Numerical studies are performed addressing the development of regions
of high
curvature and the spontaneous occurrence of cusped interfacial shapes in
two-dimensional
and axisymmetric Stokes flow. In the numerical simulations, the velocity
field is computed using a boundary-integral method, and the evolution of
the
concentration of an insoluble surfactant over an evolving interface is
computed using
an implicit finite-volume method. Three configurations are considered in
detail, and the
results are used to elucidate three different aspects of cusp formation.
In the first series,
the deformation of a two-dimensional bubble immersed in a family of straining
flows
devised by Antanovskii, and of an axisymmetric bubble immersed in an analogous
family of flows devised by Sherwood, are examined. The numerical results
indicate that
highly elongated and cusped two-dimensional shapes, and pointed or cusped
axisymmetric shapes, are unstable and should not be expected to occur in
practice. In
the second series of studies, the role of an insoluble surfactant on the
transient
deformation of bubbles subject to the Antanovskii or Sherwood flow is investigated.
Under certain conditions, the reduced surface tension at the tips raises
the local
curvature to high values and causes the ejection of a sheet or column of
gas by means
of tip streaming. In the third series of studies, the coalescence of a
polygonal formation
of five viscous columns of a fluid placed in an arrangement that differs
only slightly
from one proposed recently by Richardson is examined. The numerical results
confirm
Richardson's predictions that transient cusps may occur at a finite
time in the presence
of surface tension. The underlying physical mechanism is discussed on the
basis of
reversibility of surface-driven Stokes flow and with reference to the regularity
of the
motion driven by negative surface tension. Replacing the inviscid ambient
gas with a
slightly viscous fluid whose viscosity is as low as one hundredth the viscosity
of the
cylinders suppresses the cusp formation.
Application of the Indirect Fictitious-Boundary Integral Method to Unsteady Elastodynamic Problem. Two-Dimensional Problem.
