MST-Department of Mathematics

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    Action of Symmetric Group S5 on The Cosets of Some of Its Non-Maximal Subgroups
    (Kenyatta University, 2020-07) Omori, Kerubo Lydiah
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    The Action of Symmetry Groups of Platonic Solids on their Respective Vertices
    (Kenyatta University, 2020) Waweru, R. Brian
    Platonic solids are 3-dimensional regular, convex polyhedrons. Each of the faces are equidistant and equiangular to each other in any of the solids. They derive their name from the ancient Greek philosopher, P lato who wrote about them in his dialogue, the Timaeus as reported by Cornford (2014). The solids features have fascinated mathematicians for decades including the renown geometer, Euclid: In his Book XIII of the Elements, as rewrote by Heath et al. (1956), he successfully determined the exact number of solids that qualify to be Platonic Solids; tetrahedron, cube, octahedron, dodecahedron and icosahedron. In group theory, the symmetry group of an object is the group of all transformations under which the object remains unchanged, endowed with the group operation of composition. Due to their inherent symmetry of these solids many mathematicians have attempted to derive their symmetry groups. For instant, Foster (1990) who successfully enumerated the symmetry groups of the dodecahedron and recently Morandi (2004) attempted to compute these symmetric groups of the solids using a computer program called Maple. Although such contributions are noteworthy, a few attempts have been made to explore other features such as the symmetry groups of the platonic solids. Thus, this project investigates the properties of the group action of the symmetry groups of these platonic solids acting on their respective vertices. We embark on constructing the symmetry groups of each of the solids then employ the orbit-stabilizer and other theorems to determine the ranks and sub-degrees of each solid. The action of G on V shows that tetrahedron has a rank of 2, the octahedron has a rank of 3, dodecahedron has a rank of 6 while the cube and icosahedron have a rank of 4.
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    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.
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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.
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    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.
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    Analysis of flow in regular bend pipe
    (2012-02-01) Chirchir, Julius Kibet
    A fluid is a substance that deforms continuously on application of slight shear stress. The study of fluid mechanics deals with motion of fluids and conditions that support or after motion. Fluid mechanics can be divided into fluid kinematics and fluid dynamics. Fluid kinematics deals with forces involved in motion of fluid, whereas fluid dynamics deals with the state of fluid in motion caused by imbalances of forces acting on it. Kinematics of fluid motion uses vector quantities such as; velocity, acceleration and rate of discharge which is defined in terms of scalar quantities; length and time in some specified coding system. These depend on some boundaries of a particular system under in investigation. Fluid dynamics involves the applications of Newton's second law of motion that states:- rate of change of momentum is directly proportional to the applied force; Transfer of fluids through pipes involves changes in potentials/pressure energy between two ends of the pipes under study. This potential difference can either be caused by: - (a) Gravitational differences (g) (b) Differences in height (h) (Potential energy) (c) Density differences, (p), for example that caused by concentration differences, Beek (1985). These three factors are related by equation (i) on the next page. P=phg .........................................................(i) The flow of fluids in channels can either be turbulent or laminar. Flow is where the streamlines do not intersect such that there is no mixing up of fluids. Laminar flow occurs when fluid has; 1. Low Reynold's number (Re<2000) 2. Low viscosity (internal friction of fluid particals 3.Friction between the channel and the fluid negligible. Turbulent flow occurs where there is continuous mixing up of the fluid in motion. This type of flow occurs when the fluid has; 1. High Reinhold’s number 2. High viscosity. 3. Boundary of channel and fluid have got high friction. Fluids transfer can either occur in open or closed channel depending on the geometry of the channel. Open channels are like surface run-off, rivers and streams, dug-out drains, of which all have got a free surface while confined (closed) flow occurs in porous media, closed cracks and in pipes. Fluid flow in both types of channels have got its uses, but highlighted here are only importance of closed channel flow. That is the used both in domestic and industrial sectors. In domestic side flow in closed channels are important in areas, for example in passage of fluids for example water from one point to other. It is important to have piped water at our vicinity especially at this age of development. On the industrial side, flow in closed channels at factories is important and to mention a few examples, are in areas like; - 1. Sewage collection and disposal. 