Variational integrators for Lagrangian systems have the advantage of conserving the momenta up to machine precision, independent of the time step. While the theory of variational integrators for mechanical systems is well developed, there applications of these integrators to systems involving fluid-structure interactions have proven difficult. In this talk, we derive a variational integrator for a particular type of fluid-structure interactions, namely, simulating the dynamics of a bendable tube conveying ideal fluid that can change its cross-section (collapsible tube). First, we derive a fully three-dimensional, geometrically exact theory for flexible tubes conveying fluid. Our approach is based on the symmetry-reduced, exact geometric description for elastic rods, coupled with the fluid transport and subject to the volume conservation constraint for the fluid. Based on this theory, we derive a variational discretization of the dynamics based on the appropriate discretization of the fluidâ s back-to-labels map, coupled with a variational discretization of elastic part of the Lagrangian. We also show the results of simulations of the system with the spatial discretization using a very small number of points demonstrating a non-trivial and interesting behavior.