Diffusion of methylene blue in glass fibers – Application of the shrinking core model

Diffusion of mass in a solid cylinder with concentration dependent diffusivity (or temperature-dependent thermal conductivity in case of heat diffusion) does not admit of an analytical solution except in special cases. The ‘shrinking core model’ has been used to develop an approximate analytical solution in certain circumstances. The model, generally useful to describe heterogeneous solid–fluid reactions, is applied to theoretically analyze the adsorption–diffusion phenomena of methylene blue dye in a glass fiber in the present work. Theoretical equations have been derived for the case of diffusivity as an exponential function of concentration. The diffusivity parameters are evaluated by global minimization of the error between the experimental and the theoretical concentration history. Other forms of diffusivity, namely constant diffusivity and diffusivity varying linearly with concentration are found to involve larger errors. A parametric sensitivity analysis of the error has been done. The shrinking core model could satisfactorily interpret the experimental dye concentration profile in the substrate.     

The Asymptotic Limit for the 3D Boussinesq System

In this paper, we show the asymptotic limit for the 3D Boussinesq system with zero viscosity limit or zero diffusivity limit. By the classical energy method, we prove that as viscosity(or diffusivity) coefficient goes to zero the solutions of the fully viscous equations converges to those of zero viscosity(or zero diffusivity) equations, which extend the previous results on the asymptotic limit under the conditions of the zero parameter(zero viscosity ν = 0 or zero diffusivity η = 0) in 2D case separately.     

Two variants of magnetic diffusivity stabilized finite element methods for the magnetic induction equation

We consider the time‐dependent magnetic induction model where the sought magnetic field interacts with a prescribed velocity field. This coupling results in an additional force term and time dependence in Maxwell's equation. We propose two different magnetic diffusivity stabilized continuous nodal‐based finite element methods for this problem. The first formulation simply adds artificial magnetic diffusivity to the partial differential equation, whereas the second one uses a local projected magnetic diffusivity as stabilization. We describe those methods and analyze them semi‐discretized in space to get bounds on stabilization parameters where we distinguish equal‐order elements and Taylor‐Hood elements. Different numerical experiments are performed to illustrate our theoretical findings.     

Boundary-layer effects for the 2-D Boussinesq equations with vanishing diffusivity limit in the half plane

The main purpose of this paper is to justify the Stokes-Blasius law of boundary-layer thickness for the 2-D Boussinesq equations with vanishing diffusivity limit in the half plane, i.e., we shall prove that the boundary-layer thickness is of the value δ(ε)=εα with any α∈(0,1/2) for small diffusivity coefficient ε>0. Moreover, the convergence rates of the vanishing diffusivity limit are also obtained.     

On a generalized Fisher equation

A generalized non-linear Fisher equation in cylindrical coordinates with radial symmetry is studied from Lie symmetry point of view. In the classical Fisher equation the reaction diffusion term is replaced with a general function to accommodate more equations of this type. Moreover, the diffusivity is assumed to be a function of the dependent variable to account for many real situations. An attempt is made to classify the diffusivity function and exact solutions are obtained in some cases.     

Random perturbations of 2-dimensional hamiltonian flows

We consider the motion of a particle in a periodic two dimensional flow perturbed by small (molecular) diffusion. The flow is generated by a divergence free zero mean vector field. The long time behavior corresponds to the behavior of the homogenized process - that is diffusion process with the constant diffusion matrix (effective diffusivity). We obtain the asymptotics of the effective diffusivity when the molecular diffusion tends to zero.     

Determination of a time-dependent diffusivity from nonlocal conditions

The inverse problem of determining the temperature and the time-dependent thermal diffusivity from various additional nonlocal information is investigated. These nonlocal conditions can come in the form of an internal or boundary energy, or, in the one-dimensional case, as a difference boundary temperature or heat flux so as to ensure the uniqueness of solution for the heat conduction equation with unknown thermal diffusivity coefficient. The Ritz-Galerkin method with satisfier function is employed to solve the inverse problems numerically. Numerical results are presented and discussed.     

Boundary layers associated with the 3-D Boussinesq system for Rayleigh–Bénard convection

We investigate the boundary layer effects of the 3-D incompressible Boussinesq system for Rayleigh–Bénard convection with vanishing diffusivity limit. By adopting the multi-scale analysis and the asymptotic expansion methods of singular perturbation theory, we construct an exact approximating solution for the viscous and diffusive Boussinesq system with well-prepared initial data. In addition, we obtain the convergence result of the vanishing diffusivity limit.     

Cascade Continuum Micromechanics Model for the Effective Diffusivity of Porous Materials: Exponential Hierarchy across Cascade Levels

The effective molecular diffusivity of porous materials such as cementitious, geological or synthetic materials is strongly affected by the complexity of the pore-space which may span across multiple scales from the nanometer to the sub-millimeter range. Recently, a semi-analytical Cascade Continuum Micromechanics (CCM) Model [1,2] was proposed, which allows to compute estimates of the effective diffusivity given the porosity ϕ, the intrinsic diffusivity of molecules in the pore-fluid D and a complexity parameter n. In contrast to existing micromechanics models, the CCM model is able to predict a physically consistent percolation threshold. In this paper, the classical CCM model is generalized so as to characterize a pore-volume distribution and its influence on the effective diffusivity at a particular cascade level. The generalized CCM model shows an exponential pore-volume distribution across cascade levels that is implicitly included in the classical CCM model. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical study of pattern formation in an extended Gray–Scott model

We present the temporal evolution of noise-controlled patterns in a spatially extended Gray–Scott model firstly. We show that the model exhibits a transition from stripe-spot growth to isolated spots, and also to spiral replication. Furthermore, we establish an extended Gray–Scott model with time-varying diffusivity, and find that the patterns exhibit transition from stripe-spot growth to stripe-spot or chaos replication. Additional studies reveal that with noise and time-varying diffusivity together, a new time-dependent pattern—a few of stripes oscillate in the “red” region—emerges, which hasn’t been reported before.     

