Non-constant Diffusivity Research Articles

Fractional and tempered fractional models for Reynolds-averaged Navier–Stokes equations

Turbulence is a non-local phenomenon and has multiple-scales. Non-locality can be addressed either implicitly or explicitly. Implicitly, by subsequent resolution of all spatio-temporal scales. However, if directly solved for the temporal or spatially averaged fields, a closure problem arises on account of missing information between two points. To solve the closure problem in Reynolds-averaged Navier–Stokes equations (RANS), an eddy-viscosity hypotheses has been a popular modelling choice, where it follows either a linear or nonlinear stress–strain relationship. Here, a non-constant diffusivity is introduced. Such a non-constant diffusivity is also characteristic of non-Fickian diffusion equation addressing anomalous diffusion process. An alternative approach, is a fractional derivative-based diffusion equations. Thus, in the paper, we formulate a fractional stress–strain relationship using variable-order Caputo fractional derivative. This provides new opportunities for future modelling effort. We pedagogically study of our model construction, starting from one-sided model and followed by two-sided model applied to channel, couette and pipe flow. Non-locality at a point is the amalgamation of all the effects, thus we find the two-sided model is physically consistent. Further, our construction can also addresses viscous effects, which is a local process. Thus, our fractional model addresses the amalgamation of local and non-local process. We also show its validity at infinite Reynolds number limit. An expression for the fractional order is also found, thereby solving the closure problem for the considered cases. This study is further extended to tempered fractional calculus, where tempering ensures finite jump lengths, this is an important remark for unbounded flows. Within the context of this paper, we limit ourselves to wall bounded flow. Two tempered definitions are introduced with a smooth and sharp cutoff, by the exponential term and Heaviside function, respectively and we also define the horizon of non-local interactions. We further study the equivalence between the two definitions, as truncating the domain has computational advantages. For the above investigations, we carefully designed algorithms, notably, the pointwise version of fractional physics-informed neural network to find the fractional order as an inverse problem.

A particle-size dependent smoothing scheme for polydisperse Euler-Lagrange simulations

In Euler-Lagrange models for particulate systems, the information exchange of multiphase coupling terms, i.e. drag force, heat transfer, or mass transfer, is often smoothed from particles' location to nearby computational cells when the cell and particle sizes are comparable. The diffusion-based smoothing is among the most popular approach. However, when it comes to polydisperse systems, the state-of-the-art constant diffusivity approach does not consider variation in particle sizes, resulting in an even smoothing of exchanged information across all particles with different sizes.In this paper, a particle-size dependent smoothing scheme named “non-constant diffusivity approach” is proposed. This approach distributes the exchanged information based on the local Sauter diameter d32 such that the area of influence per particle varies by size. The one-dimensional analytical solution of the non-constant diffusivity approach is first derived and two control parameters L⁎ and S are identified, where L⁎ determines the characteristic length of the diffusivity and S determines the magnitude of the diffusivity. Together they control the characteristic length scale ℓ˜c of the smoothing operation. In the one-dimensional study, it is found that the proposed smoothing method is most suitable for the dilute particulate regime. The investigation is extended to a three-dimensional dilute particle flow based on the experiment of the CRIEPI (Central Research Institute of Electric Power Industry) pulverized coal burner. It is shown that using the non-constant diffusivity smoothing, the prediction of the O2 concentration is more accurate than when using the constant diffusivity smoothing approach, which assumes a universal smoothing length for all sizes of particles. The findings suggest that when it comes to polydisperse particulate systems, the area of influence per particle on the fluid depends on the particle size. The proposed non-constant diffusivity approach can describe that with the benefits of cheap computational cost, conservation of exchanged quantities, and easy-to-implement.

