  The dynamics of a membrane coupled to an active fluid on top of a substrate
is considered theoretically. It is assumed that the director field of the
active fluid has rotational symmetry in the membrane plane. This situation is
likely to be relevant for in vitro reconstructed actomyosin-membrane system.
Different from a membrane coupled to a polar active fluid, this model predicts
that only when the viscosity of the fluid above the membrane is sufficiently
large, a contractile active fluid is able to slow down the relaxation of the
membrane for perturbations with wavelength comparable to the thickness of the
active fluid. Hence our model predicts a finite-wavelength instability in the
limit of strong contractility, which is different from a membrane coupled to a
polar active fluid. On the other hand, a membrane coupled to an extensile
active fluid is always unstable against long wavelength perturbations due to
splay induced flows.
