MST-Department of Mathematics

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Analysis of Boundary Layer Flow Second-Grade Hybrid Nanofluid Subject to Lorentz Force
(Kenyatta University, 2024-11) Chege, Stephen Njoroge
Fluids are non-solids that usually change shape under the action of shear stress. Over the past two decades, fluid thermophysical properties have been refined by the use of nanoparticles in the field of nanotechnology. The addition of nanoparticles has been a sure way of steadily improving fluid behaviour. Hybrid nanofluids have been of major interest to researchers. This is because more advancement in fluid behaviour has been achieved by the use of two dissimilar nanoparticles in a base fluid compared to the use of just one nanoparticle. Since major progress has been achieved, a variety of issues have also arisen like what would happen to the fluid properties when the stress tensor to strain tensor relationship is up to derivatives of order two. Multiple studies have been done on first-grade hybrid nanofluid flow (a subclass of Newtonian fluids) with little emphasis on second-grade hybrid nanofluids (a subclass of non-Newtonian fluids) research. So far, no researcher has considered the influence of Lorentz force on a second-grade hybrid nanofluid flow. To bridge this gap, this study analyses the boundary layer flow of second-grade hybrid nanofluid subject to Lorentz force. The nanoparticles used are TiO_2 and MoS_2 due to their great lubricating and efficient heat transfer properties. This study’s outcome will provide theoretical information to industries dealing with electronic and automotive cooling systems on how to improve their heat transfer efficiency. This will be done by indicating how to adjust the parameters of interest for maximum yield at the end of this study. The flow is on a surface of uniform thickness. The surface is linearly stretching horizontally and the fluid flow is experiencing perpendicular magnetic influence. The governing non-linear equations are formulated and rendered dimensionless via similarity variables. The resulting boundary condition equations are transformed to initial condition equations by use of shooting technique in MATLAB bvp4c. The IVP is then numerically solved by Runge Kutta (4) method in MATLAB bvp4c. The parameters of interest in the study are the second-grade fluid and magnetic strength parameters. These parameters are simulated and the results are presented graphically. Fluid’s velocity profile rises with increasing volume fraction and fluctuates with increasing fluid parameters and magnetic strength. The temperature profile grows with the Prandtl number and magnetic field and decreases with the increase in volume fraction and the second-grade fluid parameter.
Analysis of a 3D Heat Transfer of Magnetohydrodynamics Cu-H_2 O and Al_2 O_3-H_2 O Nanofluid over an Exponentially Stretching Plate
(Kenyatta University, 2024-11) Rutto, Celestine C.
Biomedical sensors, such as eye-imaging systems, and drug delivery mechanisms, heavily rely on magnetohydrodynamic (MHD) flow for effective operation. This study investigates the heat transfer characteristics in MHD nanofluid flow over an exponentially stretching surface, focusing on copper (Cu) and alumina (Al_2 O_3) nanoparticles suspended in water as the base fluid. The governing equations, which include the continuity, momentum, and energy equations, are formulated under the assumptions of steady, incompressible, and laminar flow. These equations are then made dimensionless using a Similarity Transformation, which reduces the partial differential equations (PDEs) to a system of ordinary differential equations (ODEs). The resulting system is numerically solved using the MATLAB package bvp4c, which is designed for solving boundary value problems. The study emphasises the impact of varying the nanoparticle volume fraction on the rate of heat transfer and skin friction. The results reveal that the Cu-water nanofluid exhibits higher heat transfer rates and lower skin friction compared to the Al_2 O_3-water nanofluid, highlighting its potential for enhanced thermal management in biomedical applications.
