Variable Diffusion Research Articles

Stable determination of coefficients in semilinear parabolic system with dynamic boundary conditions

In this work, we study the stable determination of four space-dependent coefficients appearing in a coupled semilinear parabolic system with variable diffusion matrices subject to dynamic boundary conditions which couple intern-boundary phenomena. We prove a Lipschitz stability result for interior and boundary potentials by means of only one observation component, localized in any arbitrary open subset of the physical domain. The proof mainly relies on some new Carleman estimates for dynamic boundary conditions of surface diffusion type.

Analysis of a scheme which preserves the dissipation and positivity of Gibbs' energy for a nonlinear parabolic equation with variable diffusion

In this work, we design and analyze a discrete model to approximate the solutions of a parabolic partial differential equation in multiple dimensions. The mathematical model considers a nonlinear reaction term and a space-dependent diffusion coefficient. The system has a Gibbs' free energy, we establish rigorously that it is non-negative under suitable conditions, and that it is dissipated with respect to time. The discrete model proposed in this work has also a discrete form of the Gibbs' free energy. Using a fixed-point theorem, we prove the existence of solutions for the numerical model under suitable assumptions on the regularity of the component functions. We prove that the scheme preserves the positivity and the dissipation of the discrete Gibbs' free energy. We establish theoretically that the discrete model is a second-order consistent scheme. We prove the stability of the method along with its quadratic convergence. Some simulations illustrating the capability of the scheme to preserve the dissipation of Gibb's energy are presented.

Flow patterns in Rayleigh-Benard convection with variable thermal conductivity

The letter presents a numerical study of Rayleigh-Benard convection under strong non-Boussinesq conditions caused by variable thermal diffusivity. For Prandtl number in the range [0.001,10], bifurcations between hexagonal and roll flows controlled by the Prandtl number are investigated in the vicinity of critical Rayleigh number. The impact of the non-Boussinesq effects and their strength on the transition values of Prandtl number is analyzed. New non-integral criterion to select physically realizable states of motion is deduced from a comparative study of full-scale 3D simulation and model 2D runs.

On the estimation of effectiveness factor in metallic monoliths presenting non-uniform coatings

The one dimensional two-zone model (1D-2Z) to estimate effectiveness factor in non-uniform coatings of metallic monolith channels with sinusoidal cross-section is thoroughly tested in this contribution. The model relies on the idea of splitting the cross-section of the coating in two zones that can be modelled independently taking into account the diffusion and simultaneous reaction in a single direction (1D). One of the zones is considered as a slab, while the previously developed variable diffusivity model (1D-VD) is applied to the other. The predictive capabilities of the 1D-2Z model is assessed for three kinetic expressions: a first order reaction and two more challenging cases defined by Langmuir-Hinshelwood-Hougen-Watson expressions: one of them with a moderate self-inhibition effect and the other with a strong self-inhibition effect, close to the onset of the multiple steady states behaviour. Simultaneously, different degrees of non-uniform coatings are considered. The 1D-2Z model allows obtaining a remarkable precision in the estimation of the effectiveness factors for the complete range of Thiele modulus with maximum errors below 5.6%.

Determination of constant and variable thermal diffusivity of cashew pulp during heating: Experimentation, optimizations and simulations

Cashew nut is used by the industry in the production of roasted kernels, whose flavor is appreciated in various parts of the world. However, the production of 1 ton of roasted kernels generates up to 15 tons of cashew apple, which is a highly perishable by-product. To increase its shelf life, cashew apple can be transformed into pulp, which is then pasteurized. Designing efficient pasteurizers requires knowledge of the thermal diffusivity of this product, in the heating step. Thus, an experiment was carried out with heating of this pulp placed in a metallic cylinder (from 25.6 °C to 63.1 °C), in a water bath (at 63.1 °C), with a thermocouple fixed at the central point of the product. Thermal diffusivity was determined through the inverse method, using analytical and numerical solutions of the diffusion equation. The use of a variable thermal diffusivity, given as a function of the local temperature, expressed by α(T) = a1cosh(a2T2), produced excellent fit of the simulation to the experimental data, with random distribution of errors, and with χ2 = 0.1626 and R2 = 0.999984.

