Z. Angew. Math. Phys. (2017) 68:104
2017 Springer International Publishing AG
published online August 21, 2017
Zeitschrift f¨ur angewandte
Mathematik und Physik ZAMP
Energy stability of droplets and dry spots in a thin ﬁlm model of hanging drops
Ka-Luen Cheung and Kai-Seng Chou
Abstract. The 2-D thin ﬁlm equation describing the evolution of hang drops is studied. All radially symmetric steady states
are classiﬁed, and their energy stability is determined. It is shown that the droplet with zero contact angle is the only global
energy minimizer and the dry spot with zero contact angle is a strict local energy minimizer.
Mathematics Subject Classiﬁcation. Primary 76A20; Secondary 35B35, 35K55.
Keywords. Energy stable solution, Radial symmetry, The thin ﬁlm equation, Droplets with zero contact angle.
In this paper, we study the ultimate patterns of a mathematical model which describes the motion of
a thin liquid ﬁlm hanging from the underside of a rigid plate. In the lubrication approximation to the
governing system, the (normalized) model in one dimension is described by the thin ﬁlm type equation
where the y-axis points downward and h(·,t) is the height of the thin ﬁlm at time t. The same equation
also describes several diﬀerent models concerning viscous thin liquid ﬂows such as the motion of the thin
ﬁlm of a light ﬂuid pinched by a plate from below and a heavy ﬂuid from above  as well as thin ﬂuid
on the inner part of a cylinder . The numerical results in these works show that single droplets or
conﬁguration of droplets are the most commonly observed patterns.
To carry out an analytic study of these patterns, we adjunct (1.1) with the periodic condition h(· +
L, t)=h(·,t) for some positive L. Neumann condition can be used instead of the periodic condition and
yields similar results. The divergence structure of (1.1) shows that the area is conserved, that is,
h(x, t)dt =
Furthermore, we have the energy dissipative relation
E(h(·, 0)) = E(h(·,t)) +
where the energy is given by
As there is no guarantee that a positive solution remains positive at the end, the dissipative relation
suggests that the steady states of the equation be deﬁned as those nonnegative functions which satisfy