We consider the statics of compound droplets made of two immiscible fluids on a
rigid substrate, in the limit when gravity is dominated by capillarity. In particular,
we show that the merging of four phases along a single contact line is a persistent
and robust phenomenon from a mechanical and thermodynamic perspective; it can
and does occur for a range of interfacial energies and droplet volumes. We give an
interpretation for this in the context of the macroscopic Young–Laplace law and
its microscopic counterpart due to van der Waals, and show that the topological
transitions that result can be of either a continuous or discontinuous type depending
on the interfacial energies in question.

When a bubble of air rises to the top of a highly viscous liquid, it forms a
dome-shaped protuberance on the free surface. Unlike a soap bubble, it bursts
so slowly as to collapse under its own weight simultaneously, and folds into a
wavy structure. This rippling effect occurs for both elastic and viscous sheets,
and a theory for its onset is formulated. The growth of the corrugation is
governed by the competition between gravitational and bending (shearing)
forces and is exhibited for a range of densities, stiffnesses (viscosities), and
sizes—a result that arises less from dynamics than from geometry, suggesting
a wide validity. A quantitative expression for the number of ripples is presented,
together with experimental results that support the theoretical predictions.

We investigate the axial instability of the free-surface front of a viscous fluid in a horizontal cylinder
rotating about its longitudinal axis. A simplified model equation for the evolution of the free surface
is derived and includes the effects of gravity, capillarity, inertia, and viscosity. This equation is
solved numerically to determine the base state with no axial variation, and a numerical linear
stability analysis is carried out to examine the onset of unstable axial modes. Various computational
results are presented for the wavelength of the axial instability. Inertia is found to play an important
role in the onset of the instability and the wavelength of the instability l satisfies the power law
l;g1/3, where g is surface tension. Finally some numerical simulations of the simplified evolution
equation are presented to show that they can capture the steady shark-teeth patterns observed in
recent experiments @R. E. Johnson, in Engineering Science, Fluid Dynamics: A Symposium to Honor
T. Y. Wu ~World Scientific, Singapore, 1990!, pp. 435–449; S. T. Thoroddsen and L. Mahadevan,
‘‘Experimental studies of the instabilities in a partially filled horizontal rotating cylinder,’’ Exp.
Fluids 23, 1 ~1997!#.

We describe a number of different phenomena seen
in the free-surface flow inside a partially filled circular cylinder
which is rotated about its horizontal axis of symmetry. At low
angular velocities the flow settles into a steady two-dimensional flow with a front where the coating film coalesces with
the pool at the bottom of the cylinder. This mode becomes
unstable at higher angular velocities, initially to a sloshing
mode on the rising side of the coating film and then to an axial
instability on the front. The undulations that appear on the
front grow into large-amplitude stationary patterns with
cusp-like features for some parameter values. At still higher
angular velocities and volume fractions, a number of different
inertial instabilities and patterns appear. We present a phase
diagram of the various transitions and characterize some of the
more prominent instabilities and patterns in detail, along with
some possible mechanisms for the observed behaviour.