This thesis considers two closely related problems. First, the influence of insoluble surfactant at a moving contact line is considered. This work is mostly motivated by the air entrainment during the coating process where there is a three-phase contact point (e.g., air, liquid and solid). For moving contact line problems, when the fluid is assumed to be an incompressible Newtonian fluid and a no-slip boundary condition is enforced at the solid boundary, the non-integrable stress singularity arises at the contact line, which is physically unrealistic. The contact angle of 180° is considered as a special case in which the singularity is absent. The previous work showed that there exists a non-singular local solution in the vicinity of the contact line for any capillary number. A simplified asymptotic model is used here to find a global solution with a 180° contact angle. Also the effects of insoluble surfactant are checked and numerical results show there exists a critical capillary number above which there is no steady state solution.

The second problem is viscous droplets rolling down a non-wettable inclined plane. The recent experiments show that the contact angle is very large (close to 180°) when a droplet rolls on a super-hydrophobic surface. The biharmonic boundary element method (BBEM) is used to implement numerical simulations of rolling motion. The numerical results agree well with the experimental results and theoretical prediction. Numerical evidence is also found that the stress singularity at the contact line is alleviated with a 180° contact angle. For droplets with insoluble surfactant on the surface, the finite volume method is used to track the evolution of surfactant. It shows that the rolling motion is retarded because of Marangoni force due to nonuniform concentration distribution of surfactant.