Coefficient inverse problem for variable order time-fractional diffusion equations from distributed data

Coefficient inverse problem for variable order time-fractional diffusion equations from distributed data

An L1 type difference/Galerkin spectral scheme for variable-order time-fractional nonlinear diffusion–reaction equations with fixed delay

A linearized spectral Galerkin/finite difference approach is developed for variable fractional-order nonlinear diffusion–reaction equations with a fixed time delay. The temporal discretization for the variable-order fractional derivative is performed by the L1-approximation. An appropriate basis function in terms of Legendre polynomials is used to construct the Galerkin spectral method for the spatial discretization of the second-order spatial operator. The main advantage of the proposed approach is that the implementation of the iterative process is avoided for the nonlinear term in the variable fractional-order problem. Convergence and stability estimates for the constructed scheme are proved theoretically by discrete energy estimates. Some numerical experiments are finally provided to demonstrate the efficiency and accuracy of the theoretical findings.

A Lagrangian particle-based numerical model for surfactant-laden droplets at macroscales

Atmospheric aerosols can consist of inorganic and organic substances, including surfactants at a significant concentration. Importantly, the latter can reduce the surface tension at the liquid–vapor surfaces, where they preferentially adsorb due to their amphiphilic structure. As a result, processes such as droplet coalescence, development of precipitation, and ultimately cloud lifetime may depend on the presence of surfactants in the aerosols. Here, we present a numerical model for cloud droplet formation, which is based on the Lagrangian particle-based microphysics-scheme superdroplet method and takes into account the presence of the surfactant in the droplets. Our results show that surfactant facilitates cloud formation by increasing the number and size of activated droplets, which concentrate at the bottom of the cloud, while the largest droplets are concentrated at the top of the cloud. This indicates a circulation of droplets that involves activation and growth processes from the bottom of the cloud toward the top. Moreover, our conclusions are independent of the particular approach used for modeling the diffusion of Eulerian variables due to the subgrid-scale turbulence. We anticipate that our results will enrich our understanding of the role of surfactants in the behavior of atmospheric aerosols and, importantly, will pave the way for further developments in the numerical modeling of systems with surfactants at macroscopic scales.

MHD Free Convective Heat and Mass Transfer Flow from a Vertical Porous Surface with Variable Thermal Conductivity, Variable Mass Diffusivity and Thermal Diffusion Including Viscous Dissipation and Chemical Reaction

An analysis is carried out to study chemically reactive, viscous dissipative effects of an incompressible and electrically conducting fluid with MHD free convection adjacent to a vertical surface with variable thermal conductivity (VTD) and variable mass diffusivity (VMD). An approximate numerical solution for the steady laminar boundary layer flow over a wall of the surface in the presence of species concentration and thermal mass diffusion has been studied. Using numerical techniques the governing boundary layer equations are solved to get the exact solution. Numerical calculations are carried out for different values of dimensionless parameters. The results are exhibited through various graphs and it is observed from the analysis of the results that the velocity field is appreciably influenced by the magnetic effect, porous effect, chemical reaction and buoyancy ratio between the species and thermal diffusion at the wall of the surface.

CO2 solubility and diffusivity in 1-ethyl-3-methylimidazolium cation-based ionic liquids; isochoric pressure drop approach

In this study, an experimental approach based on the isochoric pressure drop principle was implemented to measure (at low equilibrium pressure, 1.3–3.6 bar) the solubility and binary diffusivity of CO2 (constant and variable) in four imidazolium-based room-temperature ionic liquids (RTILs), namely, 1-ethyl-3-methylimidazolium acetate ([emim][Ac]), 1-ethyl-3-methylimidazolium dicyanamide ([emim][DCA]), 1-ethyl-3-methylimidazolium methyl sulfate ([emim][MeSO4]) and 1-ethyl-3-methylimidazolium ethyl sulfate ([emim][EtSO4]). The molar fraction of the dissolved CO2 and Henry's law constant were measured from experimental data. Constant binary diffusion coefficients were measured from the transient thin-film and semi-infinite volume models. In the original approach, the variable diffusivity coefficient model was used to measure the concentration-based variable diffusion coefficient. Anion interaction of the RTILs was found to have a strong influence on the CO2 solubility, and Henry's law constant and the studied anions were ranked as [Ac]− >> [EtSO4]− > [DCA]− > [MeSO4]−. Increasing temperature is not likely for CO2 solubility while it favors CO2 diffusivity. Thin-film model diffusivity measurements were found to be more suitable for [emim][MeSO4] and [emim][EtSO4], while semi-infinite volume model was suitable for [emim][Ac] and [emim][DCA]. Variable diffusion coefficients were found not to be a monotonous function of time, which increased in early stages and decreased after a certain time. Gibbs free energy, enthalpy, and entropy values indicate the strength of the favorable interaction between CO2 and RTILs, mildly exothermic in nature and falling in the range of physisorption.

