Diffusion Boundary Conditions Research Articles

Stability of solutions to the logistic equation with delay, diffusion and nonclassical boundary conditions

Stability of solutions to the logistic equation with delay, diffusion and nonclassical boundary conditions

Fractional diffusion limit for a kinetic Fokker–Planck equation with diffusive boundary conditions in the half-line

Fractional diffusion limit for a kinetic Fokker–Planck equation with diffusive boundary conditions in the half-line

A design optimization method for rarefied and continuum gas flows

A design optimization method applicable to both rarefied and continuum gas flows with good efficiency is proposed in the framework of gas-kinetic theory. The method follows the methodology of topology optimization where the areas of gas and solid are marked by the material density, based on which a fictitious porosity model is used to reflect the effect of the solid on the gas and mimic the diffuse boundary condition on the gas-solid interface. The formula of this fictitious porosity model is modified to make the model work well from the free-molecular flow regime to the continuum flow regime. To find the optimized material density, a gradient-based optimizer is adopted and the design sensitivity is obtained by the continuous adjoint method. To solve the primal kinetic equation and the corresponding adjoint equation, efficient multiscale numerical schemes are constructed. Several airfoil optimization problems are solved to demonstrate the performance of the present design optimization method in wide ranges of flow speed and gas rarefaction. It is found that the thickness of the optimized airfoil first increases and then decreases as the Knudsen number increases, and on the supersonic condition the optimized airfoil has quite different shapes of leading edge for different degrees of gas rarefaction.

Global asymptotical stability and Hopf bifurcation for a three‐species Lotka‐Volterra food web model

In this article, we consider a delayed three‐species Lotka‐Volterra food web model with diffusion and homogeneous Neumann boundary conditions. We proved that the positive constant equilibrium solution is globally asymptotically stable for the system without time delays. By virtue of the sum of time delays as the bifurcation parameter, spatially homogeneous and inhomogeneous Hopf bifurcation at the positive constant equilibrium solution are proved to occur when the delay varied through a sequence of critical values. In addition, we consider the effect of cross‐diffusion on the system in the case that without time delays. By taking cross diffusion coefficients as the bifurcation parameter, our model undergoes inhomogeneous Hopf bifurcation around a positive constant equilibrium solution when the bifurcation parameter is varied through a sequence of critical values. A common feature in the most existing research work is that the bifurcation factor that induces Hopf bifurcation appears in the reaction terms (such as time delay) rather than diffusion terms. Our results demonstrate that the inhomogeneous Hopf bifurcation can be triggered by the effect of cross diffusion factors.

Numerical analysis of a gas flow in a square cavity driven by spanwise lid motion on the basis of kinetic theory: Behavior of the gas near a sharp corner

A gas flow in a square cavity driven by a lid sliding in the direction of its line of contact with the cavity wall is considered. The steady behavior of the gas is numerically investigated based on the linearized Bhatnagar–Gross–Krook kinetic equation and the diffuse reflection boundary condition. When one applies the Stokes equation and the no-slip boundary condition to the system considered here, the flow velocity becomes multivalued at the corner between the lid and the cavity wall, and the shear stress diverges at the corner inversely proportionally to the distance from there, which is known as the so-called corner singularity. In the present work, the behavior of the gas near the corner is examined based on numerical results obtained from the kinetic theory. Although the range of the flow velocity value in the kinetic solution is limited due to the significant velocity slip near the corner, the flow velocity is, nevertheless, multivalued at the corner. The shear stress varies inversely proportionally to the distance from the corner up to the position that is a few tens of mean free paths away from there. The increase in the stress is suppressed at positions closer to the corner and its magnitude remains bounded. Thus, the total forces acting on the lid and the side cavity walls are bounded as well. Due to the distinctive behavior of the stress near the corner, the resulting nondimensional total forces behave with an unconventional rate Kn ln Kn for small Knudsen numbers Kn.

Influence of boundary conditions on non-equilibrium heat transport under ultrafast laser action based on the lattice Boltzmann method

Based on the lattice Boltzmann method, this paper simulated the non-equilibrium heat transfer process in nano-silicon thin film under ultrafast laser irradiation. The influence of boundary conditions on heat transport was investigated. Results show that under rebound and diffuse boundary conditions, the energy distribution within the film is non-uniform due to interface effects. However, the specular boundary condition is equivalent to eliminating the interface, resulting in a smooth energy distribution. Under convective boundary conditions, the thermal wave phenomenon disappears owing to the open interface. When the energy tends to be stable, the energy density under convective boundary conditions is reduced to 50% of that under adiabatic boundary conditions. As the film size decreases, the differences between boundary conditions become more significant. Particularly when the film size is smaller than the phonon mean free path, the influence of boundary conditions cannot be neglected. Therefore, boundary conditions and size effects are important for the design of nanodevices. Furthermore, compared with the results obtained by the Cattaneo-Vernotte model, it is found that at smaller Knudsen numbers, the Cattaneo-Vernotte model has a better match with the lattice Boltzmann method.

