Diffusion Boundary Conditions Research Articles

On the nonlinear thermal stress, thermal creep, and thermal edge flows in triangular cavities

Here, rarefied thermally driven flow is investigated in two-dimensional equilateral triangular cavities with different uniform wall temperatures. We used three different solvers, i.e., the direct simulation Monte Carlo solver, discrete unified gas kinetic scheme solver, and continuum set of equations of a slow non-isothermal flow solver. Two main cases were considered; in the first case, the cavity's base is considered hot, and the other sides were set cold. In the second case, the right half of the bottom wall was regarded as a diffuse reflector with high temperature, while the left half of the bottom border was set as a specular reflector. The adjacent side walls were set cold with diffuse reflector boundary conditions. The imposed temperature difference/wall boundary condition induces various vortices in the geometry. In case 1, we observe that principal vortices appearing in the triangle are due to nonlinear thermal stress effects, and the thermal creep effects cause other smaller, confined ones. In case 2, a thermal edge flow is set up from the specular wall on the way to the diffusive hot wall, creating a large vortex in the geometry. As the Knudsen number decreases, another small vortex appears near the left cold border.

Regularity of Stationary Boltzmann Equation in Convex Domains

The higher regularity estimate has been a challenging question regarding the Boltzmann equation in bounded domains. Indeed, it is well-known to have “the non-existence of a second order derivative at the boundary” in Guo et al. (Invent Math 207:115–290, 2017) even for symmetric convex domains such as a disk or sphere. In this paper, we answer this question in the affirmative by constructing the \(C^{1,\beta }\) solutions away from the grazing boundary, for any \(\beta <1\), to the stationary Boltzmann equation with the non-isothermal diffuse boundary condition in strictly convex domains, as long as a smooth wall temperature has small fluctuation pointwisely.

Dissipative Phase Transition with Driving-Controlled Spatial Dimension and Diffusive Boundary Conditions.

We investigate theoretically and experimentally a first-order dissipative phase transition, with diffusive boundary conditions and the ability to tune the spatial dimension of the system. The considered physical system is a planar semiconductor microcavity in the strong light-matter coupling regime, where polariton excitations are injected by a quasiresonant optical driving field. The spatial dimension of the system from 1D to 2D is tuned by designing the intensity profile of the driving field. We investigate the emergence of criticality by increasing the spatial size of the driven region. The system is nonlinear due to polariton-polariton interactions and the boundary conditions are diffusive because the polaritons can freely diffuse out of the driven region. We show that no phase transition occurs using a 1D driving geometry, while for a 2D geometry we do observe both in theory and experiments the emergence of a first-order phase transition. The demonstrated technique allows all-optical and insitu control of the system geometry, providing a versatile platform for exploring the many-body physics of photons.

Modeling and Numerical Simulation of the Thermal Interaction between Vegetation Cover and Soil

In this work, we propose a mathematical model representing the thermal interaction between vegetation cover and the soil underneath it. This model consists of a one-dimensional reaction–diffusion equation describing the evolution of the temperature in the vegetation cover coupled with a two-dimensional reaction–diffusion equation to represent the evolution of the temperature in the soil. The thermal interaction between the vegetation cover and the soil is studied and the distribution of temperatures in the soil with depth is also obtained. The vegetation cover acts in this model as a dynamic and diffusive boundary condition for the soil. The developed model takes into account the latent heat of fusion, which appears when the transformation of ice into liquid water or vice versa occurs inside the soil. The numerical approach for the solution of the mathematical model conducted in this work is based on the finite volume method with Weighted Essentially Non-Oscillatory technique for spatial reconstruction and the third-order Runge–Kutta Total Variation Diminishing numerical scheme is used for time integration, which is very efficient to obtain the numerical solution of this type of model. Some numerical examples are solved to obtain the distribution of temperature both in the vegetation cover and the soil.

Controllability of a Stokes system with a diffusive boundary condition

We are interested by the controllability of a fluid-structure interaction system where the fluid is viscous and incompressible and where the structure is elastic and located on a part of the boundary of the fluid domain. In this article, we simplify this system by considering a linearization and by replacing the wave/plate equation for the structure by a heat equation. We show that the corresponding system coupling the Stokes equations with a heat equation at its boundary is null-controllable. The proof is based on Carleman estimates and interpolation inequalities. One of the Carleman estimates corresponds to the case of Ventcel boundary conditions. This work can be seen as a first step to handle the real system where the structure is modeled by the wave or the plate equation.

