This talk gives a brief overview of mathematical models simulating compressible two-phase flows and incompressible pollutant transport in shallow flows. In the first part of the talk, we give a brief overview of mathematical models describing compressible two-phase flows. In two-phase flows two materials of different densities are mixed together. Such flows are encountered in various scientific and engineering fields related to environmental research, chemical engineering processes, nuclear energy and advanced heat transfer systems. The modeling and simulation of such flows are the most challenging tasks in computational fluid dynamics. The high-resolution finite volume schemes are applied to solve the model equations. Several cases studies are considered to verify the performance of suggested numerical algorithm.

The second part of the talk provides an insight to the space-time dynamics of pollutant transport which is mandatory for accurately predicting consequences on the surroundings of coastal regions and rivers and to design appropriate policies for combating pollution and environmental protection. Here, we are focused on the numerical approximation of a two-dimensional (2D) model of pollution transport in shallow water. The model contains an advection-dispersion equation for transport of pollutants coupled with the system of shallow water equations that consider varying bottom topography and pollutant source. Approximations of such models are highly difficult for numerical solution methods. A third order well-balanced 2D finite volume WENO scheme is extended and applied to solve the model equations. The cardinal aspect of this method is that it doesn’t demand an exact Riemann solver for estimating the cell interface fluxes at each time step. Furthermore, the method correctly forecasts the steady state situation without compromising on the high order accuracy. A few test problems are formulated considering a variety of physical situations.

Abstract: The spectrum of a graph is set of eigenvalues of the adjacency matrix of the graphs, together with their multiplicities. In 1974, Harary and Schwenk inititate the study of graphs with integral spectra, that is graphs whose eigenvalues are all integral. In this thesis we actually take the integral graphs then apply on different operations between integral graphs to get integral graphs. It means, we actually want to find the integral graphs by using different operations. Then we find line and compliment of the graphs which are also integral. We are finding some new integral graphs through adjacency and as well Laplacian matrix. The behavior of graphs after deleting or adding edges or vertices will also discuss later. In this research, we are discussing the Diophantine equations, some important results and discussions on Diophantine equation arising in graph theory.

Abstract: In this draft, a mathematical model of forward roll coating of a thin film of a viscoplastic fluid onto a moving porous web is developed when the web passes through a small gap between the two rigid rolls. The conservation equations in the light of LAT (lubrication approximation theory) are non- dimensionalized and solutions for the velocity profile, flow rate, pressure distribution are calculated numerically. It is found that by changing (increasing/decreasing) the value of material parameters, one can really control the engineering quantities like velocity distribution, flow rate, pressure distribution, and penetration depth. These results are a quick reference for the engineer working in coating industries and to compare the results with experimental data. Some results are shown graphically. It is found that the material parameter is a device to control flow rate, coating thickness, and pressure distribution.

Among many other factors that affect the preventive interventions to any infectious disease, not reporting timely in a hospital is also one of the catastrophic behavior of human beings in any society. Similarly, masses who do not report make it difficult for healthcare researchers to measure the actual data and develop prevention strategies, accordingly. Therefore, there is a critical need to structure a potential epidemic model with the unreported class of individuals. This novel idea is deliberated in this paper to study the profiles of the epidemic model of virulent diseases due to the individuals that report timely and those who don't report in hospitals for any reason. Mathematically, a system of seven equations is taken into consideration, which describes the susceptible individuals, exposed, do not report to the hospital, report to the hospital, quarantined, infected, and recovered. So, with the consideration of new compartments, the conventional SIR epidemic model expands to SEURRPQIR. The innovative design is made more realistic by utilizing proportional fractional-order differential equations with time delay. A special simplified expansion of this derivative reduces its computational cost and produces the results with fractional index, which helps to predict each fractional change. In addition, an optimal control methodology is also carried out to analyze the effectiveness of the awareness campaign in shifting the unreported individuals to the reported class, with optimal cost function for the unreported cases. Discussions are supported through the very recent deadly pandemic as an example to conclude the practical advantage of the model. The sensitivity analysis of basic reproduction numbers based on effective awareness campaigns is also the part of this study, which infers public awareness campaigns may be devised to motivate and guide such individuals to approach any healthcare center or a hospital.

Heat transfer enhancement has gained much attention in current years, because of its several uses in engineering and industrial zones. In most applications, heat transfer fluids are utilized as cooling fluids, such as water, paraffin oil, diathermic oil, ethylene glycol, naphthenic mineral oil, vegetable oil, etc. However, these common fluids have poor thermal conductivity, which minimizes the amount of heat transfer. Thus, an innovative method of improving heat transfer in fluids by adding nano-sized particles in the different base fluids have been introduced.Nanofluids possess a significantly higher thermal conductivity than regular fluids. Therefore, nanofluids are widely used for a wide range of applications, such as nuclear reactor cooling, cooling of vehicles and machinery, electronic device cooling, biomedicine, etc. Present study deals to thermal enhancement and numerical solution of Au and Ag-water based nanofluid flow past on inclined surface utilizing different types of important flow parameters as well as the Nan particles volume fractions. The governing equations in form of the partial differential equations are transferred into ordinary differential equations by using appropriate similarity transformations. The couple system of equations is numerically solved through shooting method in Maple software. The influences of the specified parameters on the temperature, velocity, heat transfer rate and the skin friction coefficient are discussed and presented graphically. Due to existence of the multiple solutions, stability analysis has also been performed. Moreover, to check the accuracy of the code the obtained results are compared with previously obtained results present in literature.