2. Passage of oil, for example the oil pipeline from Mombasa to Eldoret 3. Passage of raw materials in chemical processes, for example breweries 4. Passage of gases in geothermal plants like in Olkaria In this work, only confined type of flow will be considered in a uniform circular cross sectional bend of a pipe. The fluid here is a liquid, for example water. And since the fluid delivered need not necessarily follow a straight path definitely there will be bends in pipes. Shapes such as this might be met in area like: Sprinkler nozzles. Nozzles of sprayers and spouts of various utensils like kettles. Detailed knowledge on fluid flow and the prevailing suitable conditions for laminar flow must be well defined as given earlier. In this work, however. We seek to give a clear picture on: - 1. Pressure distribution. 2. Velocity distribution and profile where necessary. 3. Forces acting on the pipe at certain parts. We therefore consider in detail how flow factors are affected by the shape of the region of the pipe considered. The proper interaction balance of these factors play a role in flow mechanism. Earlier, various analyses on straight pipe of some geometry Douglas (1995). The writer has analysed flows by application of methods of differential calculus and coupled with a well-defined boundary conditions have given practically near accurate results. On basis of the results obtained by this method it has become evident that complete, continuity, momentum and energy (Bernoulli) equations have to be well understood. In most of flows in pipes, velocity distribution, pressure distribution and force analysis are investigated by use of calculus modified by trigonometric ratios with successful results. Differential models has been powerful in predicting a number of flow problems in diverse fields of flow in the past decade.
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    Analysis of Heat and Mass Transfer on Magnetohydrodynamics (Mhd) Nanofluids with Thermal Radiation and Brownian Motion Over A Heated Vertical Plate
    (Kenyatta University, 2019-04) Ndungu, Elizabeth Wambui
    This study investigates the effect of heat and mass transfer on Magnetohydrodynamics (MHD) nanofluid with thermal radiation and brownian motion over a heated vertical plate. The Magnetohydrodynamics (MHD) nanofluid flow have different electrical conductivities and behave differently in presence of thermal radiation, magnetic field, thermophoresis and brownian motion. The rate of heat and mass transfer on nanofluid along the vertical plate under the influence of a magnetic field with thermal radiation and Brownian motion leads to change in the fluid motion. The diverse applications of nanofluids in engineering and industries it is of this great importance hence the need to investigate the effects of thermal radiation, thermophoresis and Brownian motion on nanofluids and magneto hydrodynamics. Nanofluids are considered as potential working fluids to be used in high heat flux systems such as electronic cooling systems, solar applications, heat pipes, and nuclear reactors. As secondary fluids, they can be applied in commercial refrigeration, chiller and solar panels in absorption systems. They provide much more energy for a given weight of fuel than any technology in use , at the same time reducing thermal pollution. The governing non-linear boundary layer equations are formulated and transformed into ordinary differential equations using the similarity transformation. The resulting ordinary differential equations are solved numerically using the fourth order Runge-Kutta method. The numerical results for dimensionless parameters as well as the skin-friction coefficient and nusselt number, are presented graphically and analysed quantitatively. We note that increasing magnetic field, radiation, thermophoresis and Brownian motion parameters leads to an increase in the fluid temperature resulting in a reduction in the Nusselt number and Sherwood number
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    Analysis of Magnetohydrodynamic Convective Heat Transfer of Casson Nanofluid Flow Over a Heated Stretching Vertical Plate
    (Kenyatta University, 2022) Kigio, John Kinyanjui; Winifred Nduku Mutuku
    Casson fluid has many industrial and engineering applications. However, Casson nanofluid possesses superior electrical and thermal conductivities compared with the original Casson fluid. Hence, the Casson nanofluid is considered in this project. The equation governing the natural convective magnetohydrodynamics flow of Casson nanofluid across a a convectively heated vertical plate is formulated. The equation is reformulated into a system of ordinary differential equations (ODEs) using similarity transformations. Thereafter, a numerical solution of the resulting ODEs is obtained using the Runge-Kutta-Gills method. It is found that flow temperature profiles increase with increasing Eckert number, Biot number and magnetic field strength while flow velocity decreases as the flow becomes Newtonian and as nanoparticle volume fraction increases.
