# Interdisciplinary Applied Mathematics

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Let us consider a bubble in a micropipe immersed in a wetting liquid, assuming also that gravity is negligible (i.e., small Bond number limit). The bubble is then axisymmetric within the capillary, and a wetting annular film is formed around the bubble and between the capillary walls. The film thickness h is very small, so we can employ the standard lubrication limit to approximate the flow in the film. Thus, the velocity profile is

y d3h

u(y) = —1-—y(y/2-hl

y dx3

where y is measured from the wall of the micropipe; also, the pressure is constant across the film thickness but varies in the longitudinal direction x. In the presence of (insoluble) surfactants the interface between the bubble and the film behaves, as rigid wall, and thus in this case the velocity profile should be modified accordingly, i.e.,

u(y)

7 d3h f у h

~plh?V V2 ~~ 2J ’ so it satisfies no-slip boundary conditions. Typically, the bubble is elongated so the length to diameter ratio lb/d is grater than 1, and the bubble consists of two caps and a middle parallel section, as shown in the sketch of Figure 8.14. If the bubble is symmetric, i.e., the front and back caps are identical, then there is no capillary pressure difference, and thus the bubble is not moving. Therefore, for mobile bubbles there is a large pressure in the back cap and a smaller pressure in the front, and this pressure difference pushes the bubbles into the annular liquid wetting film.

In order to estimate the corresponding pressure drop we employ the Laplace-Young equation. The two curvatures in this problem are the curvature in the caps, which is approximately 1/R (with R = d/2 the capillary radius), and    the    film    (axial)    curvature    hxx,    which is nonzero    in    the    transi

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