On the dynamics of liquid spreading on solid surfaces

Abstract

Our main objective is to identify a boundary-value problem capable of describing the dynamics of fluids having moving contact lines. A number of models have been developed over the past decade and a half for describing the dynamics of just such fluid systems. We begin by discussing the deficiencies of the methods used in some of these investigations to evaluate the parameters introduced by their models. In this study we are concerned exclusively with the formulation of a boundary-value problem which can describe the dynamics of the fluids excluding that lying instantaneously in the immediate vicinity of the moving contact line. From this perspective, many of the approaches referred to above are equivalent, that is to say they give rise to velocity fields with the same asymptotic structure near the moving contact line. Part of our objecive is to show that this asymptotic structure has only one parameter. A substantial portion of our investigation is devoted to determining whether or not the velocity field in a particular experiment has this asymptotic structure, and to measuring the value of the parameter.More specifically, we use the shape of the fluid interface in the vicinity of the moving contact line to identify the asymptotic structure of the dynamics of the fluid. Experiments are performed in which silicone oil displaces air through a gap formed between two parallel narrowly-spaced glass microscope slides sealed along two opposing sides. Since we were unable to make direct measurements of the shape of the fluid interface close to the moving contact line, an indirect procedure has been devised for determining its shape from measurements of the apex height of the meniscus. We find that the deduced fluid interface shape compares well with the asymptotic form identified in the studies referred to above; however, systematic deviations do arise. The origin of these deviations is unclear. They could be attributed to systematic experimental error, or, to the fact that our analysis (valid only for small values of the capillary number) is inadequate at the conditions of our experiments.

Wall energy relaxation in the Cahn–Hilliard model for moving contact lines

“…2 It is based on similar ideas for the slip model due to Ngan and Dussan V. 30 and others. The key element is equating the Cahn-Hilliard dynamic contact angle D with the intermediate contact angle ¯S of Ngan and Dussan V. 30 Previously, the Cahn-Hilliard theory has been related with the slip models via the correspondence between the diffusion and slip length: l s = 2.5l d . The wall relaxation provides another connection.…”

“…Therefore, different slip models can be used to produce the same fluid dynamics on larger scales. 26 Then Ngan and Dussan V. 30 parametrized the boundary-value problem in the outer region in terms of an intermediate contact angle, different from ˜S , measured at an intermediate distance ͑larger than l s ͒ from the contact line. Shen and Ruth 31 adopted this strategy in their finite-element calculations of moving contact lines between parallel plates and achieved good agreement with experiment.…”

“…͑23͒, ͑25͒, and ͑26͒ yields the apparent contact angle A , 33 suggested. Note that although ¯S is the interface angle at l s , it is imposed on the wall in the parametrization of Ngan and Dussan V. 30 This is valid because the interface has little curvature inside the slip region, 2 and the distance l s is infinitesimal when viewed from the outer solution.…”

Numerical Simulation of Moving Contact Line Problems Using a Volume-of-Fluid Method

“…The magnitude of slip can be quantified as a length scale, which is known as the slip length. Results in the literature [3,4,7,8,15] indicate that the magnitude of this slip length is the most essential factor in contact line spreading, while the details of the slip law have little effect on the overall flow. In numerical simulations, the physics of the problem is only resolved to the scale of the mesh, and we can expect an effective slip length on the order of the mesh size.…”

“…This is a paradox, since the force provided by surface tension is finite and, on the other hand, it is believed to be the force which is responsible for moving the contact line. In considering moving contact line problems, it is therefore important to include slip (see for instance, [3,4,7,8,15,18]). Moreover, the observed contact angle in a moving contact line is generally not equal to the static contact angle; in addition, the macroscopic contact angle is different from the actual contact angle at the wall, and determining the relationship is a complicated problem of matched asymptotics.…”