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 investigates the suitability of trapezoidal cross-section with segment base in drainage system design. The study has considered steady uniform open channel fluid flow. The saint-Venant partial differential equations of continuity and momentum governing free surface flow depth and the manning coefficient on the flow velocity were investigated. The flow variables are velocity and the flow depth while the flow parameters are across section area of flow ,channel radius, slpoe of the channel and manning coefficient. The study established that increase in cross section area of flow led to decrease in flow velocity. Further, increase in channel and manning coefficient. The study established that increase in cross section area of flow led to a decrease in flow velocity.Further, increase in channel radius and cross section area of flow led to a decrease in flow velocity and increase in roughness coefficient which causes the flow velocity to decrease.Additionally, increase in flow channel were applied to conservation equations to arrive at specific governing equations. The results of the study have been presented graphically. The study recommends further research on the effect of energy on flow velocity and turbulent flow on the same cross section.