Numerical Approach toward Ternary Hybrid Nanofluid Flow Using Variable Diffusion and Non-Fourier's Concept.

In the current study, the pseudoplastic model is used to analyze the mass and energy transmission through trihybrid nanofluid flow across a stretched permeable surface. The Darcy–Forchheimer relation is employed in the momentum equation to examine the influence of porosity. Energy and mass diffusion expressions are obtained by employing the double diffusion theories, which were proposed by Cattaneo and Christov and is broadly used by several researchers. The thermal efficiency of the trihybrid nanocrystals is evaluated by integrating them with a pseudoplastic substrate. The study of titanium dioxide (TiO2), cobalt ferrite (CoFe2O4), and magnesium oxide (MgO) nanocomposite base hybrid nanofluids across a stretchable sheet is receiving considerable interest in innovation and research due to their extensive spectrum of applicability. For this reason, the phenomena are modeled in the form of a system of PDEs with the effects of a heat source, magnetic field, natural convection, and chemical reaction. Through resemblance substitutions, these are reduced to an ODE system. The resultant first-order differential equations are further processed using the computational approach PCM. For authenticity and reliability, the values are reviewed against the existing literature. The findings are displayed through figures. When compared to the simple nanofluid, the hybrid and trihybrid nanofluid have a greater tendency for fluid energy and velocity propagation rate. The velocity and heat transition rate enhance 11.73% by varying nanoparticles’ values from 0.01 to 0.04, while the thermal conductivity of base fluid boosts with the addition of hybrid and trihybrid nanocomposites, up to 32% and 61%, respectively.

Hygrothermal Modelling of the Differences between Single versus Variable Relative Humidity Vapour Diffusion Resistivity Properties of Pliable Membranes

The study investigates through hygrothermal modelling the effect of different boundary conditions and varying measured vapour diffusion resistivity values on the hygrothermal performance of five pliable membranes. Previously, this research quantified the variable water vapour diffusion resistivity properties of five different pliable building membranes. The membranes were assessed under varying humidity conditions using the gravimetric wet and dry cup test method. The varying humidity conditions better represent the boundary conditions experienced by materials in the building envelope. The pliable membranes include two permeable, two impermeable, and one variable products, which are commonly used to provide air and vapour control layers in the construction of framed external wall systems. This article focusses on the transient hygrothermal modelling of each of these membranes as a component of a typical timber-framed, clay brick veneer external wall system. The simulations were completed for three different climate types, namely, hot and humid, temperate, and cool-temperate with snow, and with a northern and western orientation. The results from hygrothermal and bio-hygrothermal simulations highlighted different responses subject to climate type and orientation. These results show that there are significant differences in simulated moisture and mould growth risk between the results of pliable membranes with single vapour resistance factor value and pliable membranes with multipoint vapour resistance factor values.