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    Analysis of Magnetohydrodynamic Heat and Mass Transfer with Carbon Nanotubes-Graphene Casson Hybrid Nanofluid
    (Kenyatta University, 2022) Juma, Belindar Atieno; W.N.Mutuku
    Fluids such as liquids and gases are amassed into either Newtonian or non-Newtonian groups. Most fluids fall under the non-Newtonian category and, as a result, various models like Casson fluid and Williamson fluid model have been proposed to deal with the non-Newtonian fluid behaviour. Due to its ability to model the flow of blood, Casson fluid model is of major medical importance. A development on ordinary fluids are nanofluids, which posses enhanced thermophysical properties. Hybrid nanofluid, obtained when two non-identical nanoparticles are dispersed in a fluid, is an improvement on the novel nanofluids. It has superior thermal conductivity when compared with nanofluids or ordinary fluids. Multiple research have shown that the shape of the nanoparticles used during the hybridization process has significant impact on the thermal properties of the hybrid formed. Applications of hybrid nanofluid include refrigerators, electronic devices, and cancer treatments. In the study of hybrid nanofluids, the focus has been placed on the dynamic properties and heat transfer rate. In consensus, the superiority of the hybrid’s properties are emphasized. Carbon being the most abundant product, hardest, strongest and stable known compound, it is an excellent thermal conductor. CNTs and graphene are allotropes of Carbon. In the HAMT research, no researcher has explored the impact of suspending a combination of CNTs and graphene nanoparticles on a Casson base fluid. To bridge this gap, this study is designed to analyse the HAMT rate of a 2-D magnetohydrodynamic hybrid Casson nanofluid. The nanoparticles are Carbon nanotubes and Graphene. The flow is across a surface stretching exponentially. Volume fraction, nanoparticle size and other pertinent parameters are investigated on the HAMT rate.The governing equations are converted to their non - dimensional form using similarity variables, and subsequently to an ODEs. The RK4 with Shooting Technique is adopted as a method of solution. Simulation of the model and investigation of the HAMT rate is carried out using MATLAB bvp4c. The primary velocity is reduced with Casson fluid parameter but enhanced with the radiation parameter. The temperature profiles boost with Casson fluid parameter, magnetic and radiation parameters. The local skin friction increases with Casson fluid parameter and radiation parameter but decreases with magnetic field strength. HAMT rate is enhanced with increasing Grashof number but decreases with Casson fluid parameter and magnetic field strength.
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    Analysis of Magnetohydrodynamic Stagnation Point Flow Due to a Fluid towards a Convectively Heated Permeable Stretching Sheet
    (Kenyatta University, 2020-10) Mwangi, Njoroge Kelvin
    Stagnation-point flow of an electrically conducting fluid over a continuously stretching surface in presence of magnetic fields is significant in many industrial processes such as the metallurgy, polymer processing, glass blowing, filaments drawn through quiescent electrically conducting fluid subject to magnetic fields, cooling of metallic plate, hot rolling, wire drawing, aerodynamic extrusion of plastic sheets, crystal growing. In these applications of stagnation point flow, the desired output depends largely on the rate of heating and the velocity of the fluid on the surface. The required rate will be achieved by variation of various thermophysical parameters such as suction parameter, Grashoff number, Hartmann number and buoyancy parameter. The resulting nonlinear partial differential equations governing this flow were reduced to nonlinear ordinary differential equations using similarity transformations and the resulting equations solved numerically using the fourth order Runge-Kutta scheme with a shooting technique. Graphical results were presented and discussed quantatively with respect to the effects of thermophysical parameters on both velocity and temperature profiles of the fluid. From the study we note that an increase in Grashoff number (Gr), Hartmann number (Ha) and suction parameter resulted to a corresponding increase in fluid velocity. The fluid temperature also increased with increase in Gr and Ha but decreased with increase in suction parameter and buoyancy parameter.