Numerical Investigation of Turbulent Convection Flow in a Rectangular Closed Cavity
(Kenyatta University, 2024-11) Manoti, Geofrey Moturi
Natural turbulent convection in closed cavities has many practical applications in the field of engineering such as the design of electronic computer chips, atomic installation and industrial cooling among others. In particular, it enables in achieving a desired micro-climate and efficient ventilation in a building. Recent studies show that turbulent flow is affected by variations in Rayleigh numbers, aspect ratio and heater position among others. Temperature is kept constant in all these studies hence inadequate literature on the effects of temperature on a turbulent flow. In this study, aspect ratio and Rayleigh numbers are kept constant at 2 and 1012 respectively and natural turbulent convection flow in a closed rectangular cavity is investigated numerically as the operating temperature is varied from 285.5K to 293K. The rectangular cavity’s lower wall was heated and cooling done at the top face wall while the rest of the vertical walls were kept in adiabatic condition. Material properties such as density of the fluid kept on changing at any given temperature. The thermal profile data generated influenced the nature of the turbulent flow. The non-linear averaged continuity, momentum, and energy equation terms were modeled by the SST k-ω model to generate streamlines, isotherms and velocity magnitude for a different operating temperature and presented graphically. The finite difference method and FLUENT were used to solve two SST k-ω model equations, vortices and energy with boundary conditions. It was discovered that, as the operating temperature increased turbulence decreased due to a decrease in the velocity of the elements and vortices became more parallel and smaller.
Mathematical Modelling of the Impact of Misinformation on the Spread of Covid-19
(Kenyatta University, 2024-08) Thiong’o, John
COVID-19, which is caused by SARS-CoV-2 is a viral disease of the respiratory system that emerged in 2019 and spread across all continents. The virus swiftly spread because of the unprecedented speed at which information, particularly false information, has spread in this century. This study established a mathematical model to analyze how misinformation influenced the transmission rate of COVID-19. The study employs SEIR mathematical model that integrates misinformation factor to analyze its impact on transmission rates. The obtained basic reproductive number (R0), which is the average number of new infections caused by infected person was used to assess the illiness contagiousness. The study also derived stability conditions for the equilibrium points of the model and discussed the long-term dynamics of the disease. The model was then solved numerically using the fourth-order Runge-Kutta method to investigate the effects of changing parameters as well as the simulation of various scenarios. The Results demonstrated that elevating the rates of misinformation had a positive correlation with both infected and recovered compartmet. The susceptible population first declined and then rose with the increase in the rate of misinformation. These findings pointed out understanding how misinformation affects the spread of COVID-19 will help researchers develop strategies that will effectively combat misinformation and limit the disease’s spread, thereby reducing the overall impact of the disease.
Effects of Thermal Radiation on Darcy Forchheimmer Flow of a Casson Nanofluid
(Kenyatta University, 2024-11) Loco, Valerie Sasha
Fluids are non-solid states of matter which deform continuously when an external force is subjected to them. They can be classified as either Newtonian or non-Newtonian fluids. Since most fluids fall within the category of non-Newtonian fluids, models like the Casson fluid model have been developed. Owing to Casson nanofluids’ enhanced thermophysical properties, it has a wide range of applications in the fields of mining, drilling operations, material science, metallurgy, food manufacturing, and nanotechnology and bio-engineering. Casson fluid is frequently modelled since the model agrees with the rheological evidence about human blood. Some of its applications in technological, industrial, mechanical, and scientific disciplines include; grain storage, geothermal energy production, designing warm protectors, artificial dialysis, catalytic converters, circulation of water in reservoirs and fermentation processes. On the other hand, radiation heat transfer mechanism has an immense impact in industries, engineering, technological fields where apparatus operate at extremely high temperatures. This study aims to investigate the effects of thermal radiation on Darcy Forchheimer flow of a two- dimensional, steady, incompressible flow of Casson Nanofluid over a linear stretching surface. The equations governing the fluid flow are formulated, then transformed to a system of ordinary differential equations using similarity variables. The resulting ordinary differential equations are solved using the fourth order Runge- Kutta Method. The model is simulated using MATLAB bvp4c to demonstrate the impact of pertinent parameters on the temperature, velocity, and concentration of the fluid. It was observed that an increase in the thermophoretic parameter leads to an increase of the temperature at the boundary. The concentration of the fluid decreases with an increase in porosity parameter value. Increasing the Brownian Motion parameter leads to an increase in concentration. An increase in Schmidt number leads to a decrease in temperature. Schmidt number increases with a decrease in concentration. The velocity and temperature profiles decrease with an increase in porosity parameter and an increase in thermal radiation leads to an increase of heat energy on the surrounding system leading to a decrease in fluid’s flow temperature.
