Closed-form solutions for a reaction-diffusion SIR model with different diffusion coefficients

The processes responsible for the effective longitudinal transport of solar energetic particles (SEPs) are still not completely understood. We address this issue by simulating SEP electron propagation using a spatially 2D transport model that includes perpendicular diffusion. By implementing, as far as possible, the most reasonable estimates of the transport (diffusion) coefficients, we compare our results, in a qualitative manner, to recent observations at energies of 55–105 keV, focusing on the longitudinal distribution of the peak intensity, the maximum anisotropy, and the onset time. By using transport coefficients that are derived from first principles, we limit the number of free parameters in the model to (i) the probability of SEPs following diffusing magnetic field lines, quantified by , and (ii) the broadness of the Gaussian injection function. It is found that the model solutions are extremely sensitive to the magnitude of the perpendicular diffusion coefficient and relatively insensitive to the form of the injection function as long as a reasonable value of a = 0.2 is used. We illustrate the effects of perpendicular diffusion on the model solutions and discuss the viability of this process as a dominant mechanism by which SEPs are transported in longitude. Lastly, we try to quantity the effectiveness of perpendicular diffusion as an interplay between the magnitude of the relevant diffusion coefficient and the SEP intensity gradient driving the diffusion process. It follows that perpendicular diffusion is extremely effective early in an SEP event when large intensity gradients are present, while the effectiveness quickly decreases with time thereafter.

Calculation of gas concentration-dependent diffusion coefficient in coal particles: Influencing mechanism of gas pressure and desorption time on diffusion behavior

The effects of the irreversible boundary reaction and the particle drag on mass transfer are studied analytically in concentric annulus flows. The solution of mathematical model, based on the generalized dispersion model brings out the mass transport following by the insertion of catheter on an artery in terms of the three effective transport coefficients, viz., the exchange, convection and diffusion coefficient. A general expression is derived which shows clearly the time dependent nature of the coefficients in the dispersive model. The complete time dependent expression for the exchange coefficient is obtained explicitly and independent of velocity distribution in the flow; however it does depend on the initial solute distribution. Because of the complexity of the problem only asymptotic large time evaluations are made for the convective and diffusion coefficients, but these are sufficient to give the physical insight into the nature of the problem of the effects of drag and absorption parameters. It is found that as absorption parameter increases exchange and convection coefficients will be enhanced, but diffusion coefficient will be reduced. After certain period of time exchange coefficient will be constant for different values annular gap. As the drag parameter increases convection and diffusion coefficients will be reduced. With the enhancement of catheter radius i.e., the annular gap will be reduced then the convection and diffusion coefficients will be decreased.

This study indicates two issues of available time-dependent diffusion coefficient function; non-smoothness of diffusion coefficient decay, and inconsistency of stable time of diffusion coefficient. A naturally logarithmic apparent diffusion coefficient function is thus developed for closed-form solutions of chloride transport model. The developed model is validated with experimental data, and its generality is ensured by comparing with the finite difference approach. From the study, the stable time of the developed diffusion coefficient appears 2.87-3.21 years after exposure, and the stable time of surface chloride appears 5 years after exposure. Such early appearance of these stable times behaves different from other studies, causing different long-term chloride prediction and concrete service life. Using the developed model, the influence of cover depth and percent fly-ash is determined in service life prediction. Additionally, this study develops a model to predict environmental impact in terms of eutrophication potential, currently considered as an emerging global issue. The developed eutrophication potential model shows that the increase of fly-ash replacement of 0% to 50% reduces such eutrophication potential due to concrete production by as much as 38%. Moreover, the relationships between the service life and the eutrophication potential for normal and fly-ash concrete tend to be linear.