The role of non-linear diffusion in non-simultaneous blow-up

We study a parabolic system of two non-linear reaction-diffusion equations completely coupled through source terms and with power-like diffusivity. Under adequate hypotheses on the initial data, we prove that non-simultaneous blow-up is sometimes possible; i.e., one of the components blows up while the other remains bounded. The conditions for non-simultaneous blow-up rely strongly on the diffusivity parameters and significant differences appear between the fast-diffusion and the porous medium case. Surprisingly, flat (homogeneous in space) solutions are not always a good guide to determine whether non-simultaneous blow-up is possible.     

等离子体反常输运性质的量纲分析

反常输运现象是实验室等离子体和空间等离子体中的重要过程．该文运用量纲分析方法研究了无碰撞磁化等离子体的反常输运性质,得出反常电导率、反常扩散系数、反常热导系数和背景物理量之间的明显关系式．结果表明,反常扩散系数具有玻姆扩散系数形式,反常热导系数具有Ｎ_αＴ_α／ｅＢ形式,电导率表达式也符合实验结果．     

Global regularity for the 2D Boussinesq equations with partial viscosity terms

In this paper, we prove the global in time regularity for the 2D Boussinesq system with either the zero diffusivity or the zero viscosity. We also prove that as diffusivity (viscosity) tends to zero, the solutions of the fully viscous equations converge strongly to those of zero diffusion (viscosity) equations. Our result for the zero diffusion system, in particular, solves the Problem no. 3 posed by Moffatt in [R.L. Ricca, (Ed.), Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001, pp. 3-10].     

Stable spike clusters for the precursor Gierer–Meinhardt system in $$\mathbb {R}^2$$

We consider the Gierer–Meinhardt system with small inhibitor diffusivity, very small activator diffusivity and a precursor inhomogeneity. For any given positive integer k we construct a spike cluster consisting of k spikes which all approach the same nondegenerate local minimum point of the precursor inhomogeneity. We show that this spike cluster can be linearly stable. In particular, we show the existence of spike clusters for spikes located at the vertices of a polygon with or without centre. Further, the cluster without centre is stable for up to three spikes, whereas the cluster with centre is stable for up to six spikes. The main idea underpinning these stable spike clusters is the following: due to the small inhibitor diffusivity the interaction between spikes is repulsive, and the spikes are attracted towards the local minimum point of the precursor inhomogeneity. Combining these two effects can lead to an equilibrium of spike positions within the cluster such that the cluster is linearly stable.     

Convection–diffusion equation with space–time ergodic random flow

We prove the homogenization of convection-diffusion in a time-dependent, ergodic, incompressible random flow which has a bounded stream matrix and a constant mean drift. We also prove two variational formulas for the effective diffusivity. As a consequence, we obtain both upper and lower bounds on the effective diffusivity. Received: 17 December 1996/Revised revision: 9 February 1998     

Nonclassical Symmetries for Nonlinear Diffusion and Absorption

Nonclassical symmetry methods are used to study the nonlinear diffusion equation with a nonlinear source. In particular, exponential and power law diffusivities are examined and we obtain mathematical forms of the source term which permit nonclassical symmetry reductions. In addition to the known source terms obtained by classical symmetry methods, we find new source terms which admit symmetry reductions. We also deduce a class of nonclassical symmetries which are valid for arbitrary diffusivity and deduce corresponding new solution types. This is an important result since previously only traveling wave solutions were known to exist for arbitrary diffusivity. A number of examples are considered and new exact solutions are constructed.     

Global well-posedness for the 2-D Boussinesq system with the temperature-dependent viscosity and thermal diffusivity

We prove the global well-posedness for the 2-D Boussinesq system with the temperature-dependent viscosity and thermal diffusivity.     

An inverse problem in determining the acid and salt diffusivities simultaneously for polymer solution in a wet spinning process

The wet spinning process depends strongly on the acid and salt diffusivity coefficients in the fiber; however, these two coefficients are normally function of concentration of salt and are difficult to be measured directly. For this reason the technology for the inverse mass transfer problems need be applied to resolve these two concentration-dependent diffusivities simultaneously. An Iterative Regularization Method (IRM) using Conjugate Gradient Method (CGM) is applied in this study to determine simultaneously the unknown diffusivities of acid and salt for polymer solution in a wet spinning process by using measurements of concentration components. The accuracy of this inverse mass transfer problem is examined by using the simulated exact and inexact concentration measurements in the numerical experiments. Results show that the diffusivity of acid can be estimated more accurately than the diffusivity of salt and the estimation of the diffusivities can be obtained in a very short CPU time on a HP d2000 2.66 GHz personal computer.     

A method and a computer program for determining the thermal diffusivity in a solid slab

The authors describe a method for computing the thermal diffusivity of a solid, based on a computer assisted evaluation of the solution of the transient inverse heat conduction problem.The program computes either the unknown diffusivity or simulates the one-dimensional unsteady heat transfer problem. The user may model the boundary conditions by a choice of different functions.The program provides instruction and information at all stages of input and provides tabular output of results. It may be used by anybody wishing to solve or simulate heat transfer processes.     

非线性各向异性椭圆方程的均匀化

通过结合各向异性Sobolev空间与经典的补偿紧性技巧,得到了一类非线性各向异性椭圆方程的均匀化结果.