  In this work we analyze the stability and convergence properties of a
loosely-coupled scheme, called the kinematically coupled scheme, and its
extensions for the interaction between an incompressible, viscous fluid and a
thin, elastic structure. We consider a benchmark problem where the structure is
modeled using a general thin structure model, and the coupling between the
fluid and structure is linear. We derive the energy estimates associated with
the unconditional stability of an extension of the kinematically coupled
scheme, called the $\beta$-scheme. Furthermore, for the first time we present
\textit{a priori} estimates showing optimal, first-order in time convergence in
the case when $\beta=1$. We further discuss the extensions of our results to
other fluid-structure interaction problems, in particular the fluid-thick
structure interaction problem. The theoretical stability and convergence
results are supported with numerical examples.
