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and the amplitude grows. We conclude, therefore, that the adjustments provided by
the child on the swing must be at twice the frequency of the oscillation of the swing—
which is what we learnt as children.
E5.3 Meniscus on a circular tube
The phenomenon of a liquid rising in a small-diameter tube that penetrates (vertically)
the surface of the liquid is very familiar, as are the menisci that form inside and out-
side the tube. In this problem, we determine a first approximation to the shape of
the surface (inside and outside) in the case when the surface tension dominates (or,
equivalently, the tube is narrow). The basic model assumes that the mean curvature
at the surface is proportional to the pressure difference across the surface (which is
maintained by virtue of the surface tension). With the two principal curvatures of radii
written as and then this assumption can be expressed as
where z is the vertical coordinate and the pressure difference is proportional to the
(local) height of the liquid above the undisturbed level far away from the tube; this
relation is usually referred to as Laplace’s formula. In detail, written in non-dimensional
form, this equation (for cylindrical symmetry) becomes
where the surface is r is the radial coordinate with r = 0 at the centre of
the tube, and the tube wall (of infinitesimal thickness) is at r = 1; the liquid surface
satisfies as (see figure 9). The non-dimensional parameter is inversely
proportional to the surface tension in the liquid and proportional to the square of the
tube radius (and is usually called the Bond number). We will examine the problem of
solving equation (5.13) for with the boundary conditions
where is the given contact angle between the meniscus and the tube (measured
relative to the upward vertical side of the tube). For wetting, then we have
which we will assume is the case for our liquid. We solve the interior and
the exterior (r > 1) problems independently. The discussion that we present for the
exterior problem is based on Lo (1983); another description of both the interior and
exterior problems is given in Lagerstrom (1988). As we will see, this problem results in
the construction of a uniformly valid expansion (interior), and a scaling and matching
problem in the exterior (cf. §§2.4, 2.5).