A Model for the Dengue Virus Transmission Incorporating Educational Campaigning and Quarantining in Mombasa County, Kenya
(Kenyatta University, 2024-09) Munene, Antony Murimi
Mathematical Modelling of Tuberculosis-Covid-19 Co-Infection
(Kenyatta University, 2024-07) Githinji, Mary Ng’endo
It was noted that the spread of tuberculosis reduced significantly during the COVID-19 pandemic. This reduction has been associated with the preventive measures placed to combat the spread of COVID-19. Research showed a strong correlation between the spread of COVID-19 and the spread of tuberculosis in any population. It is worth noting that tuberculosis and COVID 19 are among the leading most deadly communicable diseases in the world today. The correlation in their spread also leaves us to believe that the spread of one can enhance the spread of the other. Hence, we proposed a situation where a population was co-infected with both COVID-19 and tuberculosis. By employing the conservative laws, the mathematical model was formulated, and the resulting model analysed both qualitatively and numerically. The equilibrium points of the model were obtained and the reproduction number calculated. The condition for stability of the co-infection at the equilibrium point was determined as R0<1. The non-negativity conditions for the solution were established. A numerical simulation was carried out for the model and the dynamics of the diseases studied as the parameters vary. It was found that the rate of contracting tuberculosis posessed a more significant impact on the possibility of coinfection than COVID-19 and the effects of migration from the tuberculosis sub-population were more significant than migration from COVID-19.
Analysis of Dynamics of HTLV Type 1 Infection on Cd4+ T-Cells with Cell-to-Cell and Mitotic Transmissions Using Fractional Order Model
(Kenyatta University, 2024-07) Chepng’eno, Mary; Isaac Chepkwony
Human T lymphotropic virus-1 which attacks CD4+ T-cells is a serious epidemic throughout the world. Even though research has been done extensively on the virus, it is still a threat in various parts of the world. In this research project, we formulate a fractional order model of Human T lymphotropic virus type 1 infection on CD4 cells. The model is made up of three nonlinear differential equations with fractional derivatives defined using caputo. The main aim is to develop and to explore the dynamics of infection of CD4 cells by the virus using fractional order model. The uniqueness of solution was discussed and positivity of solution provided using generalized fractional mean value theorem. Making use of the next generation matrix mathematical method, the basic reproduction number, Ro, is calculated. Model eqilibria are determined. The Routh Hurwitz stability requirement and the LaSalle’s invariance principle are used to investigate the stability of model equilibria. The global stability of equilibria is determined using the Lyapunov functional method. From the investigation done on stability, both endemic equilibrium point and the equilibrium point free of disease were discovered to be globally and locally asymptotically stable whenever the number of reproductions is more than one and when it is less than one respectively. To acquire numerical results, we used a numerical methodology that involves writing the differential equations with fractional order as an infinite system of ordinary differential equations of the first order. Then by using relatively small number of terms, the solutions are obtained by use of Runge-Kutta method of fourth order applied with the help of python. Finally, we presented the results obtained for various values of alpha graphically. The findings point to the need to control mitotic transmission during therapeutic intervention as well as the benefits of employing fractional order to model viral infection on CD4 cells.
Analysis of a 3D Heat Transfer of Magneto hydrodynamics Cu-H20 and Al2O3-H20 Nonoflluid over an Exponationally Stretching Plate
(Kenyatta University, 2023-08) Ruto, Celestine C.