Changes in Poleward Atmospheric Energy Transport over a Wide Range of Climates: Energetic and Diffusive Perspectives and A Priori Theories

Abstract The midlatitude poleward atmospheric energy transport increases in radiatively forced simulations of warmed climates across a range of models from comprehensive coupled general circulation models (GCMs) to idealized aquaplanet moist GCMs to diffusive moist energy balance models. These increases have been rationalized from two perspectives. The energetic (or radiative) perspective takes the atmospheric energy budget and decomposes energy flux changes (radiative forcing, feedbacks, or surface fluxes) to determine the energy transport changes required by the budget. The diffusive perspective takes the net effect of atmospheric macroturbulence to be a diffusive energy transport down-gradient, so transport changes can arise from changes in mean energy gradients or turbulent diffusivity. Here, we compare these perspectives in idealized moist, gray-radiation GCM simulations over a wide range of climates. The energetic perspective has a dominant role for radiative forcing in this GCM, with cancellation between the temperature feedback components that account for the GCM’s nonmonotonic energy transport changes in response to warming. Comprehensive CMIP5 simulations have similarities in the Northern Hemisphere to the idealized GCM, although a comprehensive GCM over several CO2 doublings has a distinctly different feedback evolution structure. The diffusive perspective requires a non-constant diffusivity to account for the idealized GCM-simulated changes, with important roles for the eddy velocity, dry static stability, and horizontal energy gradients. Beyond diagnostic analysis, GCM-independent a priori theories for components of the temperature feedback are presented that account for changes without knowledge of a perturbed climate state, suggesting that the energetic perspective is the more parsimonious one.

On a question of non-constant thermal diffusivity of soils

The general heat conductivity equation includes time- and depth-dependent soil properties (soil heat capacity and thermal conductivity). The simplified form of the heat conductivity equation contains only the soil thermal diffusivity parameter. Numerical solutions of the general and simplified equations were compared to quantify the possibility of equation reduction. Two test runs for soils with different compositions were done. The thermal regime for both peat soil and dark chestnut soil does not change significantly after using a simplified heat equation according to model estimations. The maximal soil temperature discrepancy was about 0.5 °C for peat soil and 2.2-3.3 °C for dark chestnut soil, which results in 4-6% error in methane efflux estimations.

Calibration of boundary conditions correlated to the diffusivity of chloride ions: An accurate study for random diffusivity

For reinforced concrete (RC) structures, the corrosion of the reinforced steel bars from chloride ions is a periodic risk. The chloride ions diffusion is governed by Fick’ laws, however, these equations can be applied under the condition that the diffusivity is constant in the time and space. In this way, the diffusion process and the relative scenario could be greatly simplified. In this paper, a more complete non-constant multifactorial diffusivity is proposed accounting the water/cement ratio, chloride binding, temperature, age, deformation, damage, humidity. To consider a non-constant diffusivity, the use of numerical solutions is necessary. For this, Hermite polynomials (HPs) have been implemented for 1D and 2D model. Specified boundary conditions for this application have been calibrated to solve partial differential equations (PDEs) for random fluctuation diffusivity. Results show that traditional approaches bad estimate the chloride concentration for long scenarios at certain depths.

Representative activation energy and solar biomemtic pump with blood flow having variable thermo physical properties

Peristaltic phenomena is expanded broadly in biomedical applications in recent years. Solar energy system is very beneficial to humanity and these are progressively being restyled to get advantage from fluids. Most solar collector's schemes include inflexible geometries which may be parabolic, cylindrical, flat-plate or tube like categories. Current progresses in biomimetic have acknowledged that irregular channels, tubes, pipes, etc. may be valuable for energy systems. Inspired by these characteristics, the present work presents an original model for fashioning a biomimetic heat and mass transference peristaltic solar fluid based pump. The thermo-physical physical properties (mass diffusivity, viscosity and conductivity) are assumed to be changeable due to heat and mass diffusions. The uniqueness of the current paper is the consolidation of peristaltic motion, variable fluid properties, activation energy and blood flow in a planetary thrust system. Heat is transferred through the walls and due to vertical flow buoyancy effects are considered. Study of heat/mass transfer are accomplished by assuming the effects of non-constant thermal conductivity, mass diffusivity, viscosity and activation energy. To implement solar filling conditions, an emitting radiation unstable model is also used. In order to create an activation energy, a chemical reaction is taken at rational rate, the temperature of the system is assumed to be so high that it creates number of molecules in the form of energy known as activation energy. Additionally, the noticeable characteristics of pumping and trapping phenomena are addressed concisely. For High values of Weissenberg number, the pumping and augmented pumping regions have same behavior. Pre-exponential factor and chemical reaction causes increase in concentration distribution.

