  Viscoelastic surface rheology plays an important role in multiphase systems.
A typical example is the actin cortex which surrounds most animal cells. It
shows elastic properties for short time scales and behaves viscous for longer
time scales. Hence, realistic simulations of cell shape dynamics require a
model capturing the entire elastic to viscous spectrum. However, currently
there are no numerical methods to simulate deforming viscoelastic surfaces.
Thus models for the cell cortex, or other viscoelastic surfaces, are usually
based on assumptions or simplifications which limit their applicability. In
this paper we develop a first numerical approach for simulation of deforming
viscoelastic surfaces. To this end, we derive the surface equivalent of the
upper convected Maxwell model using the GENERIC formulation of nonequilibrium
thermodynamics. The model distinguishes between shear dynamics and dilatational
surface dynamics. The viscoelastic surface is embedded in a viscous fluid
modelled by the Navier-Stokes equation. Both systems are solved using Finite
Elements. The fluid and surface are combined using an Arbitrary
Lagrange-Eulerian (ALE) Method that conserves the surface grid spacing during
rotations and translations of the surface. We verify this numerical
implementation against analytic solutions and find good agreement. To
demonstrate its potential we simulate the experimentally observed tumbling and
tank-treading of vesicles in shear flow. We also supply a phase-diagram to
demonstrate the influence of the viscoelastic parameters on the behaviour of a
vesicle in shear flow. Finally, we explore cytokinesis as a future application
of the numerical method by simulating the start of cytokinesis using a
spatially dependent function for the surface tension.