Mathematical Modeling and Analysis of Corruption Dynamics in Kenya
(Kenyatta University, 2024-04) Muriithi, F. Muthoni
Corruption, which can be defined as the abuse of public office for private gains, is a complex and multifaceted problem that has negative impacts on a country’s economy, development, and governance. In the presence of corrupt practices, the affected countries have witnessed an upsurge in poverty levels, political instability, limited employment opportunities, the proliferation of debts (old and new), and a host of other challenges. Although some countries have made commendable strides trying to combat corruption, others have achieved minimal progress, and regrettably, despite efforts aimed at eradicating corruption, it still remains remains a persistent issue and especially in Kenya. It’s for this reason that a better understanding of its dynamics is needed to design effective policy interventions to reduce its prevalence and impact. The goal of this study is to use mathematical modeling and analysis to better understand the dynamics of corruption in Kenya, specifically by modeling the spread and dynamics of corruption using an epidemiological approach. The study aims to investigate the existence and stability of the corrupt-free and endemic equilibrium points, determine the parameters that drive corruption, and compute the reproduction number. The methods applied include the use of ordinary differential equations, linearization method by Jacobian Matrix, Lyapunov function, Next Generation Matrix, Normalized forward sensitivity index, and numerical simulation using MATLAB software. The study conducted stability analysis of the equilibrium states by applying linearization, Lyapunov function and Routh-Hurwitz criteria. The findings indicated that the corruption free equilibrium is stable both locally and globally in cases where R0 < 1 as well as the endemic equilibrium being asymptotic stable when R0 > 1. In addition, a sensitivity analysis was conducted to identify the most sensitive parameter that could be strategically manipulated to effectively combat corruption. This study will contribute to a deeper understanding of corruption dynamics in Kenya and inform policy-making and guide anti-corruption efforts. The expected output is to provide insights into the factors that influence the spread and persistence of corruption in a society. The study also identifies strengths and limitations associated with the epidemiological approach to modeling the dynamics of corruption and recommends potential ways of combining different approaches to study this complex and multifaceted problem. The study recommends policies that aim to reduce the benefits of engaging in corruption and increase the costs of engaging in corrupt behavior to effectively address the issue of corruption.
Water, Ethylene Glycol and Propylene Glycol Based Nanofluids with Copper Oxide and Magnesium Oxide for Optimal Radiator Cooling
(Kenyatta University, 2023-11) Kisengese, Hilder Mary
Automakers recognize the importance of coolants in keeping engines running smoothly by eliminating waste heat and preventing corrosion in the cooling system. The automotive industry’s cooling system is a considerable problem in producing efficient and cost-effective engines. Most integrated circuit engines use fluid cooling, which relies on liquid coolants like ethylene glycol and water with poor heat transmission properties. Nanoparticles, which have been shown to improve in thermal conductivity, are another option for enhancing their thermal physical properties. With their improved thermophysical qualities, nanofluids find utility as coolants in various mechanical and engineering contexts, including, but not limited to, the following: electronics, vehicles, transformers, computers, and electrical devices. This study compares nanofluids with three base fluids; water, Propylene glycol, and ethylene glycol— each containing copper oxide or magnesium oxide nanoparticles in order to establish the optimal coolant for a radiator. The governing equations will be nondimensionalised using appropriate similarity transformation. The resulting equations are solved using numerical method with the Runge-Kutta method of the fourth order. An in-depth discussion is given, along with graphical representations of relevant parameters, for the effects on fluid temperature, skin friction, fluid velocity and rate of heat transfer and the results discussed taking into account industrial applications. The results show that water-based nanofluid has the highest flow temperature and velocity among the three base fluids. At low magnetic field, MgO-water nanofluid has lower skin friction but CuO-water has the lower skin friction at high magnetic field.
Magnetohydrodynamic Flow in a Rotating System Over Horizontal Parallel Plates with Mass Transfer in a Porous Media.