2D/3D Numerical Analyses of Corrosion Initiation in RC Structures Accounting Fluctuations of Chloride Ions by External Actions

The diffusion of chloride ions in reinforced concrete (RC) varies in space and time, depending on many uncertain factors as material properties, temperature, aging, humidity and external loadings. In this paper, the goals are to provide different scenarios to probabilistic assessing of the corrosion initiation time and chloride ions content inside concrete considering different fluctuations. For the former, stochastic analyses helps to estimate the service life in RC; for the latter, the focus is on the nature of loading variation due to location and structural constraints that usually are not considered. The approach consist in studying the trend of chloride ion particles regarding the action of the external loadings by incorporating a function with random values. The governing equations are modified to account for the non-constant diffusivity under transient conditions. Results show that the chloride content for random diffusivity assumes lower values than for constant diffusivity. Moreover, in 2D analysis the chloride content is lesser of 78% than in 3D analysis.

Change in internal energy of viscoelastic fluid flow between two rotating parallel plates having variable fluid properties

In this study, second-grade fluid is taken into account in order to examine its rotating behavior with heat and mass transfer effects. The analysis is carried out between two parallel plates. The study of heat and mass transfer is accomplished by taking the effects of non-constant diffusivity, conductivity and viscosity. A chemical reaction is taken in the form of activation energy. Moreover, it is assume that the kinetic energy of the fluid particles is converted into internal energy. Signifying fundamental equations are tracked by constructing numerical form solutions for velocity, temperature and concentration distributions, while assuming variable fluid properties and internal energy change. Bvp4c technique is employed by considering second-grade fluid parameter and variable fluid properties to be very small. The impact of emerging parameters on velocity, temperature and concentration distributions is examined through graphical depiction. A direct relation is taken between concentration and diffusivity, temperature and thermal conductivity, whereas the indirect relation is assumed for viscosity. Heat transfer is minor for a fluid with variable conductivity and viscosity as likened to the fluid with constant conductivity and viscosity. For uplifting values of rotation parameter, the fluid velocity magnifies.

Observer‐based output feedback control for a boundary controlled fractional reaction diffusion system with spatially‐varying diffusivity

This study is concerned with observer design and observer-based output feedback control for a fractional reaction diffusion (FRD) system with a spatially-varying (non-constant) diffusion coefficient by the backstepping method. The considered FRD system is endowed with only boundary measurable and actuation available. The contribution of this study is divided into three parts: first is the backstepping-based observer design for the FRD system with non-constant diffusivity, second is the output feedback controller generated by the integration of a separately backstepping-based feedback controller and the proposed observer to stabilise the FRD system with non-constant diffusivity, and the last is the Mittag–Leffler stability analysis of the observer error and the closed-loop FRD systems. Specifically, anti-collocated location of actuator and sensor is considered in the stabilisation problem of this system with Robin boundary condition at x = 0 and the boundary feedback controller for Dirichlet actuation at x = 1 . By designing an invertible coordinate transformation to convert the observer error system into a Mittag–Leffler stable target system, the observer gains are obtained. They are used to design the output feedback control law for stabilising the closed-loop system. Finally, a numerical example is shown to validate the effectiveness of the authors' proposed method.

Backstepping-based boundary control design for a fractional reaction diffusion system with a space-dependent diffusion coefficient

This paper presents a boundary feedback control design for a fractional reaction diffusion (FRD) system with a space-dependent (non-constant) diffusion coefficient via the backstepping method. The contribution of this paper is to generalize the results of backstepping-based boundary feedback control for a FRD system with a space-independent (constant) diffusion coefficient to the case of space-dependent diffusivity. For the boundary stabilization problem of this case, a designed integral transformation treats it as a problem of solving a hyperbolic partial differential equation (PDE) of transformation's kernel, then the well posedness of the kernel PDE is solved for the plant with non-constant diffusivity. Furthermore, by the fractional Lyapunov stability (Mittag-Leffler stability) theory and the backstepping-based boundary feedback controller, the Mittag-Leffler stability of the closed-loop FRD system with non-constant diffusivity is proved. Finally, an extensive numerical example for this closed-loop FRD system with non-constant diffusivity is presented to verify the effectiveness of our proposed controller.

