The Lagrangian-eulerian method has been used to simulate two dimensional creeping flows of Newtonian fluids in various flow fields.For Poiseuille and Couette flow and flow between rotating cylinders

the numerical results have been directly examined by the analytical solutions. For the flow around a cylinder between parallel plates

the present results are compared with the previous reports given by Mitsoulis and Liu et al.. The dimensionless drag of 130.94 is consistent fair well with the values given by Mitsoulis (132.31) and by Lu et al.(132.34)

respectively.For the flow through abrupt contraction geometry

the numerical results have been compared with the experimental observations of Nguyen and the numerical prediions of Viriyayuthakorn and Caswell. The size of the upstream vortex

measured in terms of a dimensionless re-attachment length is 0.18

excellently agrees with the previous reports.For the flow through reounded re-entrant corner

excellent agreement between the present results and the experimental observations of Walters and Webster is obtained. In conclusion

the Lagrangian-Eulerian method has predicted accurate results for all cases of the flows presented in this paper.