Shyam Sundar

Effect of rain on removal of a gaseous pollutant and two different particulate matters from the atmosphere of a city

A nonlinear five-dimensional mathematical model is proposed and analyzed to study the removal of a gaseous pollutant and two different particulate matters by rain from the atmosphere of a city. The atmosphere, during rain, is assumed to consist of five interacting phases namely, the raindrops phase, the gaseous pollutant phase, its absorbed phase and the phases of two different particulate matters, one being formed by the gaseous pollutant. We assume that the gaseous pollutant is removed from the atmosphere by the processes of absorption while the two particulate matters are removed only by the process of impaction with different removal rates. By analyzing the model, it is shown that under appropriate conditions, these pollutants can be removed from the atmosphere and their equilibrium levels, remaining in the atmosphere, would depend mainly upon the rates of emission of pollutants, growth rate of raindrops, the rate of raindrops falling on the ground, etc. It is found that if the rates of conversion of gaseous pollutant into the particulate matter and rainfall are very large, then the gaseous pollutants would be removed completely from the atmosphere.

Modeling the effect of an intermediate toxic product formed by uptake of a toxicant on plant biomass

In this paper, a nonlinear mathematical model is proposed and analyzed to study the effect of an intermediate toxic product on the growth of plant biomass. It is assumed that the toxicant uptaken by plant biomass interacts with water (sap) present in it and forms an intermediate product, which then affects the biomass. To model the phenomena, it is further assumed that the intermediate product decreases the intrinsic growth rate of biomass density while the environmental concentration of toxicant decreases its carrying capacity. The model is analyzed using stability theory of differential equations and numerical simulation. It is shown that as the rate of emission of toxicant increases, the equilibrium level of plant biomass decreases, but this effect is determined by the emission rate of toxicant in the environment, the rate of its uptake as well as by the rate of formation of the intermediate toxic product.

Removal of carbon dioxide from the atmosphere to reduce global warming: a modelling study

We propose non–linear models to study the feasibility of removing CO2 from the atmosphere by introducing some external species such as liquid droplets and particulate matters in the atmosphere, which may react with this gas and get it removed by gravity. Further, this gas can also be removed by photosynthesis process upon using plantation of leafy trees around the sources of emission. The proposed nonlinear models are analysed using stability theory of differential equations and computer simulations. Model analysis suggests that the concentration of global warming gas decreases as the rates of introduction of liquid droplets and particulate matters increase. Also, this gas can be removed almost completely from the atmosphere, if the rates of introduction of these external species are very large. The concentration of CO2 also decreases as its absorption by green belt increases. It decreases further if the rate of introduction of external species increases. The numerical simulation of the models confirms these analytical results.

Modeling the removal of primary and secondary pollutants from the atmosphere of a city by rain

In this paper, we have proposed and analyzed a nonlinear mathematical model for the removal of primary and secondary pollutants from the atmosphere of an industrial city by rain. To model the phenomenon, it is assumed that the atmosphere consists of five nonlinearly interacting phases i.e. the raindrops phase, the primary pollutants phase, the secondary pollutants phase and absorbed phases of these pollutants in the raindrops. The dynamics of these phases is assumed to be governed by nonlinear differential equations with source, interaction, recycle and removal terms. The model is analyzed using stability theory of differential equations. It is shown that these pollutants can be washed out from the atmosphere completely by rain in the case of instantaneous emission of primary pollutants. However, when the primary pollutant is emitted at a constant rate, it is found that both the primary and secondary pollutants can still be washed out from the atmosphere under some appropriate conditions and the remaining equilibrium amount would depend upon the rate of emission of primary pollutants, rate of formation of secondary pollutants, rate of raindrops formation and different removal parameters. The equilibrium levels of these pollutants are much smaller after rain than its corresponding value before rain. A numerical study of the model is also performed to investigate the influence of certain key parameters on the dynamics of model system.

Modeling the effects of aerosols to increase rainfall in regions with shortage

It is well known that the emissions of hot gases from various power stations and other industrial sources in the regional atmosphere cause decrease in rainfall around these complexes. To overcome this shortage, one method is to introduce artificially conducive aerosol particles in the atmosphere using aeroplane to increase rainfall. To prove the feasibility of this idea, in this paper, a nonlinear mathematical model is proposed involving five dependent variables, namely, the volume density of water vapour, number densities of cloud droplets and raindrops, and the concentrations of small and large size conducive aerosol particles. It is assumed that two types of aerosol particles are introduced in the regional atmosphere, one of them is of small size CCN type which is conducive to increase cloud droplets from vapour phase, while the other is of large size and is conducive to transform the cloud droplets to raindrops. The model is analyzed using stability theory of differential equations and computer simulation. The model analysis shows that due to the introduction of conducive aerosol particles in the regional atmosphere, the rainfall increases as compared to the case when no aerosols are introduced in the atmosphere of the region under consideration. The computer simulation confirms the analytical results.

Modelling the Removal of Primary and Secondary Air Pollutants by Precipitation

A nonlinear mathematical model is proposed and analyzed to study the removal of primary and secondary air pollutants by precipitation in the atmosphere. The atmosphere, under consideration, consists of four interacting phases i.e. the rain droplets phase, the primary pollutants phase, the secondary pollutants phase and the combined phase of these pollutants absorbed in the rain droplets. The dynamics of these phases is governed by the ordinary differential equations with source, nonlinear interaction, conversion and removal terms. The proposed model is analyzed qualitatively using stability theory of differential equations. It is shown that under appropriate conditions, the pollutants can be removed from the atmosphere significantly and the removed amount would depend upon the rate of introduction of primary pollutants, rate of formation of secondary pollutants, rate of precipitation, rate of absorption and rate of falling rain droplets on the ground. Finally, computer simulations are performed to investigate the dynamics of model system. The results obtained in the paper are found to be in line with the experimental observations published in the literature.

Does Unemployment Induce Crime in Society? A Mathematical Study

Today unemployment has become a global phenomenon which may be instrumental in forcing unemployed persons to earn their livelihood in an illegal manner resulting in a crime. It is possible that unemployed individuals may become more prone to develop a tendency of committing a crime when they come in contact with persons involved in criminal activities but are still unexposed. Further, when unemployed individuals are exposed to have committed a crime, they are captivated and finally awarded imprisonment if the offence charged on them is proved under the existing criminal laws. In this paper, a nonlinear mathematical model is developed to study the role of unemployment in inducing crime by taking into account four dependent variables representing the unemployment class, the employment class, the criminal class and the jail class. The model analysis, using stability theory of ordinary differential equations, provides some local and nonlinear stability conditions regarding stability of equilibrium of the model system. It is inferred that the endemic equilibrium is locally asymptotically stable as well as nonlinearly stable. Numerical simulations of the model system have also been carried to support the analytical findings and showing the effect of certain key parameters on different variables. It is observed that the increase in unemployment rate induces crime in the community leading to increase the burden on jail class.