The Brusselator model is widely used to illustrate and study basic features of models of chemical reactions involving trimolecular steps. We provide the necessary mathematical theory related to Reaction-Diffusion systems in general and to the Brusselator model in particular. Specifically, the issues of local and global existence of solutions, their uniqueness and regularity are discussed in detail. The theoretical and numerical investigation presented futher in the thesis provides an insight into the asymptotic behavior of the solutions of this model, characterizing the parameter region for each of the three qualitatively different cases: homogeneous steady state, Turing pattern and bulk oscillations. Particular attention is given to the supercritical Hopf bifurcation parameter domain where no substantial theory is available. This study was largely motivated by the observations of Young, Zhabotinsky and Epstein that Turing patterns eventually (for sufficiently small ratio of the diffusion coefficients) dominate the Hopf bifurcation induced bulk oscillations. In this work we confirm this observation and further establish more precisely the shape of the boundary separating the Turing pattern domain and the bulk oscillations domain in the parameter space. The obtained results are used in revealing an essential mechanism generating oscillating patterns in the coupled Brusselator model. It can be considered as a model of the reaction sequences in two thin layers of gel that meet at an interface. Each layer contains the same reactants with the same kinetics but with different diffusion coefficients. The occurrence of oscillating patterns is due to the fact that for the same values of the parameters of the model but with different diffusion coefficients the one system can be in the Turing pattern domain while the other is in the bulk oscillations domain. Hence, roughly speaking, one layer provides a pattern while the other layer drives the oscillations.

Diffusion coefficients of supercritical CO2 in oil-saturated cores under low permeability reservoir conditions

A study of Nafion-coated and uncoated thin mercury film-rotating disk electrodes for cadmium and lead speciation in model solutions of fulvic acid

Microbial degradation and transformation of substances in soils plays a crucial role in the nutrient turnover of ecosystems. To quantify these processes, a mathematical description is needed. For this purpose, an analytical solution to a model of solute diffusion and biodegradation in soil aggregates was developed. The model is a first approach toward understanding the influence of geometric arrangement of microorganisms and substrates in structured soils. These soils are considered to consist of uniformly sized and shaped aggregates surrounded by surface films of the soil solution. The model simulates transient diffusion of finite substrate amounts from the surface films into spherical aggregates. Biodegradation is considered for the special case of unlimited microbial growth, and adsorption is assumed to follow a linear Freundlich isotherm. The system is represented by a composite sphere, the outer sphere being the solution film and the inner sphere representing the soil aggregate. The diffusion equations are solved by Laplace transformation. The model solution gives a direct relationship between the initial substrate and biomass concentrations, the diffusion coefficient, the specific growth rate, and the adsorption coefficient. Good agreement between this closed form solution and numerical solutions is obtained for diffusion with and without biodegradation. Since the substrate is exhausted by organisms close to the surface, the centers of large aggregates are not reached by the diffusing substrate. These unaffected centers become larger as the growth rate is higher, the diffusion constant is lower, and adsorption of the substrate is stronger.

The study of mass transfer during osmotic dehydration process in limited volume solutions was carried out to evaluate the diffusion coefficients of sucrose and water in the osmotic treatment of hexahedral pineapple slices. The experimental osmotic dehydration kinetics for pineapple slices of two different sizes were conducted at 25 °C using a 1:1 solution to fruit weight ratio. The analytical solution of a 3D mass transfer model considering a limited volume of osmotic solution (i.e., an osmotic media of variable solute concentration) was used for describing the mass transfer in osmotic dehydration of pineapple slices. This model was fitted to the experimental kinetics by means of nonlinear regression to obtain the diffusion coefficients. Additionally, the diffusion coefficients were evaluated considering an infinite volume of osmotic solution (i.e., an osmotic media of constant solute concentration). Results showed that the proposed model may be fitted accurately to the experimental osmotic dehydration kinetics and allows the estimation of diffusion coefficients when solute concentration in the osmotic media varies along the process.