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    Analysis of seasonal time series with missing observations
    (2012-04-11) Kihoro, John Mwaniki
    In the past, Kenya's Tourist data has been analysed by among others, Mutiso (1982) who carried out Spectral Analysis to the monthly data for a series covering 10 years (1971-1980). Mutiso came up with general conclusion on cycles experience by the tourism industry in Kenya. Onyango (1993) fitted non-linear model to the tourist data before testing it for linearity. In an effort to confirm or disapprove their findings we decided to analyse a longer series, that is, to collect and use data for the years 1971 to 1990. It is in this process that we encountered the problem of missing data. Monthly data for the year 1984 could not be located in the Kenya central Bureau of Statistics (CBS) records, but quarterly totals were available. Our immediate problem was to fill in the block of missing values and as such most of the work in this project point to this direction. In this dissertation we have used the two known segments [(1971 - 1983) and (1985 - 1990)] of tourist data to estimate the block of missing values. We have considered two methods under regression (indirect) imputation methods and imputed the missing values from eventual forecasts. We have showed how two forecasts arising from the two regression methods can be combined to come up with more accurate estimates. We have also suggested a method of adjusting the estimates to incorporate the information from the already known quarterly totals. We have proposed simpler and direct method of imputing the missing values and we have showed that the final adjusted estimates from the direct and the indirect methods are similar. Finally we modelled the two data sets arising from the direct and the indirect imputation and generated eventual forecasts, which further showed that the two methods give almost similar estimates.
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    Analysis of Vertical Transmission Dynamics of Infectious Hepatitis B Virus: Mathematical Model Involving Vaccination And Treatment in Turkana County, Kenya
    (Kenyatta University, 2022) Ghai Kuei, Geu; Gatoto
    Hepatitis B has been a major global health menace for it is a potentially life-threatening liver disease. Close to 0.25 billion persons are living with this infectious disease across the world. It’s transmitted by infected individual to uninfected person either vertically (transmission before or during birth by carrier mother to the baby) or horizontally (transmission when the bodily fluid of an infected person comes into contact with the hepatitis B virus-free person). This can happen through the sharing of non-sterilized injection syringes, tattooing objects and through sexual intercourse. This particular project studied a mathematical model that combined both vaccination and treatment as a means to controlling the hepatitis B virus (HBV). In our mathematical model, equations are derived from the flow chart representing the HBV transmission dynamics. The determination of the disease-free equilibrium state (DFE), the endemic equilibrium state (EE) and the basic reproduction number 𝑅0 were made. The stability of these points were determined and the results show that the disease-free equilibrium was both locally and globally asymptotically stable. In other words, 𝑅0 < 1. The stability analysis of endemic equilibrium point also reveals that the point is locally and globally asymptotically stable, that is, 𝑅0 > 1. The basic reproduction number 𝑅0 is computed using the next generation matrix method. The system of ordinary differential equations (ODEs), which is non-linear are solved by numerical simulation. This was achieved by use of Runge-kutta method of order four with the help of MATLAB software. These results show that either of the method, treatment or vaccination, administered is effective in alleviating the spread of HBV disease. However, when both control strategies are combined, the disease is quickly controlled and ultimately brought to eradication.
