Diffusion Boundary Conditions Research Articles

The Method of Singular Equations in Boundary Value Problems in Kinetic Theory

We develop a new effective method for solving boundary value problems in kinetic theory. The method permits solving boundary value problems for mirror and diffusive boundary conditions with an arbitrary accuracy and is based on the idea of reducing the original problem to two problems of which one has a diffusion boundary condition for the reflection of molecules from the wall and the other has a mirror boundary condition. We illustrate this method with two classical problems in kinetic theory: the Kramers problem (isothermal slip) and the thermal slip problem. We use the Bhatnagar-Gross-Krook equation (with a constant collision frequency) and the Williams equation (with a collision frequency proportional to the molecular velocity).

The Asymptotic Behavior of the Solution to a Cauchy Problem Governed by a Transport Operator

The purpose of this paper is to investigate some spectral properties of a time-dependent linear transport equation with diffusive boundary conditions arising in growing cell populations. After deriving the explicit expression of the strongly continuous semigroup {U K (t ): t ≥ 0} generated by the streaming operator, we establish the strict singularity of the operator BU K (s)B, s> 0, for a wide class of collision operators B. Making use of the Weis theorem, Theorem 4.1, this enables us to estimate the essential type of the transport semigroup from which the asymptotic behavior of the solution is derived.

Advected fields in maps: II. Dynamo action in the stretch–fold–shear map

The growth of magnetic field is considered in the stretch–fold–shear map in the limit of weak diffusion. Numerical results are given for insulating, perfectly conducting and periodic boundary conditions. The resulting eigenvalue branches and magnetic fields are related to eigenvalue branches for perfect dynamo action, obtained for zero diffusion using a complex variable formulation. Email: A.D.Gilbert@ex.ac.uk The effect of diffusion on these perfect dynamo modes depends on their structure, growth rate and the diffusive boundary conditions employed. In some cases, the effect of diffusion is a small perturbation, giving a correction going to zero in the limit of weak diffusion, with a scaling exponent given analytically. In other cases weak diffusion can entirely destroy a perfect dynamo branch. Diffusive boundary layers can also generate entirely new branches. These different cases are elucidated, and within the framework of the asymptotic approximations used (which do not constitute a rigorous proof), it is seen that for all three boundary conditions employed, the stretch–fold–shear map is a fast dynamo.

Boundary conditions for the upwind finite difference Lattice Boltzmann model: Evidence of slip velocity in micro-channel flow

We conduct a systematic study of the effect of various boundary conditions (bounce back and three versions of diffuse reflection) for the two-dimensional first-order upwind finite difference Lattice Boltzmann model. Simulation of Couette flow in a micro-channel using the diffuse reflection boundary condition reveals the existence of a slip velocity that depends on the Knudsen number ε = λ/ L, where λ is the mean free path and L is the channel width. For walls moving in opposite directions with speeds ± u w , the slip velocity satisfies u slip = 2 εu wall/(1 + 2 ε). In the case of Poiseuille flow in a micro-channel, the slip velocity is found to depend on the lattice spacing δs and Knudsen number ε to both first and second order. The best results are obtained for diffuse reflection boundary conditions that allow thermal mixing at a wall located at half lattice spacing outside the boundary nodes.

Discussion of three models used for the investigation of insertion/extraction processes by the potential step chronoamperometry technique

Three models used in electrochemical literature to fit experimental data for insertion/extraction processes investigated by the potential step chronoamperometry (PSCA) technique are examined and compared in this article. The first model (M I) gives a theoretical expression of Faradaic current transient taking reaction kinetics at the electrode surface, diffusion process in the host material and Ohmic drop effects into consideration. The ‘cell-impedance controlled diffusion’ constraint is used in the second model (M II) together with the diffusion equations and boundary conditions. Finally, the third model (M III) applies when the diffusion flux of injected/extracted species at the electrode surface is limited by interfacial charge transfer kinetics represented by the Butler–Volmer equation. The use of a finite-difference method is required to numerically solve models M II and M III. First, the theoretical predictions from models M I, M II and M III are compared when a small potential step is applied on the electrode. It is stated that the theoretical expression of current transient derived from model M I is the analytical solution of both models M II and M III under small-signal (linearization) conditions. Next, the three models are numerically compared under Langmuir isotherm conditions, assuming a constant diffusion coefficient of guest species in the host material. It is shown that the ‘cell-impedance controlled diffusion’ constraint used in model M II does not work very well when a large potential step is applied on the electrode. Finally, an alternative formulation of the boundary condition at the host material|electrolyte interface (model M III) is used to study the influence of Ohmic drop, together with the use of large potential steps.

