boundary-corrected drag force exerted on the droplet normalized by the value in the absence of the walls is found to increase monotonically with an increase in the internal-to-external viscosity ratio for any given geometry.

Two-dimensional oscillatory Stokes flows between two parallel planes

The eigen solutions of two-dimensional oscillatory Stokes flow between two parallel plane walls are studied, and the behavior of eddy structure is examined as the frequency of oscillation varies. Oscillatory Stokes flows between two planes induced by singularities such as stokeslet and rotlet are obtained in terms of eigen solutions. Transient behavior of unsteady Stokes flow after the impulsive introduction of singularity is briefly discussed.

Confinement induced immiscibility of mixtures of enantiomers

The concentration profiles and phase properties of symmetric fluid mixtures confined by parallel planar walls are analyzed for the cases of identical and symmetrically opposed fields at the walls. We focus on the occurrence of phase separation otherwise absent in bulk mixtures and find that this can take place when chemical potentials and molecular interactions are modified at the surfaces. The phenomenon may find application in the resolution of racemic mixtures of enantiomers.

Nonlinear waves in a chain of plane-parallel domain walls in ferromagnets

Taking into account the nonlinear interaction between plain domain walls (DWs) in a chain of DWs, one-and two-parameter solitons are obtained. These solitons are solitary shear waves propagating along the DW chain.

Characteristics of a Partially Cavitating Hydrofoil Spanning Parallel Plane Walls.

Guide vanes and impellers of turbomachinery operate in boundary layer due to the walls of casing and boss, and further operate in cavitating condition in high speed flow. From these circumstances, hydrodynamic characteristics are analyzed about a partially cavitating hydrofoil which is placed in boundary layer between parallel plane walls. By separation of variables for disturbance pressure, the governing equation is transformed into two problems, of which the one is that of wing sectional plane, the other that of spanwise direction. In the problem of the sectional plane, a simultaneous integral equation is derived and solved by series expansion method. The problem of spanwise direction is reduced to the eigenvalue problem of Sturm-Liouville type which is numerically solved by finite difference method. These two kinds of solution are linearly combined in order to obtain the hydrodynamic characteristics.

Structure and solvation forces in confined alkane films

We report grand canonical ensemble Monte Carlo simulations of thin films of fluid n- and iso-decane confined between two plane-parallel walls. Damped oscillatory solvation pressure profiles were observed. This can be explained by density oscillations of the confined film. Although the united atoms beads in our alkane models are mostly arranged in layers parallel with the walls, some interlayer interdigitation always exists, whatever the model used, at least for the given choice of thermodynamic state. In contrast to the previously studied atomic fluid, it is found that the variation in the average number of molecules with the walls separation is, overall, continuous. The confined alkane fluid exhibits liquid-like behaviour down to a thickness corresponding to a bilayer film. A small step is then observed which presumably corresponds to the solidification of the fluid. We thus conclude that alkane films behave rather differently from atomic (or globular shaped molecules) fluid films, although oscillations in the solvation pressure profiles are observed in both cases. Linear and branched alkanes behave quite similarly, at least for the cases studied here. Attenuated oscillations are found for branched alkanes, in agreement with the most recent experiments, but this does not reflect a huge difference between the (overall disordered) microscopic structure of the confined fluids.

Nonlinear effects of physisorption on static friction

The effects of a physisorbed film on the force of static friction in a model contact (monatomic adsorbate confined between plane-parallel walls) were investigated by Monte Carlo simulation. At fixed coverage the friction curve (shear yield stress vs normal stress) exhibits a marked nonlinearity, which results from a competition between adsorbate–wall interactions that predominate at low loads and wall–wall interactions that set in beyond a threshold load, which increases with coverage. Previous proximal-probe and computer experiments, carried out at high coverages, see only the initial (low-load) linear portion of the friction curve.

