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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).
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