Existence and Uniqueness of Solutions for the Couette Problem

We study existence and uniqueness results for the one-dimensional Boltzmann equation with inflow and diffusive boundary conditions. Our focus, partly encompasses some of the properties of the Boltzmann collision gain term which play a significant role in existence and uniqueness results. A series of estimates are proven on the collision term which is shown to produce a suitable function space in which the contraction mapping arguments are available.

Mean field kinetic theory description of evaporation of a fluid into vacuum

The evaporation of a liquid slab into vacuum is studied by numerical solutions of the Enskog–Vlasov equation for a fluid of spherical molecules interacting by Sutherland potential. The equation provides a simplified description of the microscopic behavior of the fluid but it has the capability of handling both the liquid and vapor phase, thus eliminating the necessity of postulating ad hoc models for boundary conditions at the vapor-liquid interface. This work focuses on obtaining the structure of the vapor-liquid interface in nonequilibrium conditions as well as the distribution function of evaporating molecules. The results show that the molecules crossing a properly defined vapor-liquid boundary have an almost Maxwellian distribution function and that the vapor phase is reasonably well described by the Boltzmann equation with diffusive boundary condition.

Characterization of the reduced scattering coefficient for optically thin samples: theory and experiments

Diffusion theory is currently used to characterize highly scattering media from time and/or spatially resolved measurements. However, many of the limits of validity of the diffusion approximation in its standard form have been set through assumptions taken during its derivation, and it is generally believed that the diffusion equation holds only for length scales much larger than the transport mean free path (ltr). We here present several modifications to the standard diffusion approximation to provide a simple expression for characterizing optically thin scattering slabs. The improvements concern mainly the correct modelling of the temporal and spatial distribution of the source term and the use of a frequency dependent diffusion coefficient and boundary conditions. This novel expression is put to the test against Monte Carlo simulations and experimental data taken from measurements with phantoms of known optical properties. The reduced scattering coefficient can be retrieved accurately for slab widths of the order of the mean free path.

Numerical analysis of the shear flow of a binary mixture of hard-sphere gases over a plane wall

The shear flow of a binary mixture of rarefied gases over a plane wall is investigated on the basis of the linearized Boltzmann equation for hard-sphere molecules with the diffuse reflection boundary condition. This fundamental problem in rarefied gas dynamics is analyzed numerically by a finite-difference method, in which the complicated collision integrals are computed by the extension to the case of a gas mixture of the method proposed by Sone, Ohwada, and Aoki [Phys. Fluids A 1, 363 (1989)]. As a result, the behavior of the mixture is clarified not only at the level of the macroscopic variables but also at the level of the velocity distribution function. In addition, an accurate formula of the shear-slip (viscous-slip) coefficient for arbitrary values of the concentration of a component gas is constructed by the use of the Chebyshev polynomial approximation.

Positive Steady States of a Competitor-Competitor-Mutualist Model

In this paper we deal with the positive steady states of a Competitor-Competitor-Mutualist model with diffusion and homogeneous Dirichlet boundary conditions. We first give the necessary conditions, and then establish the sufficient conditions for the existence of positive steady states.

Optimization of Solar Kiln for Drying Wood

Theoretical investigation of the physical process of solar drying of timber based on conventional heat and mass transfer equations is presented. The governing equations and boundary conditions of the mass diffusion through the wood timbers are derived; also the governing equations of the components of the solar kiln are presented. The finite difference technique is used to solve the set of these equations by means of a simulation program that is based on object-oriented approach. The simulation program is used to investigate the effect of several design parameters on the drying rate and duration of the wood timbers in order to accomplish the drying process with minimal drying defects. These parameters include the ventilation conditions that control the drying schedule inside the solar kiln, wood volume as a ratio to the solar kiln absorber area, wood timber thickness, season of drying, the drying air velocity, and the stresses that formed on the timber boards due to drying with these several parameters, leading to derive the limit of damage for a selected local wood type. The selected local wood type is Casuarina, which is common in Egypt, and it is commonly used in many simple industries.

