A Thesis Submitted in Partial Fulfillment for the Award of Masters of Science Degree in Applied Mathematics in the School of Pure and Applied Sciences of Meru University of Science and Technology.

This study considers unsteady non-uniform open channel flow in a closed conduit with horseshoe cross-section.Effects of the flow depth,channel radius,slope of the channel, manning constant and lateral inflow on the flow velocity as well as the depth at which flow velocity is maximum were investigated. The finite difference approximation method was used to solve the governing equations because of its accuracy, stability and convergence followed by a graphical presentation of results.The results indicated that for a given flow area, the velocity of flow increases with increasing depth and increase in the slope of the channel leads to an increase in flow velocity whereas increase in manning constant, radius of the channel and lateral inflow leads to a decrease in flow velocity.The results of this study agree with earlier researches done on related work of different open channel cross section.For example ,Kwanza et al.(2007) analyzed effects of channel width and slope for both trapezoidal and rectangular channels and concluded that discharges for both channels increases as the specified parameters are varied upwards. In addition, Kinyanjui et al.,(2012) studied modeling of open channel leads to decrease in flow velocity. The practical applications of this study include controlling floods, irrigation and in construction of channels such as house gutters.