Computer assisted proof of branches of stationary and periodic solutions, and Hopf bifurcations, for dissipative PDEs

We discuss an approach to the computer assisted proof of the existence of branches of stationary and periodic solutions for dissipative PDEs, using the Brussellator system with diffusion and Dirichlet boundary conditions as an example, We also consider the case where the branch of periodic solutions emanates from a branch of stationary solutions through a Hopf bifurcation.

Mutual Diffusion Coefficients and Refractive Index Increments of K2SO4(aq) at 298.15 K from Rayleigh Interferometry

Isothermal mutual diffusion coefficients (interdiffusion coefficients) were measured for K2SO4(aq) at 298.15 ± 0.005 K, at numerous concentrations ranging from dilute solutions to near saturation (0.59648 mol·dm–3; 0.61349 mol·kg–1) under free diffusion boundary conditions, using high precision Rayleigh interferometry. Under the experimental conditions these diffusion coefficients are on the volume-fixed reference frame Dv. Two series of experiments were performed, the first using the traditional experimental approach with a mercury lamp source and with the interference patterns being recorded on glass photographic plates, and the second with a He–Ne laser light source and computerized data acquisition using a photodiode array. The results from both series of experiments are in excellent agreement, and generally yield diffusion coefficients precise to at least 0.002 × 10–9 m2·s–1 (0.15% to 0.19%) and in most cases to 0.001 × 10–9 m2·s–1 (0.07% to 0.1%). These experiments also yield accurate values of the refractive index differences for the solution pairs used in the diffusion experiments. The new diffusion coefficients are compared to two sets of published values of diffusion coefficients for K2SO4(aq) which are somewhat discrepant from each other. This study extends and complements our earlier work on the diffusion coefficients of the most common brine salts: NaCl(aq), KCl(aq), MgCl2(aq), CaCl2(aq), Na2SO4(aq),]MgSO4(aq), NaHCO3(aq), and KHCO3(aq) at 298.15 K.

Study of the pressure distribution along a concentric annular channel

A study of an isothermal rarefied gas flow through a long concentric annular channel is carried out. The solution is based on the linearized BGK model with diffuse boundary conditions and it is valid in the whole range of the rarefaction parameter. The pressure distribution along the channel is obtained and investigated depending on the values of the pressure maintained at the channel ends.

Remarks on a class of integro-differential problems

A class of variational elliptic problems involving integro-differential diffusion and Dirichlet boundary condition is studied. The absence of a maximum principle is observed and sign-changing solutions are obtained when the nonlinearity is a sublinear or superlinear power of u+, depending on the size of a positive parameter multiplying the nonlocal term. More general integro-differential operators, for which the problem is no longer variational, are also considered.

Knudsen type group for time in ℝ and related Boltzmann type equations

We consider certain Boltzmann type equations on a bounded physical and a bounded velocity space under the presence of both reflective as well as diffusive boundary conditions. We introduce conditions on the shape of the physical space and on the relation between the reflective and the diffusive part in the boundary conditions such that the associated Knudsen type semigroup can be extended to time [Formula: see text]. Furthermore, we provide conditions under which there exists a unique global solution to a Boltzmann type equation for time [Formula: see text] or for time [Formula: see text] for some [Formula: see text] which is independent of the initial value at time 0. Depending on the collision kernel, [Formula: see text] can be arbitrarily small.

Incompressible Euler Limit from Boltzmann Equation with Diffuse Boundary Condition for Analytic Data

A rigorous derivation of the incompressible Euler equations with the no-penetration boundary condition from the Boltzmann equation with the diffuse reflection boundary condition has been a challenging open problem. We settle this open question in the affirmative when the initial data of fluid are well-prepared in a real analytic space, in 3D half space. As a key of this advance, we capture the Navier-Stokes equations of $$\begin{aligned} \textit{viscosity} \sim \frac{\textit{Knudsen number}}{\textit{Mach number}} \end{aligned}$$satisfying the no-slip boundary condition, as an intermediary approximation of the Euler equations through a new Hilbert-type expansion of the Boltzmann equation with the diffuse reflection boundary condition. Aiming to justify the approximation we establish a novel quantitative \(L^p\)-\(L^\infty \) estimate of the Boltzmann perturbation around a local Maxwellian of such viscous approximation, along with the commutator estimates and the integrability gain of the hydrodynamic part in various spaces; we also establish direct estimates of the Navier-Stokes equations in higher regularity with the aid of the initial-boundary and boundary layer weights using a recent Green’s function approach. The incompressible Euler limit follows as a byproduct of our framework.

Boundary layer of the Boltzmann equation in 2-dimensional convex domains

Consider the stationary Boltzmann equation in 2D convex domains with diffusive boundary condition. In this paper, we establish the hydrodynamic limits while the boundary layers are present, and derive the steady Navier-Stokes-Fourier system with non-slip boundary conditions. Our contribution focuses on novel weighted $W^{1,\infty}$ estimates for the Milne problem with geometric correction. Also, we develop stronger remainder estimates based on an $L^{2m}-L^{\infty}$ framework.