(Kenyatta University, 2023-12) Momanyi,Francis Onduso
MHD as a field has got range of applications in real-life situations such as engineering and industries, polymer processing, glass fibre production, metallurgy, paper production glass blowing, purifying molten metal and non-metal inclusion, solar energy harvesting, paper production, cooling the nuclear reactant, and plastic film. This work investigated MHD flow in a rotating system over horizontal parallel plates with mass transfer in a porous media. The flow is considered steady and at the same time, the porous plates are stationary. The sheet along the z direction was considered infinite. The governing equations and the boundary conditions were non dimensionalised. The non-dimensional numbers obtained were; modified Grashof number, magnetic parameter, Rotational number, and permeability parameter. The resulting equations were then transformed using the FD method resulting in algebraic matrices that were solved by computer software (MATLAB) and then analysed for velocity and concentration profile. The numerical data on concentration and velocity profiles were recorded and presented graphically for interpretation and discussion. Some of the results revealed that velocity decreases when the magnetic parameter and Grashof number increase and an increase in Grashof values leads to a decrease in velocity.
Mathematical Modelling of Underground Water Contamination
(Kenyatta University, 2024-04) Kibet, Aruasa
Groundwater pollution is a major cause of many health hazards in our society. The pollution comes as a result of human indifference to waste disposal, industrial effluents, chemical spills etc. One major contaminant of underground water is nitrogen; which stems from excessive application of nitrogen-containing fertilisers, chemical spillage, etc. Its transport as it percolates through the porous media of ground surface layer to the underground water can be modelled mathematically by the advection-diffusion equation; advection (transport of contaminants by a bulk of the fluid parcel) and diffusion (random movement of the solutes i.e. contaminants, during transport). A linear model that incorporates both source and sinks is formulated and then non-dimensionalised by introducing suitable dimensionless variables and parameters. A finite difference scheme is developed for the problem using the Crank-Nilcoson scheme. The resulting algebraic equations are solved simultaneously at each time step to unravel the effect of the parameters on the concentration of nitrogen at each soil layer. The findings show that raising diffusion increases in concentration of nitrogen but increase in Peclet number leads to decreases in the concentration.
Determining the Socio-Economic Classes Using Principal Component Analysis Based on Binary Data
(Kenyatta University, 2023-01) Nasokho, Ruth Simuli; Leo Odongo; Anthony Gichangi
Abstract
Numerical Simulation of Natural Turbulent Convection with Vorticity Vector Formulation
(Kenyatta University, 2023-10) FILENTINUS, OTULO ONYANGO; Kennedy Awuor
Abstract
Modeling Jiggers’ Infestation with Incomplete Recoveries Incorporating the Flea Population; A Case Study of Murang’a County, Kenya
(Kenyatta University, 2023-09) Agutu, Arthur Omondi; Kennedy Awuor
Sand flea insect thrives in hot and humid regions full of dust particles. It attacks humans leading to jiggers infection (tungiasis). Spread of jiggers has been recorded in Caribbean, South American and African countries. In Kenya, Murang'a, Homabay and Siaya Counties are among the top regions affected by tungiasis infections. Poverty, lack of sufficient awareness, improper sanitation, and poor control methods are the major reasons for the unending spread of jiggers in Kenya. Prevention and treatment measures have been put in place by the government and NGO's to combat the unending new infections, yet the recoveries are still incomplete. A number of mathematical frameworks have been put in place to unravel the cyclic behavior of this infectious disease. However, a comparative study of the dynamical behavior of the disease in both human and flea population has not been conducted. In this research, we designed a model of jiggers infestation which incorporates the human and sand flea population in Muranga County, Kenya. We derived an ODE system from SEIR-FLA mathematical model to investigate the dynamics of jiggers infestation which incorporates both the human and flea population in Muranga County, Kenya. We used the Mathematica software tools to determine the effective basic reproduction number by applying the next generation matrix method. We applied the MATLAB software to generate the solutions of equations. Results confirmed local stability of JFE when Ro = 4.9827¢ — 13 as t — oo for all the Susceptible, Exposed, Infectious, Recovered human compartments and the Egg, Larval, Adult sand flea compartments. All state variables are positive at all times ¢, and numerical analysis of the invariant region reveals that the model is well-posed. These findings confirm that treatment aid in reducing incomplete recoveries of jiggers infestation.
