  Active gel theory has recently been very successful in describing
biologically active materials such as actin filaments or moving bacteria in
temporally fixed and simple geometries such as cubes or spheres. Here we
develop a computational algorithm to compute the dynamic evolution of an
arbitrarily shaped, deformable thin membrane of active material embedded in a
3D flowing liquid. For this, our algorithm combines active gel theory with the
classical theory of thin elastic shells. To compute the actual forces resulting
from active stresses, we apply a parabolic fitting procedure to the
triangulated membrane surface. Active forces are then dynamically coupled via
an Immersed-Boundary method to the surrounding fluid whose dynamics can be
solved by any standard, e.g. Lattice-Boltzmann, flow solver. We validate our
algorithm using the Green's functions of [Berthoumieux et al. New J. Phys. 16,
065005 (2014)] for an active cylindrical membrane subjected (i) to a locally
increased active stress and (ii) to a homogeneous active stress. For the latter
scenario, we predict in addition a so far unobserved non-axisymmetric
instability. We highlight the versatility of our method by analyzing the flow
field inside an actively deforming cell embedded in external shear flow.
Further applications may be cytoplasmic streaming or active membranes in blood
flows.
