Contact-line dynamics of a diffuse fluid interface

An investigation is made into the moving contact line dynamics of a Cahn-Hilliard-van der Waals (CHW) diffuse mean-field interface. The interface separates two incompressible viscous fluids and can evolve either through convection or through diffusion driven by chemical potential gradients. The purpose of this paper is to show how the CHW moving contact line compares to the classical sharp interface contact line. It therefore discusses the asymptotics of the CHW contact line velocity and chemical potential fields as the interface thickness epsilon and the mobility kappa both go to zero. The CHW and classical velocity fields have the same outer behaviour but can have very different inner behaviours and physics. In the CHW model, wall-liquid bonds are broken by chemical potential gradients instead of by shear and change of material at the wall is accomplished by diffusion rather than convection. The result is, mathematically at least, that the CHW moving contact line can exist even with no-slip conditions for the velocity. The relevance and realism or lack thereof of this is considered through the course of the paper. The two contacting fluids are assumed to be Newtonian and, to a first approximation, to obey the no-slip condition. The analysis is linear. For simplicity most of the analysis and results are for a 90 degrees contact angle and for the fluids having equal dynamic viscosity mu and mobility kappa. There are two regions of flow. To leading order the outer-region velocity field is the same as for sharp interfaces (flow field independent of r) while the chemical potential behaves like r(-xi), xi = pi/2/max{theta(eq), pi - theta(eq)}, theta(eq) being the equilibrium contact angle. An exception to this occurs for theta(eq) = 90 degrees, when the chemical potential behaves like ln r/r. The diffusive and viscous contact line singularities implied by these outer solutions are resolved in the inner region through chemical diffusion. The length scale of the inner region is about 10 root mu kappa -typically about 0.5-5 nm. Diffusive fluxes in this region are O(1). These counterbalance the effects of the velocity, which, because of the assumed no-slip boundary condition, fluxes material through the interface in a narrow boundary layer next to the wall. The asymptotic analysis is supplemented by both linearized and nonlinear finite difference calculations, These are made at two scales, experimental and nanoscale. The first set is done to show CHW interface behaviour and to test the qualitative applicability of the CHW model and its asymptotic theory to practical computations of experimental scale, nonlinear, low capillary number hows. The nanoscale calculations are carried out with realistic interface thicknesses and diffusivities and with various assumed levels of shear-induced slip. These are discussed in an attempt to evaluate the physical relevance of the CHW diffusive model. The various asymptotic and numerical results together indicate a potential usefullness for the CHW model for calculating and modelling wetting and dewetting flows.