Condensation pressures in small pores: An analytical model based on density functional theory

Integral methods are used to derive an analytical expression describing fluid condensation pressures in slit pores bounded by parallel plane walls. To obtain this result, the governing equations of density functional theory (DFT) are integrated across the pore width assuming that fluid densities within adsorbed layers are spatially uniform. The thickness, density, and free energy of these layers are expressed as composite functions constructed from asymptotic limits applicable to small and large pores. By equating the total free energy of the adsorbed layers to that of a liquid-full pore, we arrive at a closed-form expression for the condensation pressure in terms of the pore size, surface tension, and Lennard-Jones parameters of the adsorbent and adsorbate molecules. The resulting equation reduces to the Kelvin equation in the large-pore limit. It further reproduces the condensation pressures computed by means of the full DFT equations for all pore sizes in which phase transitions are abrupt. Finally, in the limit of extremely small pores, for which phase transitions may be smooth and continuous, this simple analytical expression provides a good approximation to the apparent condensation pressure indicated by the steepest portion of the adsorption isotherm computed via DFT.

Density fluctuations and correlations of confined fluids

The density fluctuations about the equilibrium structure of fluids confined by parallel planar walls are analyzed for the cases of identical and symmetrically opposed fields at the walls. We determine the stability matrix (of the second derivatives of the free energy functional with respect to the density) for conditions both above and below the wetting transition temperature T w of the semi-infinite system and corroborate in all cases that the equilibrium configurations are stable. We identify the fluctuations close to the walls and in the middle of the slab and discuss their effect when the wall separation L diverges. For competing walls above T w the localized fluctuation with lowest eigenvalue describes the displacements of the incipient wetting films that become unimpeded interfacial translations for L→∞. Below T w the fluctuations with lowest eigenvalue correspond to stiffer deformations extended across the slab. For identical walls above T w coexisting states display incipient prewetting films and the lowest eigenvalue describes the nature of their growth as L increases. We also calculate the pair correlation function for the inhomogeneous states and, for symmetrically opposed walls, we obtain standard Ornstein–Zernike (OZ) behavior at the walls, but find significant deviations from this law at the interface-like region in the middle of the slab. To model fluids with short-ranged forces we use a ferromagnetic Ising-type Hamiltonian in mean-field approximation.

Effects of confinement on the phase behavior of supercooled water

Molecular dynamics simulations are performed to investigate phase behavior of a thin film (thickness d∼2 nm) of supercooled water confined between two parallel planar walls. In contrast to bulk supercooled water, the confined water, under fixed normal pressures of 0.1 MPa and 500 MPa, does not exhibit the high density amorphous (HDA) to low density amorphous (LDA) liquid–liquid phase transition in the course of the cooling process. A possible explanation is that when the water is confined between two walls, the bulk HDA–LDA phase boundary and the associated critical point C ′ shift to lower temperature or/and lower P zz (confinement pressure) as decreasing the film thickness.

A quasi two-dimensional bubble method for surface tension and contact angle measurements at the crystal-melt interface. I. Theoretical background

A method for simultaneous determination of the surface tension of liquids and of the contact angle between liquid and solid is proposed, in view of studying the wettability of a crystal by its own melt. The method is based on the shape of a gas bubble captured at the solid-liquid interface inside a thin cell with plane-parallel walls. This quasi two-dimensional configuration simplifies the theoretical treatment of the bubble shape with respect to that in a three-dimensional space. It is stressed that contact angle is independent of dimensionality.

Instabilities of fully developed rapid flow of a granular material in a channel

The equations of motion for ‘rapid’ flow of a granular material have fully developed solutions representing flow driven by a body force, such as gravity, along a channel bounded by plane parallel walls. The stability of these solutions to small perturbations is investigated. For given properties of the particles and the channel walls it is found that the condition of critical stability is a relation between the mean concentration of the particles and the width of the channel. When the base state is unstable the fastest growing modes are travelling waves propagating in the axial direction and these induce characteristic patterns of variations in particle concentration, as well as velocity. These instabilities are contrasted with those found for Couette shearing in an earlier publication.