A Research thesis submitted in partial fulfillment of the requirement for the award of the degree of masters of Science in Applied Mathematics of Meru University of Science and Technology

Different irrigation methods are being used in agriculture. However, due to scarcity of water, irrigation methods that use efficiently are needed. The motivation of this study is the increasing use of porous pipes to meet this requirement. When the porous pipes are laid on the ground during irrigation, the geometry they take is dependent on their alignment to the planted crops which may have an effect on the delivery of water.The objective of this study is to investigate the effect of curvature and Rynolds number on radial velocity profile of water across a porous wall of a curved pipe with a circular cross-section, constant permeability, k and porosity ,q.The pressure driven flow is studied in two dimensions. The momentum equations are written in toroidal coordinates and only characterized by 8 and Re as the only non-dimensional groups of numbers with 8(0,1) and Re(1500,2100).We also considered the flow to be fully developed, unsteady,laminar and irrotational. Darcy number being used to non-dimensionalise the permeability .The equation of pressure is built from the continuity equation through the use of the Darcy equation since there is a linear relationship between flow rate and applied pressure. The flow in the main pipe was coupled with the flow through the porous wall. The equations were solved using finite difference method. It was observed that the effect of curvature on the velocity across the pipe wall is negligible while increasing the Reynolds number in the porous medium has the effect of increasing radial velocity through the pipes' wall. The findings of this study are important in the design of porous pipes and also in their use during irrigation.