A Jump Diffusion Model with Fast Mean Reverting Stochastic Volatility for Pricing Vulnerable Options
(Kenyatta University, 2023-10) Kalekye, Nthiwa Joy; Ananda Omutokoh Kube; Cyprian Ondieki Omari
The Black-Scholes-Merton option pricing model is a classical approach that assumes the underlying asset prices follow a normal distribution with constant volatility. However, this assumption is often violated in real-world financial markets, resulting in mispricing and inaccurate hedging strategies for options. Such discrepancies may result into financial losses for investors and other related market inefficiencies. To address this issue, this study proposes a jump diffusion model with fast mean-reverting stochastic volatility to capture the impact of market price jumps on vulnerable options. The performance of the proposed model was compared under three different error distributions: Normal, Student-t, and Skewed Student-t and under different market scenarios that consist Bullish, Bearish, and Neutral markets. In a simulation study, the results show that our model under Skewed Student-t distribution performs better in pricing vulnerable options than the rest under different market scenarios. Our proposed model was fitted to S&P 500 Index by maximum likelihood estimation for the mean and volatility processes and Gillespie algorithm for the jump process. The best model was selected based on AIC and BIC. Samples of the simulated values were compared with the S&P 500 values and MSE computed at various sample sizes. Values of MSE at different sample sizes indicate significant decrease to actual MSE values demonstrating it provides the best fit for modeling vulnerable options.
Enumeration of Sigma Algebras on Sets with at Most Seven Elements
(Kenyatta University, 2023-04) Mariga, Hildbrand; Benard Kivunge
Magnetohydrodynamic Nanofluid Flow over Convectively Heated Porous Radially Stretching Sheet
(Kenyatta University, 2023-11) Alai, Kulow Alou; Lawrence Njau; Maurine Wafula
Any material containing pores is termed a Porous media. Extensive research has been conducted on MHD nanofluid flow through porous materials. Included in these experiments are parallel-rotating plates surrounded by a porous channel and the influence of rotation on unstable couette flow. MHD nanofluid flow across a plate heated by convection atop a stretching sheet immersed in a porous medium has not been taken into account in any of these research. Consequently, the study’s goal is to investigate the motion of MHD nanofluids across a convectively heated plate superimposed on a radially expanding sheet embedded in a porous media. The model is formulated and non dimensionalised using similarity variables. By employing shooting technique to transform the boundary conditions and, Runge-Kutta scheme in MATLAB bvp4c, the system of ODEs are solved. The results obtained are displayed in graphs and others in tables. The results indicate that with increasing porosity, magnetism and surface rotation, the flow primary velocity decreases while the temperature profile surges. The findings from this study will provide beneficial theoretical insight on what parameters should be varied for maximum profit in a number of sectors like, power engineering sector, aerodynamic combination, drug recovery systems and water solar heating system.
Sequential Change Point Estimation Using Empirical Likelihood for Time Series Data
(Kenyatta University, 2023-09) Machuka, Carolyne Kemunto; Ananda Kube
Sequential Change point detection has enhanced the reliance on analysis of live data as it streams into the system to support real-time decision-making processes. This has played a key role in the advancement of time series modeling and forecasting in financial time series and risk management. This domain has a growing demand to identify change points precisely and efficiently for development of automated analytical models. In this work, sequential change point estimation based on empirical likelihood test statistic is developed by maximizing the likelihood of the empirical distribution of the data subject to constraints based on the sample moments. Change point is declared when the test statistic exceeds a set threshold. The threshold is set such that it maximises the power of rejecting the null hypothesis. A stop time function is defined based on the null hypothesis. Consistency of the change point estimator has been verified through monte carlo simulations based on different sample sizes to demonstrate the empirical power of the test statistic. As the pre change historical data grew unbounded, the bias of the Mean Absolute Error (MAE) approaches zero. The estimator converges closer to the true values. The estimator was fitted on KES/USD exchange rate data from January 2017 to December 2021.

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