The Inner Disk Rim of HD 163296: Linking Radiative Hydrostatic Models with Infrared Interferometry

Previous studies of the protoplanetary disk HD 163296 revealed that the morphology of its sub-au infrared emission encompasses the terminal sublimation front of dust grains, referred to as the inner rim, but also extends into the (supposedly) dust-free region within it. Here, we present a set of radiative hydrostatic simulations of the inner rim in order to assess how much the rim alone can contribute to the observed interferometric visibilities V, half-light radii R hl, and fractional disk fluxes F in the wavelength range 1.5–13 μm. In our set of models, we regulate the cooling efficiency of the disk via the boundary condition for radiation diffusion and we also modify the shape of the sublimation front. We find that when the cooling efficiency is reduced, the infrared photosphere at the rim becomes hotter, leading to an increase in R hl sufficient to match the observations. However, the near-infrared disk flux is typically too low ( F≃0.25 at 1.5 μm), resulting in H-band visibility curves located above the observed data. We show that the match to the H-band observations up to moderate baselines can be improved when a wall-shaped rather than curved sublimation front is considered. Nevertheless, our model visibilities always exhibit a bounce at long baselines, which is not observed, confirming the need for additional emission interior to the rim. In summary, our study illustrates how the temperature structure and geometry of the inner rim need to change in order to boost the rim’s infrared emission.

An integrated mechanical degradation model to explore the mechanical response of a bioresorbable polymeric scaffold

Simulation of bioresorbable medical devices is hindered by the limitations of current material models. Useful simulations require that both the short- and long-term response must be considered; existing models are not physically-based and provide limited insight to guide performance improvements. This study presents an integrated degradation framework which couples a physically-based degradation model, which predicts changes in both crystallinity (Xc) and molecular weight (Mn), with the results of a micromechanical model, which predicts the effective properties of the semicrystalline polymer. This degradation framework is used to simulate the deployment of a bioresorbable PLLA (Poly (L-lactide) stent into a mock vessel and the subsequent mechanical response during degradation under different diffusion boundary conditions representing neointimal growth. A workflow is established in a commercial finite element code that couples both the immediate and long-term responses. Clinically relevant lumen loss is reported and used to compare different responses and the effect of neo-intimal tissue regrowth post-implantation on degradation and on the mechanical response is assessed. In addition, the effects of possible changes in Xc, which could occur during processing and stent deployment, are explored.

Finite-element simulation of ethanol diffusion in nonencapsulated subdural hematomas

Determination of the diffusion regularities of ethanol in nonencapsulated subdural hematomas. The finite-element modelling of the ethanol's concentration field for the non-lethal nonencapsulated subdural hematoma using the ELCUT 6.5 application was carried out. A two-dimensional finite-element concentration field model of a low volume subdural hematoma, taking into account the different boundary conditions of ethanol diffusion, has been developed. It has been proven that the diffusion rate of ethanol is determined by the impact of its initial concentration field, the level of ethanolemia, the geometry of hematoma, as well as the diffusion properties of bounding with hematoma tissues. The possibility of ethanol diffusion in the content of a nonalcoholic hematoma from bounding tissues was shown. The central region of hematoma has been found to have the highest ethanol concentration stability. If the diffusion's boundary conditions are not identical, the indicated area shifts to the surface of hematoma with a lower density of diffusion's flow. It is recommended to interpret the postmortem concentration of ethanol in the most diffusion-resistant region of hematoma as the minimum limit of ethanolemia during the hematoma formation.

The Boltzmann equation for plane Couette flow

In the paper, we study the plane Couette flow of a rarefied gas between two parallel infinite plates at y=\pm L moving relative to each other with opposite velocities (\pm \alpha L,0,0) along the x -direction. Assuming that the stationary state takes the specific form of F(y,v_{x}-\alpha y,v_{y},v_{z}) with the x -component of the molecular velocity sheared linearly along the y -direction, such steady flow is governed by a boundary value problem for a steady nonlinear Boltzmann equation driven by an external shear force under the homogeneous nonmoving diffuse reflection boundary condition. In the case of the Maxwell molecule collisions, we establish the existence of spatially inhomogeneous nonequilibrium stationary solutions to the steady problem for any small enough shear rate \alpha>0 via an elaborate perturbation approach using Caflisch’s decomposition together with Guo’s L^{\infty}\cap L^{2} theory. The result indicates a polynomial tail at large velocities for the stationary distribution. Moreover, the large time asymptotic stability of the stationary solution with exponential convergence is also obtained and as a consequence the nonnegativity of the steady profile is justified.

