0.9 Flow with free surface  (Page 4/6)

Surface viscosity. Adsorption of a surface-active component at an interface not only changes the surface tension or surface pressure but can also affect the surface rheology. Material adsorbed at interfaces form two-dimensional surface phases that may be gasous, expanded liquid, condensed liquid, or solid. The surface viscosity can change by more than a order of magnitude at a transition from one surface phase to another. This is analogous to the change in viscosity of bulk fluids at phase transitions. The attached figure shows vertical soap film drainage of a system is similar to that of the mobile film except that dodecanol was added to the sodium dodecyl sulfate (SDS) solution. The dodecanol screens the electrostatic repulsion of the SDS at the interface and promote the formation of a condensed liquid monolayer. This monolayer is rigid in this system and the films drains much more slowly than in the case of the mobile film. The mechanism of this difference in the drainage of foam films has been explained in terms of the surface tension gradient driven instability and the stabilizing effect of a large surface viscosity (Joye, et al., 1994, 1996).

Film drainage and deposition with laplace pressure

In Chapter 8 we modeled the gravity drainage of a film along a wall neglecting the pressure between the liquid and gas because the mean curvature of the system was very small compared to the length of the film. Now suppose that we have a film that in connected to a curved meniscus. The meniscus may be moving along a substrate and depositing a film or gathering up a deposited film, e.g., a bubble in a capillary tube. Alternatively, the substrate may be stationary with respect to the meniscus and the film is draining into the mensicus, e.g. foam or emulsion film between two bubbles or drops. For simplicity, we will assume that we have a pure system so there are no surface tension gradients or effects of surface viscosity. We will assume that the system is translational invarient. A schematic of some possible system configurations are shown below.

Configration of a bubble in a tube (Breatherton, 1961)

The continuity equation and equations of motion were specialized for lubrication and film flow in Chapter 6. The equations to $O(h_o/L)$ or $O(h_o/L)2$ are as follows.

The systems with a solid substrate will have the boundary condition of no-slip at the solid boundary and zero shear stress at the pure-fluid interface. In the case of two bubbles or drops coming into contact, the mid-plane is a plane of symmetry and has zero shear stress. It will be assumed the fluid interface is immobile in this latter case. The variable, $h$ , is the half-film thickness in this case. Since this case has zero shear on one surface and no slip on the other surface, the solution will be derived as for the cases with the solid substrate. The boundary conditions are as follows.