Boundary effects of molecular diffusion in nanoporous materials: a pulsed field gradient nuclear magnetic resonance study.

The boundary conditions of intraparticle diffusion in nanoporous materials may be chosen to approach the limiting cases of either absorbing or reflecting boundaries, depending on the host-guest system under study and the temperature of measurement. Pulsed field gradient nuclear magnetic resonance is applied to monitor molecular diffusion of n-hexane and of an n-hexane-tetrafluoromethane mixture adsorbed in zeolite crystallites of type NaX under either of these limiting conditions. Taking advantage of the thus-established peculiarities of mass transfer at the interface between the zeolite bulk phase and the surrounding atmosphere, three independent routes for probing the crystal size are compared. These techniques are based on (i) the measurement of the effective diffusivity under complete confinement, (ii) the application of the so-called NMR tracer desorption technique, and (iii) an analysis of the time dependence of the effective diffusivity in the short-time limit where, by an appropriate variation of the adsorbate and the measuring conditions, the limiting cases of reflecting and adsorbing boundaries could be considered. All these techniques are found to yield coinciding results, which are in excellent agreement with the crystal sizes determined by microscopy.

New directions in fluid dynamics: non-equilibrium aerodynamic and microsystem flows.

Fluid flows that do not have local equilibrium are characteristic of some of the new frontiers in engineering and technology, for example, high-speed high-altitude aerodynamics and the development of micrometre-sized fluid pumps, turbines and other devices. However, this area of fluid dynamics is poorly understood from both the experimental and simulation perspectives, which hampers the progress of these technologies. This paper reviews some of the recent developments in experimental techniques and modelling methods for non-equilibrium gas flows, examining their advantages and drawbacks. We also present new results from our computational investigations into both hypersonic and microsystem flows using two distinct numerical methodologies: the direct simulation Monte Carlo method and extended hydrodynamics. While the direct simulation approach produces excellent results and is used widely, extended hydrodynamics is not as well developed but is a promising candidate for future more complex simulations. Finally, we discuss some of the other situations where these simulation methods could be usefully applied, and look to the future of numerical tools for non-equilibrium flows.

Numerical analysis of thermal-slip and diffusion-slip flows of a binary mixture of hard-sphere molecular gases

The thermal-slip (thermal-creep) and the diffusion-slip problems for a binary mixture of gases are investigated on the basis of the linearized Boltzmann equation for hard-sphere molecules with the diffuse reflection boundary condition. The problems are analyzed numerically by the finite-difference method incorporated with the numerical kernel method, which was first proposed by Sone, Ohwada, and Aoki [Phys. Fluids A 1, 363 (1989)] for a single-component gas. As a result, the behavior of the mixture is clarified accurately not only at the level of the macroscopic variables but also at the level of the velocity distribution function. In addition, accurate formulas of the thermal-slip and the diffusion-slip coefficients for arbitrary values of the concentration of a component gas are constructed by the use of the Chebyshev polynomial approximation.

ROLE OF PRODUCT SHRINKAGE IN DRYING RATE PREDICTIONS USING A LIQUID DIFFUSION MODEL

Most real materials shrink during drying. The degree of shrinkage and its variation with drying conditions as well as product moisture content influences the mechanisms of heat and mass transport within the material. In the most commonly employed liquid diffusion model it is customary either to exclude shrinkage effects or to include them only in terms of a change in the physical domain for application of the diffusion equation and boundary conditions. The objective of this communication is to show quantitatively the effect of shrinkage on predictions of the drying rate and product surface temperature using a one dimensional liquid diffusion model. It is shown that literature values of the moisture diffusivity do not take into account the thickness change during drying and hence predictions using the realistic physical shrinkage lead to incorrect estimates of the drying rate. Typical results are presented using a potato slab as a model object dried convectively at low temperatures.

A Homogenization Technique for the Boltzmann Equation for Low Pressure Chemical Vapor Deposition

We present a homogenization technique for rarefied gas flow over a microstructured surface consisting of patterns of periodic features. The length scale of the model domain is comparable to the mean free path of the molecules, while the scale of the surface patterns is much smaller. The flow is modeled by a system of linear Boltzmann equations with a diffusive boundary condition at the patterned surface. The resulting homogenized boundary condition holds at a virtual flat surface and incorporates the microscopic geometry information about the surface structure on the macroscopic level. Numerical results validate the approach. The setup models low pressure chemical vapor deposition processes in the manufacturing of integrated circuits.