Lattice Boltzmann simulations for the drying of porous media with gas–side convection–diffusion boundary

One of the most significant challenges in describing the drying of porous materials is the realistic integration of internal transport phenomena into mathematical models, coupled with the external free flow. However, the intricacies of pore-scale geometry make experimentation and observation very difficult in practice, thereby making modeling techniques a useful tool for the analysis of the drying process. Among the many modeling techniques developed for drying, the lattice Boltzmann method (LBM)-based modeling approach has gained favor in recent years due to its ability to incorporate realistic geometry and transport at the pore scale. Our previous works on Shan–Chen LBM for drying of capillary porous media were based on the Bhatnagar–Gross–Krook collision operator and diffusion interface boundary conditions. This study elucidates the drying of a capillary porous medium under the influence of convection–diffusion boundary conditions at the gas side, using Shan–Chen LBM. The pore-scale effects of convection–diffusion conditions during the drying process are presented in relation to the macroscale drying kinetics. Moreover, the differences between the convection–diffusion kinetics and purely diffusion–dominated kinetics of the drying process are also presented here. This work also aims to incorporate the convection–diffusion transport phenomena into the drying process of a porous medium under the influence of an imposed thermal gradient, establishing and studying the phenomena of stabilization and destabilization of the drying front under the influence of a temperature gradient, thereby extending the lattice Boltzmann method of modeling for the simulation of convection–diffusion drying, both for the isothermal case and the imposition of a thermal gradient.

Subdiffusion–Superdiffusion Random-Field Transition

A contaminant spreading affected by a random field at boundaries in the comb geometry is considered. The physical effect of the random boundary conditions results in increasing a transport exponent such that the mean squared displacement increases with time from t12 to t12+5α/2 for real 0≤α≤1. This stochastic acceleration due to these space-time-dependent boundary conditions leads to a transition from subdiffusion to superdiffusion. Experimentally, it can be realized by controlling the boundary conditions of 2D diffusion in the comb geometry.

Hölder regularity of the Boltzmann equation past an obstacle

AbstractRegularity and singularity of the solutions according to the shape of domains is a challenging research theme in the Boltzmann theory. In this paper, we prove an Hölder regularity in for the Boltzmann equation of the hard‐sphere molecule, which undergoes the elastic reflection in the intermolecular collision and the contact with the boundary of a convex obstacle. In particular, this Hölder regularity result is a stark contrast to the case of other physical boundary conditions (such as the diffuse reflection boundary condition and in‐flow boundary condition), for which the solutions of the Boltzmann equation develop discontinuity in a codimension 1 subset (Kim [Comm. Math. Phys. 308 (2011)]), and therefore the best possible regularity is BV, which has been proved by Guo et al. [Arch. Rational Mech. Anal. 220 (2016)].

Non-similar approach on the MHD Carreau nanofluid flow with quadratic radiation and Soret-Dufour effects

The present study aims to investigate the influence of magnetohydrodynamic (MHD) Carreau nanofluid flow past a stretching cylinder with quadratic Rosseland heat radiation. This paper examines the consequences of the Soret-Dufour effects when considering the influence of thermophoresis and Brownian effects. The convective and diffusive boundary conditions have been implemented. The modeled mathematical system of non-linear partial differential equations (PDEs) is transformed into a dimensionless representation using a non-similar approach. The ensuing set of dimensionless equations are solved numerically with local non-similarity method (LNM) aided by the finite difference algorithm. The findings of the study unveil that the presence of the Dufour and Soret effect declines the heat transfer and mass transfer rates, respectively. It is also noted that flow profiles are more profound in the case of stretching cylinder configuration. Per unit increase in the hydrodynamic slip parameter augments the drag coefficient by 35.87% and 33.40% for cylinder and sheet configurations, respectively. The present study has potential applications in biomedicine, such as targeted drug delivery, hyperthermia, theranostics and cardiovascular treatments.

Continuity of attractors for singularly perturbed semilinear problems with nonlinear boundary conditions and large diffusion

We exhibit singularly perturbed parabolic problems with large diffusion and nonhomogeneous boundary conditions for which the asymptotic behavior can be described by a one-dimensional ordinary differential equation. We estimate the continuity of attractors in Hausdorff’s metric by the rate of convergence of resolvent operators. Moreover, we will show explicitly how this estimate of continuity varies exponentially with the fractional power spaces Xα for α in an appropriate interval.