Linear harmonic oscillatory rarefied gas flow with arbitrary frequency in comb finger blocks

The linear oscillatory two-dimensional rarefied gas flow in the zig-zag channel between the shuttle and stator fingers of typical comb elements, due to the harmonic motion of the shuttle finger with arbitrary frequency, is numerically investigated, via the linearized unsteady Shakhov kinetic model equation, subject to purely diffuse boundary conditions. The amplitude and phase distributions of the velocity components, normal and shear stresses, density and temperature are thoroughly investigated in the whole range of the gas rarefaction and oscillation parameters, characterizing the flow. The low, moderate and high flow oscillation regimes are defined by the ratio of the gas rarefaction over the oscillation parameters. It is shown that in the low oscillation regime the flow may be approximated by the corresponding steady-state analysis, while in the high oscillation one by the superposition of the associated oscillatory one-dimensional normal sound propagation and Couette flows. In the former case edge and compressibility effects are considerable and in the latter one the flow is characterized by gas trapping along the shuttle finger. In the moderate oscillation regime the fully oscillatory two-dimensional flow must be analysed. Here, the antiresonance and resonance states of the flow are considered. The ratios of the gas rarefaction over the oscillation parameters, where the average normal and shear stresses acting on the shuttle finger are minimized are computationally specified. Changing the main geometry parameters of the comb finger block, significantly affects the normal and shear stresses acting on the shuttle finger, mainly in the low oscillation regime. The present work may be applied to minimize the damping forces in comb finger blocks, as well as to optimize the design of new generation devices operating in moderate and high frequencies.

An integrated semigroup approach for age structured equations with diffusion and non-homogeneous boundary conditions

In this work, we consider a linear age-structured problem with diffusion and non-homogeneous boundary conditions both for the age and the space variables. We handle this linear problem by re-writing it as a non-densely defined abstract Cauchy problem. To that aim we develop a new result on the closedness of a commutative sum of two non-densely defined operators by using the theory of integrated semigroups. As an application of this abstract result, we are able to associate a suitable integrated semigroup to some age-structured problem with spatial diffusion and equipped with non-homogeneous boundary conditions. This integrated semigroup is characterized by the description of its infinitesimal generator. Further applications of our abstract result are also given to the commutative sum of two almost sectorial operators, for which we derive a closedness results.

Existence of a nonequilibrium steady state for the nonlinear BGK equation on an interval

We show existence of a non-equilibrium steady state for the one-dimensional, non-linear BGK model on an interval with diffusive boundary conditions. These boundary conditions represent the coupling of the system with two heat reservoirs at different temperatures. Our analysis is not perturbative around the equilibrium. We employ a fixed point argument to reduce the study of the model with non-linear collisional interactions to the linear BGK.

Compressible Navier-Stokes approximation for the Boltzmann equation in bounded domains

It is well known that the full compressible Navier-Stokes equations can be deduced via the Chapman-Enskog expansion from the Boltzmann equation as the first-order correction to the Euler equations with viscosity and heat-conductivity coefficients of order of the Knudsen number $\epsilon>0$. In the paper, we carry out the rigorous mathematical analysis of the compressible Navier-Stokes approximation for the Boltzmann equation regarding the initial-boundary value problems in general bounded domains. The main goal is to measure the uniform-in-time deviation of the Boltzmann solution with diffusive reflection boundary condition from a local Maxwellian with its fluid quantities given by the solutions to the corresponding compressible Navier-Stokes equations with consistent non-slip boundary conditions whenever $\epsilon>0$ is small enough. Specifically, it is shown that for well chosen initial data around constant equilibrium states, the deviation weighted by a velocity function is $O(\epsilon^{1/2})$ in $L^\infty_{x,v}$ and $O(\epsilon^{3/2})$ in $L^2_{x,v}$ globally in time. The proof is based on the uniform estimates for the remainder in different functional spaces without any spatial regularity. One key step is to obtain the global-in-time existence as well as uniform-in-$\epsilon$ estimates for regular solutions to the full compressible Navier-Stokes equations in bounded domains when the parameter $\epsilon>0$ is involved in the analysis.