Mechanisms of the sidewall facet evolution in lateral epitaxial overgrowth of GaN by MOCVD

The lateral epitaxial overgrowth of GaN was carried out by low-pressure metalorganic chemical vapor deposition, and the cross section shape of the stripes was characterized by scanning electron microscopy. Inclined {11–2n} facets (n≈1–2.5) were observed in the initial growth, and they changed gradually into the vertical {11–20} sidewalls in accordance with the process of the lateral overgrowth. A model was proposed utilizing diffusion equations and boundary conditions to simulate the concentration of the Ga species constituent throughout the concentration boundary layer. Solutions to these equations are found using the two-dimensional, finite element method. We suggest that the observed evolution of sidewall facets results from the variation of the local V/III ratio during the process of lateral overgrowth induced by the lateral supply of the Ga species from the SiNx mask regions to the growing GaN regions.

On the profile of the changing sign mountain pass solutions for an elliptic problem

We consider nonlinear elliptic equations with small diffusion and Dirichlet boundary conditions. We construct changing sign solutions with peaks close to the boundary and consider the location of the peak.

Electron-electron interactions, quantum Coulomb gap, and dynamical scaling near integer quantum Hall transitions

The effects of electron-electron interactions on tunneling into the bulk of a two-dimensional electron system are studied near the integer quantum Hall transitions. Taking into account the dynamical screening of the interactions in the critical conducting state, we show that the behavior of the tunneling density of states (TDOS) is significantly altered at low energies from its noninteracting counterpart. For the long-range Coulomb interaction, we demonstrate that the TDOS vanishes linearly at the Fermi level according to a quantum Coulomb gap form $\ensuremath{\nu}(\ensuremath{\omega}{)=C}_{Q}|\ensuremath{\omega}|{/e}^{4},$ with ${C}_{Q}$ a nonuniversal coefficient of a quantum-mechanical origin. In the case of short-range or screened Coulomb interactions, the TDOS is found to follow a power law $|\ensuremath{\omega}{|}^{\ensuremath{\alpha}},$ with \ensuremath{\alpha} proportional to the bare interaction strength. Since short-range interactions are known to be irrelevant perturbations at the noninteracting critical point, we predict that, upon scaling, the power law is smeared, leading to a finite zero-bias TDOS $\ensuremath{\nu}(\ensuremath{\omega})/\ensuremath{\nu}(0)=1+(|\ensuremath{\omega}|/{\ensuremath{\omega}}_{0}{)}^{\ensuremath{\gamma}},$ where \ensuremath{\gamma} is a universal exponent determined by the scaling dimension of short-ranged interactions. We also consider the case of quasi-one-dimensional (1D) samples with edges, i.e., the long Hall bar geometry, and find that the TDOS becomes dependent on the Hall conductance due to an altered boundary condition for diffusion. For short-range interactions, the TDOS of a quasi-1D strip with edges is linear near the Fermi level, with a slope inversely proportional to ${\ensuremath{\rho}}_{\mathrm{xx}}$ in the perturbative limit. These results are in qualitative agreement with the findings of bulk tunneling experiments. We discuss recent developments in understanding the role played by electron-electron interactions at the integer quantum Hall transitions and the implications of these results on the dynamical scaling of the transition width. We argue that for long-range Coulomb interactions, the existence of the quantum Coulomb gap in the quantum critical regime of the transition gives rise to the observed dynamical exponent $z=1.$

Poiseuille-type flow of a rarefied gas between two parallel plates driven by a uniform external force.

A unidirectional flow of a rarefied gas between two parallel plates driven by a uniform external force is investigated on the basis of kinetic theory with special interest in the behavior in the near continuum regime. The Bhatnagar-Gross-Krook (BGK) model of the Boltzmann equation and the diffuse reflection boundary condition are employed as the basic system. First, a systematic asymptotic analysis of the basic system for small Knudsen numbers is carried out, and a system of fluid-dynamic-type equations and their boundary conditions are derived up to the second order in the Knudsen number. Then, an accurate numerical analysis of the original BGK system is performed for a wide range of the Knudsen number by means of a finite-difference method. The behavior of the gas, such as the non-Navier-Stokes effects in the near continuum regime, is clarified on the basis of the fluid-dynamic-type system as well as the numerical solution of the BGK system.