Barrier properties of polyamide 12/montmorillonite nanocomposites: Effect of clay structure and mixing conditions

Hybrid films, composed of a PA12 polyamide matrix filled with 5 wt% of clays were melt processed under different mixing conditions to classically obtain montmorillonite-filled microcomposites and organo-modified montmorillonite-filled nanocomposites. The change in mixing shear rates allowed to notably modulating the fragmentation level for montmorillonite and the exfoliation level for modified montmorillonite. Barrier properties to nitrogen and water were altered with a reduction in PA12 chain mobility by confinement effect and geometrical constraints. Tortuosity effects induced by fillers were also highlighted. The water permeability and diffusivity were significantly decreased with exfoliated clays, while an increase was measured with an increased water solubility in presence of clay aggregates, both counterbalanced tortuosity effects. This trend was amplified when using low shear rates. According to filler used and mixing conditions, water sorption measurement has successfully demonstrated an increase in water solubility, due to filler affinity, which was surprisingly inconsistent with the change of water permeability. The occurrence of water plasticization phenomenon was also pointed out through the non-constant diffusivity, reflecting an increased free volume with water activity. Again, a decrease in water diffusivity was unexpectedly obtained, explained by tortuosity effects and ascribed to local water retention at the vicinity of clays, which creates water aggregates in films. These findings were also confirmed by a mathematical modeling approach of sorption process, discussing about the sorbed water access.

Nonlinear reaction–diffusion systems with a non-constant diffusivity: Conditional symmetries in no-go case

Q-conditional symmetries (nonclassical symmetries) for a general class of two-component reaction–diffusion systems with non-constant diffusivities are studied. The work is a natural continuation of our paper “Conditional symmetries and exact solutions of nonlinear reaction–diffusion systems with non-constant diffusivities” (Cherniha and Davydovych, 2012) [1] in order to extend the results on so-called no-go case. Using the notion of Q-conditional symmetries of the first type, an exhaustive list of reaction–diffusion systems admitting such symmetry is derived. The results obtained are compared with those derived earlier. The symmetries for reducing reaction–diffusion systems to two-dimensional dynamical systems (ODE systems) and finding exact solutions are applied. As result, multiparameter families of exact solutions in the explicit form for nonlinear reaction–diffusion systems with an arbitrary power-law diffusivity are constructed and their properties for possible applicability are established.

Turbulent diffusion in tall tubes. I. Models for Rayleigh-Taylor instability

Rayleigh-Taylor instability in high-aspect-ratio domains has been studied experimentally and a hierarchy of modelling approaches has been used to understand the dynamics of the problem. Part I examines the simplest case of initially homogenous layers above and below the Rayleigh-Taylor unstable interface. Part II examines the more complex case where one layer is stably stratified in density. Here, in Part I, we develop models for turbulent mixing induced by Rayleigh-Taylor instability based on a diffusion equation for density. By considering the force balance in the problem, and using Prandtl’s mixing length hypothesis, we compute a non-constant turbulent diffusivity, and this leads to a non-linear diffusion equation. We reiterate a h~t25 scaling and use this to develop a new similarity solution to the nonlinear diffusion equation in an infinite domain. To match experimental boundary conditions of a finite domain, we use numerical integration, and finally, we compare with implicit large eddy simulation.

Boundary Control of Nonlinear Distributed Parameter Systems by Input-Output Linearization

AbstractThe present study focuses on boundary control of nonlinear distributed parameter systems and deals with Dirichlet actuation. Thus, a design approach of a geometric control law that enforces stability and output tracking of a given punctual output is developed based on the notion of the characteristic index. The control performance of the proposed strategy is evaluated through numerical simulation by considering two control problems. The former concerns the control of the temperature of a thin metal rod modelled by a heat equation with a nonlinear source, and the later concerns the control of concentration of a dye in liquid medium modeled by Fick law with nonconstant diffusivity.