Impact of Degradation and Material Crystallinity on the Mechanical Performance of a Bioresorbable Polymeric Stent

Bioresorbable polymeric stents (BPS) offer possibilities to help address the long-term complications associated with permanent vascular implants, however in-vivo degradation behaviour is not yet fully understood. Here, finite element analysis (FEA) techniques based on physio-chemical reaction diffusion equations are used to predict and analyse BPS degradation behaviour. Physio-chemical degradation models for polymers, both amorphous and semi-crystalline, are incorporated into the FEA software package Abaqus/Standard allowing for BPS degradation rate predictions to be made, with a focus on poly-L-lactide (PLLA). The outputs of the degradation models are linked to mechanical behaviour via three different damage models which couple the changes in molecular weight and crystallinity with a hyperelastic constitutive model for PLLA mechanical behaviour. A simplified representation of a PLLA BPS in an artery is used as a demonstration case. The effects of applied degradation product diffusion boundary conditions on the molecular weight and crystallinity of PLLA BPS under simulated degradation are examined, and the impact of material heterogeneities and mechanical load boundary condition on the scaffolding performance and elastic properties of the degrading stent are investigated. The results suggest that the BPS performance are strongly dependent on the assumed boundary conditions, both in terms of degradation product diffusion and mechanical loading.

Mutual Diffusion Coefficients of Aqueous Na2SO4 at C = 0.02989 mol·dm–3 and 298.15 K by Using Rayleigh Interferometry with Free Diffusion Boundary Conditions: Experimental Test of the Effect of the C3/2 Concentration Dependence of Refractive Index at Low Concentration

Rayleigh interferometry is one of the most precise and accurate methods for the experimental determination of the mutual diffusion coefficients of both electrolytes and non-electrolytes in liquid solutions. For binary solutions at moderate and high concentrations where the concentration dependences of both diffusion coefficient and refractive index can be assumed to be linear or almost linear and the concentration difference ∆C between a pair of solutions undergoing free diffusion is not large compared to their average concentration, and by using proper combinations of the interference fringe positions as they vary with time, under these conditions diffusion coefficients are obtained that are independent of the size of ∆C. However, this situation becomes more complicated at low concentrations for electrolyte solutions where the concentration dependences of both the diffusion coefficient and refractive index are non-linear, with C1/2 dependence for the diffusion coefficient and C3/2 dependence for the refractive index. As a test of these effects on calculated mutual diffusion of electrolyte solutions at low concentrations as measured by Rayleigh interferometry, apparent mutual diffusion coefficients Dv,a (volume-fixed reference frame) were measured for fixed average concentrations C = (0.02989 ± 0.00001) mol·dm–3 of Na2SO4(aq) at 298.15 K while varying ∆C/C from 0.66673 to 2, a threefold variation. The dependence of the diffusion coefficient on ∆C/C was found to be linear, yielding Dv = 1.0416 × 10–9 m2·s–1 for an infinitely small concentration difference between the diffusing solutions.

Penetration of non-uniform electromagnetic field into conducting body

The study is based on the exact analytical solution for the general conjugation problem of three-dimensional quasi-stationary field at a flat interface between dielectric and conducting media. It is determined that non-uniform electromagnetic field always decreases in depth faster than uniform field. The theoretical conclusion is confirmed by comparing the results of analytical and numerical calculations. The concept of strong skin effect is extended to the case when penetration depth is small not only compare to the characteristic body size, but also when the ratio of the penetration depth to the distance from the surface of body to the sources of the external field is small parameter. For strong skin effect in its extended interpretation, the influence of external field non-uniformity to electromagnetic field formation both at the interface between dielectric and conducting media and to the law of decrease field in conducting half-space is analyzed. It is shown, at the interface the expressions for the electric and magnetic intensities in the form of asymptotic series in addition to local field values of external sources contain their derivatives with respect to the coordinate perpendicular to the interface. The found expressions made it possible to generalize the approximate Leontovich impedance boundary condition for diffusion of non-uniform field into conducting half-space. The difference between the penetration law for the non-uniform field and the uniform one takes place in the terms of the asymptotic series proportional to the small parameter to the second power and to the second derivative with respect to the vertical coordinate from the external magnetic field intensity at the interface.

Impact of Soret and Dufour on MHD Casson fluid flow past a stretching surface with convective–diffusive conditions

The intension of current inquiry is to highlight the impact of magnetohydrodynamic Casson fluid flow across a convective surface with cross diffusion, chemical reaction, non-linear radiative heat are accounted. Convection and diffusive boundary conditions are deemed. The governing flow equations for the flow fields are converted into non-dimensional form by using appropriate similarity variables. These ODEs are solved by fourth order Runge–Kutta and Shooting method. The outcomes are bestowed with plots and table for the influence of diverse flow variables on the flow fields. We observed that depreciate temperature and appreciate concentration with the upsurge in Sorret number, whereas a negative impact is perceived with Dufour number. A decrement in Nusselt number and an enhancement in Sherwood number are observed for the intensification of non-linear radiative parameter.