Accounting for the concentration dependence of electrolyte diffusion coefficient in the Sand and the Peers equations

Since the diffusioncoefficient is a key parameter tocharacterizethe diffusion rate of methane molecules, its measurement and solutionhave always been a research hotspot. The diffusion coefficient isnormally solved through analytical solutions of theoretical models,which is complex and poorly applicable. In comparison, the numericalsimulation optimization method can seek a solution easily and quickly,providing a clue for solving such problem. In this paper, first,gas desorption experiments were conducted on coal samples with differentinitial gas equilibrium pressures, coal particle sizes, and metamorphicdegrees. Combined with existing theoretical models, the numericalsimulation optimization method was adopted to solve the diffusioncoefficient of the coal particle. Furthermore, the applicability andadvantages of the numerical simulation optimization method were discussed.Finally, the variation law of the diffusion coefficients was analyzed.The results demonstrate that the numerical simulation optimizationmethod can not only solve the diffusion coefficient easily and quicklybut also reveal the law of diffusion concentration with time. The d values between the solution results and the experimentaldata under different conditions are all smaller than 0.2, which provesthe effectiveness and accuracy of the simulation optimization method.The diffusion coefficient of gas from coal particles is unrelatedto the initial gas equilibrium pressure, yet it has a Z-shaped relationshipwith the coal particle size and a V-shaped relationship with the metamorphicdegree.

The computational studies of a predictive mathematical model for the extraction of interstitial (ISF) for transdermal and non-invasive diagnosis using biodegradable and hollow microneedle patch is presented in this paper. Rapid Diagnostic Tests diagnosis, which is non-invasive, affordable, straightforward, and provides results promptly and reliably, has increased access to parasite-based analysis on a global scale. Microneedle arrays are a rapidly evolving and promising technology for transdermal interstitial fluid extraction, which is used for many clinical diagnostic procedures. Hence, a developed mathematical predictive model used to optimize the design of microneedle patch for transdermal ISF extraction and subsequent diagnosis using dissolvable microneedle arrays was applied in this study. The model's solutions were obtained using the Differential Transform Method. The numerical Runge-Kutta method of fourth order was used to validate it. An experimental test result was also used to further validate the analytical results in the absence of the extracted velocity parameter. And there was a good agreement among them. Influence of dissolution rate constant, microneedle height, diffusion coefficient, velocity of ISF, microneedle ISF drug load, and density of the microneedle; were investigated. Increase in diffusion coefficient and density led to an increase in concentration of ISF extracted over time, an increase in dissolution rate led to a decrease in concentration extracted, while decrease in drug load and height, led to increase in ISF concentration extracted. A negligible effect was observed by varying the velocity of ISF extracted. The approximate analytical approach utilized in the current work has given us a more precise strategy for creating a mathematical model that predicts how ISF will be extracted from skin for use in transdermal and non-invasive rapid diagnostic tests.

A mechanistic study of griseofulvin dissolution into surfactant solutions in laminar flow conditions

Pharmacologic Vitreolysis is a new therapy intended to enzymatically liquefy vitreous and separate the posterior vitreous cortex from the retina. There are presently no non- invasive methods with which to quantify vitreous liquefaction. This study evaluated non-invasive Dynamic Light Scattering (DLS) to assess vitreous liquefaction in model vitreous solutions. Vitreous of 5 bovine eyes was measured by DLS. Methodology involved the addition 50 IU/ml of hyaluronidase and 5 mg/ml of collagenase to solutions of 0.1 mg/ml of hyaluronan and collagen with polystyrene beads (30 nm diameter), respectively. Solutions were incubated at 37 degrees Celsius for 24 hours and DLS measurements were made at various time intervals. Results showed that the diffusion coefficients and particle size distributions determined from DLS measurements in the 5 bovine vitreous specimens were consistent and reproducible. Adding enzyme to the model vitreous solutions increased diffusion coefficients in the collagen as well as the HA solutions. These results suggest that DLS is a useful non-invasive method to measure diffusion coefficients in both model solutions and bovine vitreous. This approach may provide a quantitative means to assess different pharmacologic vitreolysis agents in vitro, and may also serve as a clinical tool for monitoring the effects of pharmacologic vitreolysis in vivo.

Dissolution of Weak Acids Under Laminar Flow and Rotating Disk Hydrodynamic Conditions: Application of a Comprehensive Convection–Diffusion–Migration–Reaction Transport Model