On the non-linear Maxwell–Cattaneo equation with non-constant diffusivity: Shock and discontinuity waves

A numerical study on a non-linear hyperbolic diffusion equation is proposed. The Hartree hybrid method combining finite difference techniques with the method of characteristics is used in the presence of discontinuities between initial and boundary conditions. The technique proved to be an useful tool to overcome oscillation problems and spurious solutions in case of strong non-linearities related to both attractive or repulsive interactions between diffusing species. Two different expressions for the diffusion coefficient are used in order to compare our results with the ones obtained in previous studies relying upon the Laplace transform technique and the MacCormack predictor–corrector method. Finally, an analytic approach based on the singular surface theory is proposed to motivate the numerical results and to clarify some controversial aspects concerning the penetration depth of a diffusive front in the presence of interactions.

Mixture Diffusion in Nanoporous Adsorbents: Development of Fickian Flux Relationship and Concentration-Swing Frequency Response Method

A frequency response method using concentration variation is developed theoretically and experimentally and applied to investigate mixture diffusion in nanoporous adsorbents. The method is based on periodically time-varying species feed concentrations with a constant total molar inlet flow rate. It can be used without the need of a carrier gas. A mathematical model is formulated considering nanopore diffusion, surface barrier resistance, external film resistance, and axial dispersion. The related analytical solutions for frequency response are derived. For nanopore diffusion, a theory for nonconstant mixture Fickian diffusivity with cross-terms is developed from irreversible thermodynamics and shows that mixture Fickian diffusivities can be expressed as the product of corrected diffusivities and a thermodynamic factor that accounts for concentration dependence. The number of unknown variables for Fickian diffusivities is the same as the number for Onsager coefficients or Maxwell−Stefan diffusivities. Adso...

Application of Petrov–Galerkin methods to transient boundary value problems in chemical engineering: adsorption with steep gradients in bidisperse solids

Petrov–Galerkin methods are known to be versatile techniques for the solution of a wide variety of convection–dispersion transport problems, including those involving steep gradients, but have hitherto received little attention by chemical engineers. We illustrate the technique by means of the well-known problem of simultaneous diffusion and adsorption in a spherical sorbent pellet comprised of spherical, non-overlapping microparticles of uniform size and investigate the uptake dynamics. Solutions to adsorption problems exhibit steep gradients when macropore diffusion controls or micropore diffusion controls, and the application of classical numerical methods to such problems can present difficulties. In this paper, a semi-discrete Petrov–Galerkin finite element method for numerically solving adsorption problems with steep gradients in bidisperse solids is presented. The numerical solution was found to match the analytical solution when the adsorption isotherm is linear and the diffusivities are constant. Computed results for the Langmuir isotherm and non-constant diffusivity in microparticle are numerically evaluated for comparison with results of a fitted-mesh collocation method, which was proposed by Liu and Bhatia (Comput. Chem. Engng. 23 (1999) 933–943). The new method is simple, highly efficient, and well-suited to a variety of adsorption and desorption problems involving steep gradients.

Reaction waves and non-constant diffusivities

AbstractA reaction-diffusion equation with non-constant diffusivity,ut = (D(x, t)ux)x + F(u),is studied for D(x, t) a continuous function. The conditions under which the equation can be reduced to an equivalent constant diffusion equation are derived. Some exact forms for D(x, t) are given. For D(x, t) a stochastic function, an explicit finite difference method is used to numerically determine the effect of randomness in D(x, t) upon the speed of the reaction wave solution to Fisher's equation. The extension to two spatial dimensions is considered.

Double-Diffusive Interleaving: The Influence of Nonconstant Diffusivities

The influence of nonconstant diffusivity and viscosity on double-diffusive interleaving is examined using a simple model. The analysis allows a prediction of the slope, vertical wavenumber, and growth rate of the fastest-growing interleaving mode for specified background gradients of salinity and temperature. Allowing the salt diffusivity to be a function of the density ratio Rp leads to a larger growth rate and a lower vertical wavenumber than in the constant diffusivity case considered by Toole and Georgi, while the cross-front slope of the intrusions is essentially unaffected. The nonconstant viscosity included in the model formulation is found to have no effect on the form of the solutions. The larger vertical scales and growth rates predicted by the model can be traced to an enhanced “effective diffusivity” resulting from the diffusivity gradients associated with the growing intrusions. A transformation is found that converts the system with generalized diffusivities examined here to the simpler system considered by Toole and Georgi.