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    Aspects of controllability of autonomous systems in locally convex topological spaces
    (2012-05-16) Oduor, Shem Richard
    In this project, two theorems in (6) on controllability of autonomous systems defined with the help of ordinary Differential Equations (ODE) in finite dimensional vector spaces is genralized to infinite dimensional locally convex topological spaces. This has been achieved with the help of the idea of a derivative, much more general than Frechet's derivative and an integral both defined in (12)
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    Bioconvection of a Newtonian Nanofluid Over a Vertical Plate in Presence of Gyrotactic Microorganisms
    (Kenyatta University, 2019-12) Kibe, John Ngugi
    Bioconvection induced by gyrotactic microorganisms in a Newtonian nanofluid past a permeable vertical plate is studied. Addition of motile microorganisms to a suspension of nanoparticles in a basefluid enhances mass transfer and mixing in most microsystems in addition to the enhancement of the convectional properties of the nanofluid. This concept has solved many heating problems in various areas including civil engineering, chemical engineering and mechanical engineering. The present study looks into the movement of motile microorganisms present in a nanofluid over a vertical plate for purposes of saving energy through the enhancement of heat transfer. The objectives are to formulate and solve the mathematical equations governing bioconvection of a Newtonian Nanofluid flow along a vertical plate, investigate the effects of gyrotactic microorganisms on the temperature and velocity and temperature profiles of the nanofluid and to analyse the effects of nanoparticles on microorganisms concentration, temperature and velocity profile. Ordinary differential equations are obtained from the governing partial differential equations by use of similarity variables. To numerically solve the ODE’s, the Runge-Kutta Ferhlberg method is used with the shooting technique. Further, an investigation on the effects of controlling parameters on several numbers and dimensionless quantities of our interest. It is found that the Nusselt Number, the Sherwood Number and Skin Friction are strongly affected by nanofluid and bioconvection parameters
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    Buoyancy-Induced Mhd Stagnation Point Flow of Williamson Fluid with Thermal Radiation Over a Stretching Surface
    (Kenyatta University, 2020) Ouru, John Olum; Winfred N. Mutuku
    The current study investigates the buoyancy-induced MHD stagnation point flow of Williamson fluid with thermal radiation over a stretching sheet. A system of nonlinear partial differential together with the boundary conditions governing the fluid flow are formulated, then transformed to ordinary differential equations using appropriately identified similarity variables. The resulting equations are solved numerically by combination of shooting technique and Runge-Kutta fourth order integration scheme. The results are depicted graphically to illustrate the effects of thermal radiation parameter, buoyancy and electromagnetic forces on both the fluid velocity and temperature. The results revealed that an increase in buoyancy leads to an increase in the overall velocity of the flow but a decrease in the temperature of the flow.
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    Comparison of the performance of estimators in estimation of finite population total
    (2012-02-01) Muema, Benjamin K.
    In this project we compare the performance of two different estimators of the population total. One estimator is model-based and the other one is model assisted. We look at model-based properties of the two estimators. We observed that under the general model, the biases of the two estimators are different.
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    Coriolis Effects on Air, Nanofluid and Casson Fluid Flow Over a Surface with Nonuniform Thickness
    (Kenyatta University, 2022) Samuel, Oke Abayomi; Winifred N. Mutuku
    The trajectories of moving objects in a rotating frame is usually deflected by Coriolis force and this leads to many of the natural phenomena on Earth. Some of the natural outcomes of Coriolis effect include the tropical storms, cyclones, and hurricanes. Understanding the Coriolis effect on Newtonian and non-Newtonian fluid flow is therefore vital in determining the right actions needed to forestall these natural disasters. The invention of mass-flow meter to measure the density of fluid and the rate of mass-flow in a tube and the invention of bioreactors are some of the many practical uses of Coriolis force. The path followed by the air (Newtonian fluid) and nanofluid (fluid suspending some nanoparticles) heated by the sun to reach the Earth is usually deflected by the Coriolis force. It becomes worthwhile to investigate the influence of Coriolis force on the flow of these fluids. Also, the effect of a rotating frame of reference on the Brownian diffusivity and thermophoretic diffusivity of nanofluids is important in industries for its role in cooling/heating processes. Casson fluid possesses characteristics that closely agree with the nature of blood; with applications in bioengineering. More so, the Earth’s surface is neither horizontal nor vertical, and it is neither inclined. The Earth surface is geoid and so, a more appropriate surface to model it is one with non-uniform thickness. This thesis investigates and gives a comprehensive insight into the significance of rotational force on the flow of Newtonian and non-Newtonian fluids over a surface with non-uniform thickness. A suitable set of equations for the flow is obtained by introducing the Coriolis force as a body force into the Navier-Stokes equations. Appropriate similarity variables are used to reduce the set of governing partial differential equations to a system of non-linear ordinary differential equations. The transformed set of equations is numerically solved via shooting technique alongside Runge-Kutta algorithm. The results are depicted graphically and in tables to show the effects of pertinent parameters on the flow. It is observed that rotation enhances primary flow velocity but impedes secondary flow velocity. Rotation also has an increasing effect on temperature profiles but decreasing effects on the heat transfer rate of Casson and Newtonian fluids.