Impact of surface catalyst and variable diffusion coefficients on a comparative appraisal of hybrid and nanofluid flows

A comparison of the hybrid and nanofluid flows performance amidst two spinning disks accompanying variable thermal conductivity and viscosity is appraised in this study. The hybrid nanofluid is comprised of single-wall-multiwall carbon nanotubes/water combination and a simple nanofluid consists of single-wall carbon nanotubes (multiwall carbon nanotubes)/water amalgamation. The selection of carbon nanotubes is based on their matchless characteristics including lightweight, extraordinary thermal conductivity attributes, and stability. The envisaged model is supported with catalysis on the surface to influence the homogeneous-heterogeneous reactions and trigger the process in a relatively lesser time. The cases of clockwise and anticlockwise rotation for both disks are undertaken. The numerical MATLAB-based bvp4c technique is betrothed. The graphical illustrations depicting the association of the prominent parameters with the allied profiles are given with logical arguments. The fallouts revealed that the hybrid nanofluids are more effective than the simple nanofluids as far as thermal efficiency is concerned. It is also heeded that the rate of mass transfer is proliferated for the surface catalyzed parameter. A comparison table in a limiting case is also supplemented to jazz up the justification of the presented model.

Generalized Thermo-Diffusion Interaction in an Elastic Medium under Temperature Dependent Diffusivity and Thermal Conductivity

The purpose of this work is to investigate, within the context of extended thermo-diffusion theory, the transient thermo-diffusion responses for a half-space with variable thermal conductivity and diffusivity. The half-bounding space’s surface is traction-free and exposed to a time-dependent thermal shock, but the chemical potential is believed to be a known function of time. Because the nonlinear equations are complicated, the finite element technique is applied to solve these equations. Numerical outcomes are produced and graphically illustrated. The effects of varying thermal conductivity and diffusivity on the response are studied using parameter studies. Using the results of this study, researchers hope to understand better how thermo-mechanical fields interact in real materials. By ignoring the new parameter, a comparison of numerical results and analytical cases is produced, and the behavior of physical quantities for numerical solutions is studied to ensure that the proposed technique is accurate.

Two-grid finite element methods for nonlinear time fractional variable coefficient diffusion equations

In this article, an efficient two-grid finite element method is proposed for solving the nonlinear time fractional variable coefficient diffusion equations. This algorithm firstly solves a nonlinear system to get the numerical solution uHn on the coarse grid with size H, then based on the initial iterative solution uHn on the coarse grid, the linearized finite element system is solved on the fine grid with size h to get the numerical solution Uhn, in which the temporal direction is approximated by the L2−1σ scheme. Besides, the stability and priori error estimates of standard finite element method and two-grid method are given. Finally, the validity and efficiency of the two-grid algorithm are verified by two numerical experiments.

Evaporation Dynamics of Sessile Saline Microdroplets in Oil.

The occurrence of concentration and temperature gradients in saline microdroplets evaporating directly in air makes them unsuitable for nucleation studies where homogeneous composition is required. This can be addressed by immersing the droplet in oil under regulated humidity and reducing the volume to the picoliter range. However, the evaporation dynamics of such a system is not well understood. In this work, we present evaporation models applicable for arrays of sessile microdroplets with dissolved solute submerged in a thin layer of oil. Our model accounts for the variable diffusion distance due to the presence of the oil film separating the droplet and air, the variation of the solution density and water activity due to the evolving solute concentration, and the diffusive interaction between neighboring droplets. Our model shows excellent agreement with experimental data for both pure water and NaCl solution. With this model, we demonstrate that assuming a constant evaporation rate and neglecting the diffusive interactions can lead to severe inaccuracies in the measurement of droplet concentration, particularly during nucleation experiments. Given the significance of droplet evaporation in a wide array of scientific and industrial applications, the models and insights presented herein would be of great value to many fields of interest.

Calculation and Use of Variable Diffusivity for the Analysis of Transient Liquid Phase Bonding Behavior

Calculation and Use of Variable Diffusivity for the Analysis of Transient Liquid Phase Bonding Behavior

Dissemination of a long-term care planning tool, PlanYourLifespan.org, through community-based stakeholder leaders.

