The emergence of a fluid flow in a closed chamber driven by dynamical deformations of an elastic sheet is investigated. The problem is formulated for both in-viscid and viscous fluids filling the chamber. For inviscid fluid the linear stability analysis of the problem formulated for the potential flow is performed both analytically and numerically. The results obtained by the analytical solution derived within the limit of the small-amplitude approximation are successfully reproduced by the numerical analysis not imposing any limitations on the amplitude magnitude. The numerical analysis utilizes the Finite Element (FE) and the finite difference discretizations for the potential flow and the non-linear elastica equations, respectively. The numerical formulation is based on the monolithic approach, i.e. both potential flow and non-linear elastica equations are treated implicitly. The linear stability of the linearized operator is performed in a shift and inverse mode allowing to conduct the comprehensive parametric study with respect to all the non-dimensional parameters governing the system. In the next step the time evolution of a dynamic elastic sheet within viscous fluid is investigated. To resolve the fluid-structure interactions (FSI) of the system, the immersed boundary (IB) method is used. As previously, the monolithic approach is utilized, which means that all the fluid and the sheet fields are fully coupled and treated implicitly. The time integration is performed by utilizing the Newton-Raphson method for linearizing the fully coupled non-linear operator at each computational step. A number of further directions aimed at improving the computational efficiency of the developed numerical methodology are discussed.