Analysis of the absorbing boundary conditions for anomalous diffusion in comb model with Cattaneo model in an unbounded region

A fractional governing equation is derived from the two-dimensional anomalous diffusion in the comb model with the Cattaneo model by using rigorous derivation. One strategy to deal with the unbounded region is to create acceptable truncation. We abandon the traditional method of approximating infinite boundaries by enormous value and use the artificial boundary method to construct absorbing boundary conditions using the Laplace transform. The absorbing boundary conditions with the Mittag-Leffler function are obtained, and the stability is demonstrated. We discretise the governing equation by using the finite difference method, and the accuracy of the numerical method is confirmed by comparing with the exact solution, which is generated by introducing a source term. The particle distributions and the mean square displacement under the absorbing boundary conditions are in good agreement with the exact expressions which are superior to the conventional direct truncation boundary conditions. Additionally, the particle distributions under different parameters are examined and explained graphically.

On application of the regularized lattice Boltzmann method for isothermal flows with non-vanishing Knudsen numbers

Despite the advances in microscale modelling capabilities, physically observed non-zero Knudsen number (Kn) effects still elude the lattice Boltzmann method (LBM). This article details various Regularized LBM (RLBM) strategies used to model microscale flows and presents a thorough analysis on their applicability for the force-driven Poiseuille flow problem. In particular, we compare the formulations of eight schemes: Projected RLBM (P-RLBM), standard and Galilean Invariant Filtered Collision RLBM, Kn-dependant P-RLBM, Recursively Regularized LBM (R-RLBM), Hybrid R-RLBM, Central-Moment RLBM, and Onsager-Regularized LBM. We demonstrate, at different Kn with discrete Maxwell’s diffuse kinetic wall boundary conditions, the numerical equivalence of the considered RLBM schemes for hard sphere molecules on the D2Q9 lattice. Thereafter, we include a Kn-dependant wall-modified mean-free-path and extend the characterization onto higher-order D2Q17, D2Q21 and Q2Q25-Zero-One-Three, Q2Q49-Zero-One-Two-Three lattices. The study indicates that RLBM schemes are Navier-Stokes solvers which fail to entirely capture all the non-equilibrium phenomenon of slip/transition regime flows.

Space-angle discontinuous Galerkin method for 2D RTE with diffusively and specularly reflective boundary conditions

A space-angle discontinuous Galerkin (saDG) method for solving radiative transfer problems in arbitrary 2D domains is proposed. The space-angle domain is fully discretized by the DG formulation. The angular decomposition scheme is adopted to help accelerate the process by solving the radiative transfer equation iteratively. The performance of the method is analyzed in the square geometry with different types of boundary conditions. We found that the number of iteration or convergence rate of the iteration for the specular reflection problems is not only effected by the number of the reflective surfaces in the domain, but also by the scattering and extinction properties of the media. the uniformly and directionally diffuse reflective boundary conditions are compared in the same medium. In addition, the approach is applied to the simulation of radiative transfer in a parabolic reflector to obtain the localization at the focal point.

Analysis of absorbing boundary conditions for the anomalous diffusion in comb model on unbounded domain by finite volume method

The diffusion in the comb model is an important kind of anomalous diffusion which is described by a governing equation containing the Dirac delta function and the solution’s domain is infinite. Two key problems are solved to analyze the mass transfer mechanism in the comb model. One is to construct the appropriate and reasonable boundaries for treating the infinite boundaries. The exact absorbing boundary conditions with the Caputo’s fractional derivative are deduced by the Laplace transform technique. The other one is applying the finite volume method by integrating the governing equation over a governed volume to obtain the numerical solution with the advantage that the effects of the singular Dirac functions disappear. In addition, by introducing a source term, the exact solution is deduced and the comparison between the numerical solution and the exact solution is presented, which verifies the effectiveness and reasonableness of the numerical method. The particle distributions with different parameters are analyzed and discussed. By comparing with the traditional truncated boundary method, it is noteworthy that the mean square displacement for the absorbing boundary conditions has a good agreement with the exact expression.

Exponential mixing of Vlasov equations under the effect of gravity and boundary

In this paper, we study exponentially fast mixing induced/enhanced by gravity and stochastic boundary in the kinetic theory of Vlasov equations. We consider the Vlasov equations with and without a vertical magnetic field inside a horizontally-periodic 3D half-space equipped with a non-isothermal diffusive reflection boundary condition of bounded continuous boundary temperature at the bottom. We construct both stationary solutions and global-in-time dynamical solutions in L∞. We prove that moments of a dynamical fluctuation around the steady solutions decay exponentially fast in L∞. As a key of this proof, we establish a uniform bound of so-called residual measures independently of the bouncing number of stochastic characteristics, by constructing a continuous stationary outgoing boundary flux which is strictly positive almost everywhere.