  When a droplet (which size is characterized by R) is confined between two
parallel planes, its morphology will change accordingly to either varying the
volume of the droplet or the separation (characterized by h) between the
planes. We are aiming at investigating how such a geometric confinement affects
the wetting behaviours of a droplet. Our focus lies on two distinguished
regimes: (1) a pancake shape in a Hele-Shaw cell when the droplet is highly
compressed (i.e. h/R << 1), in which particular attention is paid on nonwetting
and wetting cases, respectively; and (2) a liquid Hertzian contact rendered by
a slight confinement (i.e. h/(2R) --> 1) in a nonwetting case. To realize this
aim, we first develop strict analytical expressions of the shape of the droplet
which are available for arbitrary contact angles between the liquid and the
solid planes, but in which the elliptic integrals indicate that these solutions
are implicit. By employing asymptotic methods, we are able to give expressions
of relevant geometrical and physical parameters (the Laplace pressure, droplet
volume, surface energy and external force) in terms of sole functions of R and
h in an explicit manner. Comparisons suggest that over a large range of h/R,
our asymptotic results quantitatively agree well with the numerical solutions
of the analytical expressions. This systematic study of the parameter space
allows a comprehensive understanding of the geometric confinement on wetting,
especially a wide existence of logarithmic behaviours in a liquid Hertzian
contact, to be identified.
