After the 80 s, the use of directional wells, both horizontal and long range wells, became frequent in the development of oil fields, which allowed great progress in exploration. This scenario, however, brings great challenges to operations related to wellbore cleaning. Wellbore cleaning consists in the removal of cuttings from within the annular through the circulation of drilling fluid. Due to the inclination of the well, the cuttings undergo the action of gravity and more to the bottom of the channel, as they have higher density than the liquid phase. They may be transported with a speed less than the speed of the liquid flow. When deposited on the channel, the cuttings form a stationary bed, blocking part of the annular and decreasing the flow rate that causes problems, such as reducing the rate of penetration, premature wear of the bit, high torque and drag, trapping column, loss of circulation and others, which may cause the loss of the well. Therefore, the fundamental understanding of particle sedimentation and transport in a suspending flowing liquid is necessary for drilling operation design and optimization. This research studies the laminar two-dimensional flow of solid particles suspended in a liquid due to a pressure gradient between two parallel plates, representing a simplified description of the flow that occurs in an annular during the drilling process. The profile of the particle bed along the channel and the flow rate pressure difference relationship for different flow conditions are determined. The mathematical formulation leads to a coupled system of three differential equations: mass and momentum conservation and transport equation, which includes the effects of particle diffusion due to the frequency of interaction between the particles, the gradient of viscosity and the difference density between the liquid and the particles. The system is solved by the finite element method Galerkin. The results will be of extreme importance in the development of more accurate models that describe the solids transport process in annular space.