The deformation of a liquid capsule enclosed by an elastic membrane
in an infinite
simple shear flow is studied numerically at vanishing Reynolds numbers
using a
boundary-element method. The surface of the capsule is discretized into
quadratic
triangular elements that form an evolving unstructured grid. The elastic
membrane
tensions are expressed in terms of the surface deformation gradient, which
is evaluated
from the position of the grid points. Compared to an earlier formulation
that
uses global curvilinear coordinates, the triangular-element formulation
suppresses
numerical instabilities due to uneven discretization and thus enables the
study of large
deformations and the investigation of the effect of fluid viscosities.
Computations are
performed for capsules with spherical, spheroidal, and discoidal unstressed
shapes
over an extended range of the dimensionless shear rate and for a broad
range of the
ratio of the internal to surrounding fluid viscosities. Results for small
deformations of
spherical capsules are in quantitative agreement with the predictions of
perturbation
theories. Results for large deformations of spherical capsules and deformations
of
non-spherical capsules are in qualitative agreement with experimental observations
of
synthetic capsules and red blood cells. We find that initially spherical
capsules deform
into steady elongated shapes whose aspect ratios increase with the magnitude
of the
shear rate. A critical shear rate above which capsules exhibit continuous
elongation is
not observed for any value of the viscosity ratio. This behaviour contrasts
with that
of liquid drops with uniform surface tension and with that of axisymmetric
capsules
subject to a stagnation-point flow. When the shear rate is sufficiently
high and the
viscosity ratio is sufficiently low, liquid drops exhibit continuous elongation
leading
to breakup. Axisymmetric capsules deform into thinning needles at sufficiently
high
rates of elongation, independent of the fluid viscosities. In the case
of capsules in
shear flow, large elastic tensions develop at large deformations and prevent
continued
elongation, stressing the importance of the vorticity of the incident flow.
The long-time
behaviour of deformed capsules depends strongly on the unstressed shape.
Oblate
capsules exhibit unsteady motions including oscillation about a mean configuration
at low viscosity ratios and continuous rotation accompanied by periodic
deformation
at high viscosity ratios. The viscosity ratio at which the transition from
oscillations
to tumbling occurs decreases with the sphericity of the unstressed shape.
Results on
the effective rheological properties of dilute suspensions confirm a non-Newtonian
shear-thinning behaviour.
Large deformation contact mechanics of long rectangular membranes. I. Adhesionless contact
