There are different ways to overcome this difficulty. A first one is to consider that a thin film made by one of the phases covers the surface so that the interface does not reach the surface []. Then the apparent contact line can be studied using classical continussical continuum mechanics but the difficulty arises again at the tip of the film.
The second one, the most popular, is to relax the no-slip condition by introducing a new parameter, the slip length [] [][][]. This parameter as well as the dynamic contact angle must be determined experimentally. They may depend on the velocity of the contact line. The difficulty is to make a distinction between the contact angle and an apparent contact angle [][]
The third one, frequently used, is to assume that one of the phases is a perfect fluid. The paradox disappears. The flow in the viscous phase near the contact line is a rolling-like motion. But, however small the viscosity coefficients of the phases may be, the dissipation is infinite. Assuming that one of these coefficients vanishes means passing carelessly to the limit in a singular perturbation problem. Some important phenomena may be missed.
Finally, the continuum model used may be questioned. It is itself an approximation which may not be valid in the vicinity of the contact line. Then one may use molecular simulations [] or a continuum model able to describe the interface as a layer of finite thickness. This is the way chosen in this paper. We use the Cahn-Hilliard model (or the Van der Waals model). This model was not built in order to solve the contact line paradox; its parameters can be estimated from situations irrelevant to the problem. It is clear that it is a simpleat it is a simple model likely superseded in most practical situations but it is the first continuum model which describes the flow at a moving contact line.
The statics of the Cahn-Hilliard fluid is well known. But the nature of the internal forces in such a fluid is not trivial. One needs the second gradient theory [] or the theory of continua with edge forces [] to understand it. In the following section we establish the equations and boundary conditions for a viscous isothermal Cahn-Hilliard fluid. We restrict our study to the vicinity of a contact line. We define the zone (the inner zone) where the Cahn-Hilliard model will be used. In an intermediate zone where the classical model can be used we develop an analytic solution. Then we show how this intermediate solution matches the external and inner solutions and fixes the boundary conditions for the inner problem.
We point out the dimensionless parameters for the inner problem. This problem is nonlinear. We study it by numerical simulation where we split the problem into two minimization problems. The minimization of the dissipation is a linear problem solved in a classic way while the minimization of the energy is a nonlinear problem solved by a steepest-descent method.
The dependence of the dynamic contact angle upon the velocity of the contact line is clearly exhibited.
