  The deformation of an elastic micro-capsule in an infinite shear flow is
studied numerically using a spectral method. The shape of the capsule and the
hydrodynamic flow field are expanded into smooth basis functions. Analytic
expressions for the derivative of the basis functions permit the evaluation of
elastic and hydrodynamic stresses and bending forces at specified grid points
in the membrane. Compared to methods employing a triangulation scheme, this
method has the advantage that the resulting capsule shapes are automatically
smooth, and few modes are needed to describe the deformation accurately.
Computations are performed for capsules both with spherical and ellipsoidal
unstressed reference shape. Results for small deformations of initially
spherical capsules coincide with analytic predictions. For initially
ellipsoidal capsules, recent approximative theories predict stable oscillations
of the tank-treading inclination angle, and a transition to tumbling at low
shear rate. Both phenomena have also been observed experimentally. Using our
numerical approach we could reproduce both the oscillations and the transition
to tumbling. The full phase diagram for varying shear rate and viscosity ratio
is explored. While the numerically obtained phase diagram qualitatively agrees
with the theory, intermittent behaviour could not be observed within our
simulation time. Our results suggest that initial tumbling motion is only
transient in this region of the phase diagram.