Geriatrics research generally cumulates in academic journal publications, with variable diffusion to patients and communities. PlanYourLifespan.org is a free, evidence-based tool that assists older adults, and their loved ones, to better understand and plan for their long-term support needs. There is a need to effectively disseminate geriatrics research, such as PlanYourLifespan.org, to communities that may directly benefit from this research. To leverage community-based stakeholder leaders, utilizing a train-the-trainer program, to disseminate PlanYourLifespan.org and evaluate the extent of the dissemination. Using a train-the-trainer strategy, community stakeholder leaders from the original study paired up with newly recruited community stakeholder leaders. New community stakeholder leaders were trained on dissemination, using a "how-to-disseminate" web-based toolkit-developed as part of this project. Newly trained community stakeholder leaders subsequently trained additional community stakeholder leaders who conducted and tracked dissemination activities in their communities. Google Analytics tracked newly created PlanYourLifespan.org accounts, login sessions, and daily website visitors. Five newly trained community stakeholder leaders disseminated PlanYourLifespan.org over a three-month period. Cumulatively, on the day of the dissemination activity, there were 11,361 PlanYourLifespan.org log-ins (average: 378.7 log-ins/activity day), 89,068 log-ins (average: 2969 log-ins/activity week) one-week after the activity, and 319,154 log-ins (average: 10,638 log-ins/activity month) one month after the dissemination activity. Approximately 9.4 new PlanYourLifespan.org accounts were created one-week post dissemination activity and over 1100 new accounts in the one-month period thereafter. Wide dissemination of PlanYourLifespan.org occurred by leveraging a train-the-trainer approach with community stakeholder leaders. Researchers should consider collaborating early on with community stakeholders to meaningfully disseminate results.

Mixed convectional and chemical reactive flow of nanofluid with slanted MHD on moving permeable stretching/shrinking sheet through nonlinear radiation, energy omission

Hybrid nanofluids are remarkable functioning liquids that are intended to reduce the energy loss while maximizing the heat transmission. In the involvement of suction and nonlinear thermal radiation effects, this study attempted to explore the energy transmission features of the inclined magnetohydrodynamic (MHD) stagnation flow of CNTs-hybrid nanofluid across the nonlinear permeable stretching or shrinking sheet. This work also included some noteworthy features like chemical reactions, variable molecular diffusivity, quadratic convection, viscous dissipation, velocity slip and heat omission assessment. Employing appropriate similarity components, the model equations were modified to ODEs and computed by using the HAM technique. The impact of various relevant flow characteristics on movement, heat and concentration profiles was investigated and plotted on a graph. Considering various model factors, the significance of drag friction, heat and mass transfer rate were also computed in tabular and graphical form. This leads to the conclusion that such factors have a considerable impact on the dynamics of fluid as well as other engineering measurements of interest. Furthermore, viscous forces are dominated by increasing the values of λ p, δ m and δ q, and as a result, F′(ξ) accelerates while the opposite trend is observed for M and ϕ. The drag friction is boosted by the augmentation M, λ p and ϕ, but the rate of heat transfer declined. According to our findings, hybrid nanoliquid effects dominate that of ordinary nanofluid in terms of F′(ξ), Θ(ξ) and ϕ(ξ) profiles. The HAM and the numerical technique (shooting method) were found to be in good agreement.

Thermal and concentration diffusion impacts on MHD Maxwell fluid: A generalized Fourier's and Fick's perspective

In this article, a new approach to study the fractionalized Maxwell fluid is described by the Prabhakar fractional derivative near an exponentially accelerated vertical plate together with exponentially variable velocity, energy and mass diffusion through a porous media is critically examined. The phenomena has been described in forms of partial differential equations along with heat and mass transportation effect taken into account. The Prabhakar fractional operator which was recently introduced is used in this work together with generalized Fick's and Fourier's law. The fractionalized model is transfromed into non-dimensional form by using some suitable dimensionless quantities. The non-dimensional developed fractional model for momentum, thermal and diffusion equations based on Prabhakar fractional operator has been solved analytically via Laplace transformation method and calculated solutions expressed in terms of Mittag-Leffler special functions. Graphical demonstration are made to characterized the physical behavior of different parameters and significance of such system parameters over the momentum, concentration and energy profiles. Moreover, for results validation, comparative study among limiting models derived from fractionalized Prabhakar Maxwell fluid such as fractional and classical fluid models for Maxwell and Newtonian are performed. Further, it is observed from the graphs the valocity curves for classical fluid models relatively higher as compared to fractional fluid models, and fractional approach is more realistic and convenient as compared to classical approach.